L 1 estimates for oscillating integrals and their applications to semi-linear models with σ -evolution like structural damping
aa r X i v : . [ m a t h . A P ] O c t L estimates for oscillating integrals and their applications to semi-linearmodels with σ -evolution like structural damping Tuan Anh Dao a,b , Michael Reissig b, ∗ a School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, No.1 Dai Co Viet road,Hanoi, Vietnam b Faculty for Mathematics and Computer Science, TU Bergakademie Freiberg, Pr¨uferstr. 9, 09596, Freiberg, Germany
Abstract
The present paper is a continuation of our recent paper [5]. We will consider the following Cauchy problemfor semi-linear structurally damped σ -evolution models: u tt + ( − ∆) σ u + µ ( − ∆) δ u t = f ( u, u t ) , u (0 , x ) = u ( x ) , ku t (0 , x ) = u ( x )with σ ≥ µ > δ ∈ ( σ , σ ]. Our aim is to study two main models including σ -evolution modelswith structural damping δ ∈ ( σ , σ ) and those with visco-elastic damping δ = σ . Here the function f ( u, u t )stands for power nonlinearities | u | p and | u t | p with a given number p >
1. We are interested in investigatingthe global (in time) existence of small data Sobolev solutions to the above semi-linear model from suitablefunction spaces basing on L q space by assuming additional L m regularity for the initial data, with q ∈ (1 , ∞ )and m ∈ [1 , q ). Keywords: structural damped σ -evolution equations, visco-elastic equations, σ -evolution like models,oscillating integrals, global existence, Gevrey smoothingMSC (2010): 35B40, 35L76, 35R11
1. Introduction
In the present paper, we study the following two Cauchy problems: u tt + ( − ∆) σ u + µ ( − ∆) δ u t = | u | p , u (0 , x ) = u ( x ) , u t (0 , x ) = u ( x ) , (1)and u tt + ( − ∆) σ u + µ ( − ∆) δ u t = | u t | p , u (0 , x ) = u ( x ) , u t (0 , x ) = u ( x ) (2)with σ ≥ µ > δ ∈ ( σ , σ ]. The corresponding linear model with vanishing right-hand side is u tt + ( − ∆) σ u + µ ( − ∆) δ u t = 0 , u (0 , x ) = u ( x ) , u t (0 , x ) = u ( x ) . (3)A lot of papers (see, for example, [1, 4, 18]) focused on studying the special case σ = 1 in (3) with δ ∈ (0 , u tt − ∆ u + µ ( − ∆) δ u t = 0 , u (0 , x ) = u ( x ) , u t (0 , x ) = u ( x ) . (4)More in detail, in the case δ ∈ (0 ,
1) in [18] the authors studied L estimates for oscillating integrals toconclude L p − L q estimates not necessarily on the conjugate line for solutions to (4). In the case of semi-linear structurally damped wave models (1) with σ = 1 and δ ∈ (0 ,
1] (see [4]), the authors proved the global ∗ Corresponding author.
Email addresses: [email protected] (Tuan Anh Dao), [email protected] (Michael Reissig)
Preprint submitted to Elsevier October 9, 2018 in time) existence of small data solutions in low space dimensions by using classical energy estimates.In particular, they proposed to distinguish between “parabolic like models” δ ∈ (0 , ) (see also [16]) and“hyperbolic like models” δ ∈ ( ,
1) (see also [8]) from the point of decay estimates. Moreover, in [1] someglobal (in time) existence results for small data solutions were presented for “parabolic like models” related to(1) with σ = 1 and δ ∈ (0 , ). More general, if we interested in studying (1) and (2) with δ ∈ [0 , σ ], then wewant to mention the paper [2] as another approach to obtain sharp L p − L q estimates with 1 < p ≤ q < ∞ forthe solutions to the linear model (3). In detail, here the authors found an explicit way to get these estimatesby using the Mikhlin-H¨ormander multiplier theorem (see, for instance, [14, 22]) for kernels localized at highfrequencies. Then, there appeared some L q estimates for the solutions and some of their derivatives, with q ∈ (1 , ∞ ), to prove the global (in time) existence of small data solutions to the semi-linear models (1) and(2). In order to look for these results two different strategies were used due to the lack of L − L estimatesfor solutions to (3). They took account of additional L ∩ L ∞ regularity and additional L η ∩ L ¯ q regularityfor any small η and large ¯ q , respectively, in the first case with δ = σ and in the second case with δ ∈ (0 , σ ).Recently, in [3] the use of L − L estimates for solutions to (3) by assuming additional L regularity forthe data was investigated to study semi-linear σ -evolution models (1) and (2) with δ = σ .Moreover, another interesting model related to (4) is that with visco-elastic damping δ = 1 (or strongdamping, see also [10, 11]). It was considered in detail in [25]. The author obtained a potential decayestimate for solutions localized to low frequencies, whereas the high frequency part decays exponentiallyunder the requirement of a suitable regularity for the data by application of the Marcinkiewicz theorem(see, for example, [13, 27]) to related Fourier multipliers. The case of semi-linear visco-elastic damped wavemodels (1) and (2) with σ = δ = 1 was studied in several recent papers such as [4] and [20]. In [6] the authorsmentioned some different interesting models related to (4), namely those with σ = δ = 2, well-known asthe visco-elastic damped plate models. Some decay estimates of the energy and qualitative properties ofsolutions as well were studied.The present paper is a continuation of our recent paper [5], in which the global (in time) existence ofsmall data Sobolev solutions by mixing of additional L m regularity for the data on the basis of L q − L q estimates, with 1 ≤ m < q < ∞ , is proved to the semi-linear models (1) and (2). Here we remark that theproperties of the solutions to (1) and (2) change completely from (0 , σ ) to ( σ , σ ]. In particular, we want todistinguish between “parabolic like models” ( δ ∈ [0 , σ )) (see [5]) and “ σ -evolution like models” ( δ ∈ ( σ , σ ])according to expected decay estimates. To do this, the first step of the present paper is to develop some L estimates relying on several techniques from [18] for oscillating integrals in the presentation of solutions to(3) by using theory of modified Bessel functions. It is also reasonable to apply Fa`a di Bruno’s formula (see,for instance, [17, 23]) since the connection to Fourier multipliers appearing for wave models used in [18] failsto σ -evolution models for σ > L p − L q estimates not necessarily on the conjugate line for the solutions to (3), with 1 ≤ p ≤ q ≤ ∞ ,in the case of structural damping δ ∈ ( σ , σ ). In the second step of this paper, we obtain L q − L q estimates,with 1 ≤ q ≤ ∞ , for the solutions to (3) by assuming suitable regularity for the data and applying theMikhlin-H¨ormander multiplier theorem for high frequencies in the remaining case of visco-elastic damping δ = σ , for any σ ≥
1. Finally, having L q − L q estimates by assuming additional L m regularity for the dataand some developed tools from Harmonic Analysis in [21] (see also [4, 9, 17]) play a fundamental role toprove our global (in time) existence results.The organization of this paper is as follows:In Section 2, we state the main results for the global (in time) existence of small data Sobolev solutions to(1) and (2). We present estimates for the solutions to (3) in Section 3. In particular, we provide estimatesfor solutions in the case of structural damping δ ∈ ( σ , σ ) in Section 3.1 including the proof of L estimates, L ∞ estimates and L r estimates as well. Section 3.2 is devoted to derive estimates of solutions in the caseof visco-elastic damping δ = σ . Then, in Section 3.3 we state L q − L q estimates by assuming additional L m regularity for the data with q ∈ (1 , ∞ ) and m ∈ [1 , q ). Then, we prove our global (in time) existence resultsto (1) and (2) in Section 4. Finally, in Section 5 we state some concluding remarks and open problems.Throughout the present paper, we use the following notations.2 otation 1. We write f . g when there exists a constant C > f ≤ Cg , and f ≈ g when g . f . g . Notation 2.
We denote [ s ] + := max { s, } as the positive part of s ∈ R , and ⌈ s ⌉ := min (cid:8) k ∈ Z : k ≥ s (cid:9) . Notation 3.
The spaces H a,q and ˙ H a,q , with a ≥ q >
1, denote Bessel and Riesz potential spacesbased on L q . As usual, (cid:10) D (cid:11) a and | D | a stand for the pseudo-differential operators with symbols (cid:10) ξ (cid:11) a and | ξ | a , respectively. Notation 4.
We introduce the space A sm,q := (cid:0) L m ∩ H s,q (cid:1) × (cid:0) L m ∩ H [ s − δ ] + ,q (cid:1) with the norm k ( u , u ) k A sm,q := k u k L m + k u k H s,q + k u k L m + k u k H [ s − δ ]+ ,q , for s ≥ . Notation 5.
We fix the constants κ := 1+(1+[ n ])(1 − σ δ ) (cid:0) q − m (cid:1) and κ := (2+[ n ])(1 − σ δ ) (cid:0) q − m (cid:1) .
2. Main results
In the first case, we obtain solutions to (1) from energy space on the base of the space L q . Theorem 2.1.
Let q ∈ (1 , ∞ ) be a fixed constant and m ∈ [1 , q ) in (1). We assume the condition p > { mδ (1 + κ ) , n − mq n + 2 mδ } n − mδκ . (5) Moreover, we suppose the following conditions: p ∈ h qm , ∞ (cid:17) if n ≤ qδ, or p ∈ h qm , nn − qδ i if n ∈ (cid:16) qδ, q δq − m i . (6) Then, there exists a constant ε > such that for any small data ( u , u ) ∈ A δm,q satisfying the assumption k ( u , u ) k A δm,q ≤ ε, we have a uniquely determined global (in time) small data energy solution (on the base of L q ) u ∈ C ([0 , ∞ ) , H δ,q ) ∩ C ([0 , ∞ ) , L q ) to (1). The following estimates hold: (cid:13)(cid:13) u ( t, · ) (cid:13)(cid:13) L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) k ( u , u ) k A δm,q , (7) (cid:13)(cid:13) | D | σ u ( t, · ) (cid:13)(cid:13) L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) − σ δ k ( u , u ) k A δm,q , (8) (cid:13)(cid:13) u t ( t, · ) (cid:13)(cid:13) L q . (1 + t ) (2+[ n ])(1 − σ δ ) r − n δ (1 − r ) k ( u , u ) k A δm,q , (9) (cid:13)(cid:13) | D | δ u ( t, · ) (cid:13)(cid:13) L q . (1 + t ) (1+[ n ])(1 − σ δ ) r − n δ (1 − r ) k ( u , u ) k A δm,q , (10) where q = r + m . In the second case, we obtain Sobolev solutions to (1).
Theorem 2.2.
Let q ∈ (1 , ∞ ) be a fixed constant, m ∈ [1 , q ) in (1) and < s < δ . We assume thecondition p > { mδ (1 + κ ) , n − mq n + ms } n − mδκ . (11) Moreover, we suppose the following conditions: p ∈ h qm , ∞ (cid:17) if n ≤ qs, or p ∈ h qm , nn − qs i if n ∈ (cid:16) qs, q sq − m i . (12)3 hen, there exists a constant ε > such that for any small data ( u , u ) ∈ A sm,q satisfying the assumption k ( u , u ) k A sm,q ≤ ε, we have a uniquely determined global (in time) small data Sobolev solution u ∈ C ([0 , ∞ ) , H s,q ) to (1). The following estimates hold: (cid:13)(cid:13) u ( t, · ) (cid:13)(cid:13) L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) k ( u , u ) k A sm,q , (13) (cid:13)(cid:13) | D | s u ( t, · ) (cid:13)(cid:13) L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) − s δ k ( u , u ) k A sm,q , (14) where q = r + m . Remark 2.1.
We want to underline that due to the flexibility of parameter q ∈ (1 , ∞ ), we really get aresult for arbitrarily small positive s in Theorem 2.2. In particular, if we take any small positve s = ε , thenwe also choose for example a sufficiently large q = ε in order to guarantee the existence of both the spacedimension n and the exponent p satisfying the required conditions in Theorem 2.2.In the third case, we obtain solutions to (1) belonging to the energy space (on the base of L q ) with asuitable higher regularity. Theorem 2.3.
Let q ∈ (1 , ∞ ) be a fixed constant, m ∈ [1 , q ) in (1) and δ < s ≤ δ + nq . We assume theexponent p > ⌈ s − δ ⌉ satisfying the condition p > { mδ (1 + κ ) , n − mq n + ms } n − mδκ . (15) Moreover, we suppose the following conditions: p ∈ h qm , ∞ (cid:17) if n ≤ qs, or p ∈ h qm , qδn − qs i if (cid:16) qs, qs + 2 mqδq − m i . (16) Then, there exists a constant ε > such that for any small data ( u , u ) ∈ A sm,q satisfying the assumption k ( u , u ) k A sm,q ≤ ε, we have a uniquely determined global (in time) small data energy solution u ∈ C ([0 , ∞ ) , H s,q ) ∩ C ([0 , ∞ ) , H s − δ,q ) to (1). The following estimates hold: k u ( t, · ) k L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) k ( u , u ) k A sm,q , (17) (cid:13)(cid:13) | D | s u ( t, · ) (cid:13)(cid:13) L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) − s δ k ( u , u ) k A sm,q , (18) k u t ( t, · ) k L q . (1 + t ) (2+[ n ])(1 − σ δ ) r − n δ (1 − r ) k ( u , u ) k A sm,q , (19) (cid:13)(cid:13) | D | s − δ u t ( t, · ) (cid:13)(cid:13) L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) − s δ k ( u , u ) k A sm,q , (20) where q = r + m . Remark 2.2.
Let us explain the conditions for p and n in Theorems 2.1 to 2.3. The conditions (5), (11) and(15) imply the same decay estimates for the solutions to (1) as for the solutions to the corresponding linearCauchy problem (3). Hence, we can say that the nonlinearity is interpreted as a small perturbation. Theother conditions (6) and (12) come into play after we apply fractional Gargliardo-Nirenberg inequality. Inaddition, the upper bound for n arises from the corresponding set of admissible parameters p to guaranteethat this range is non-empty. Employing fractional chain rule leads to the condition p > ⌈ s − δ ⌉ inTheorem 2.3. Eventually, the condition (16) appears as an interplay between fractional Gargliardo-Nirenberginequality and fractional chain rule. 4inally, we obtain high regular solutions to (1) by using the fractional Sobolev embedding. Theorem 2.4.
Let s > δ + nq . Let q ∈ (1 , ∞ ) be a fixed constant and m ∈ [1 , q ) in (1). We assume thatthe exponent p > s − δ satisfies the condition p > { mδ (1 + κ ) , n − mq n + ms } n − mδκ . (21) Moreover, we suppose the following conditions: p ∈ h qm , ∞ (cid:17) and n > mδκ . (22) Then, there exists a constant ε > such that for any small data ( u , u ) ∈ A sm,q satisfying the assumption k ( u , u ) k A sm,q ≤ ε, we have a uniquely determined global (in time) small data energy solution u ∈ C ([0 , ∞ ) , H s,q ) ∩ C ([0 , ∞ ) , H s − δ,q ) to (1). Moreover, the estimates (17) to (20) hold. Finally, we obtain large regular solutions to (2) by using the fractional Sobolev embedding.
Theorem 2.5.
Let s > δ + nq . Let q ∈ (1 , ∞ ) be a fixed constant and m ∈ [1 , q ) in (2). We assume thatthe exponent p > s − δ satisfies the condition p > { mδ (1 + κ ) , n − mq n + m ( s − σ ) } n − mδκ . (23) Moreover, we suppose the following conditions: p ∈ h qm , ∞ (cid:17) and n > mδκ . (24) Then, there exists a constant ε > such that for any small data ( u , u ) ∈ A sm,q satisfying the assumption k ( u , u ) k A sm,q ≤ ε, we have a uniquely determined global (in time) small data energy solution u ∈ C ([0 , ∞ ) , H s,q ) ∩ C ([0 , ∞ ) , H s − δ,q ) to (2). Moreover, the estimates (17) to (20) hold. Remark 2.3.
Let us turn to interpret the conditions for p and n in Theorems 2.4 and 2.5. Since we want touse fractional powers, the condition p > s − δ is necessary. Moreover, the conditions (21) and (23) bringthe same decay estimates for the solutions, respectively, to (1) and (2) as for solutions to the correspondingCauchy problem (3). Hence, the nonlinearity can be considered as a small perturbation. Finally, theremaining conditions (22) and (24) come from applying fractional Gargliardo-Nirenberg inequality andfractional Sobolev embedding. Remark 2.4.
In comparison with all the theorems in our previous paper [5], we want to underline that thesolutions from all the above theorems have no loss of regularity (see also [2, 15, 19]) with respect to the initialdata. Loss of regularity of the solutions appearing in [5] is due to the singular behavior of time-dependentcoefficients in the estimates of solutions to the linear models localized to high frequencies as t −→ +0 with δ ∈ (0 , σ ). Meanwhile, this phenomenon does not appear in the case δ ∈ ( σ , σ ] (see later, Proposition 3.9).5 emark 2.5. Let us compare our results with some known results from [4]. By choosing σ = 1, q = 2 and m = 1 we see that, on the one hand, the admissible exponents p in the cited paper are somehow better thanthose in Theorem 2.1 for low space dimensions. On the other hand, we want to underline that Theorem 2.1allows some flexibility for both p and n because of the flexible choice of parameters σ , δ , q and m (see alsosome of the examples below). Example 2.1.
In the following examples, we choose m = 1, q = 4, σ = , δ = 1 and n = 4: • Using Theorem 2.1 we obtain p ∈ [4 , ∞ ). • If s = , then using Theorem 2.2 we obtain p ∈ [4 , ∞ ). • If s = , then using Theorem 2.3 we obtain p ∈ [4 , ∞ ). • If s = , then using Theorem 2.4 we obtain p ∈ (cid:0) , ∞ (cid:1) . • If s = 4, then using Theorem 2.5 we obtain p ∈ [4 , ∞ ). Example 2.2.
In the following examples, we choose m = 1, q = 4, σ = δ = and n = 5: • Using Theorem 2.1 we obtain p ∈ (cid:0) , ∞ (cid:1) . • If s = 2, then using Theorem 2.2 we obtain p ∈ [4 , ∞ ). • If s = , then using Theorem 2.3 we obtain p ∈ (cid:0) , ∞ (cid:1) . • If s = , then using Theorem 2.4 we obtain p ∈ (cid:0) , ∞ (cid:1) . • If s = 4, then using Theorem 2.5 we obtain p ∈ [4 , ∞ ).
3. Decay estimates for solutions to linear Cauchy problems
The goal of this section is to obtain decay estimates for the solution and some its derivatives to (3).These estimates play an essential role to prove the global (in time) existence results to (1) and (2) in thenext section. First, using partial Fourier transformation to (3) leads to the following Cauchy problem for v ( t, ξ ) := F x → ξ (cid:0) u ( t, x ) (cid:1) , v ( ξ ) := F x → ξ (cid:0) u ( x ) (cid:1) and v ( ξ ) := F x → ξ (cid:0) u ( x ) (cid:1) v tt + µ | ξ | δ v t + | ξ | σ v = 0 , v (0 , ξ ) = v ( ξ ) , v t (0 , ξ ) = v ( ξ ) . (25)Without loss of generality we can choose µ = 1 in (25). The characteristic roots are λ , = λ , ( ξ ) = 12 (cid:16) − | ξ | δ ± q | ξ | δ − | ξ | σ (cid:17) . The solution to (25) can be written as follows: v ( t, ξ ) = λ e λ t − λ e λ t λ − λ v ( ξ ) + e λ t − e λ t λ − λ v ( ξ ) =: ˆ K ( t, ξ ) v ( ξ ) + ˆ K ( t, ξ ) v ( ξ ) . (26)Here we assume λ = λ . Taking account of the cases of small and large frequencies separately, we get1 . λ , ∼ −| ξ | δ ± i | ξ | σ , λ − λ ∼ i | ξ | σ for small | ξ | , (27)2 . λ ∼ −| ξ | σ − δ ) , λ ∼ −| ξ | δ , λ − λ ∼ | ξ | δ for large | ξ | . (28)We now decompose the solution to (3) into two parts localized separately at low and high frequencies, thatis, u ( t, x ) = u χ ( t, x ) + u − χ ( t, x ) , where u χ ( t, x ) = F − (cid:0) χ ( | ξ | ) v ( t, ξ ) (cid:1) and u − χ ( t, x ) = F − (cid:16)(cid:0) − χ ( | ξ | ) (cid:1) v ( t, ξ ) (cid:17) , with a smooth cut-off function χ = χ ( | ξ | ) equal to 1 for small | ξ | and vanishing for large | ξ | .6 .1. Estimates for oscillating integrals in the case of structural damping: δ ∈ ( σ , σ ) L estimates for small frequencies Proposition 3.1.
The following estimates hold in R n for any n ≥ : (cid:13)(cid:13) F − (cid:0) | ξ | a ˆ K χ ( | ξ | ) (cid:1) ( t, · ) (cid:13)(cid:13) L . ( for t ∈ (0 , ,t (2+[ n ])(1 − σ δ ) − a δ for t ∈ [1 , ∞ ) , (cid:13)(cid:13) F − (cid:0) | ξ | a ˆ K χ ( | ξ | ) (cid:1) ( t, · ) (cid:13)(cid:13) L . ( t for t ∈ (0 , ,t n ])(1 − σ δ ) − a δ for t ∈ [1 , ∞ ) , for any non-negative number a . To derive the desired estimates for the norm of the Fourier multipliers localized to small frequencies, wewrite | ξ | a ˆ K ( t, ξ ) = e − | ξ | δ t | ξ | a cos (cid:16) | ξ | σ r − | ξ | δ − σ t (cid:17) + e − | ξ | δ t | ξ | a +2 δ sin (cid:0) | ξ | σ q − | ξ | δ − σ t (cid:1) | ξ | σ q − | ξ | δ − σ , (29)and | ξ | a ˆ K ( t, ξ ) = e − | ξ | δ t | ξ | a sin (cid:0) | ξ | σ q − | ξ | δ − σ t (cid:1) | ξ | σ q − | ξ | δ − σ . (30)For this reason, we will split our proof into two steps. In the first step we derive L estimates for thefollowing oscillating integrals: F − (cid:16) e − c | ξ | δ t | ξ | β sin( c | ξ | σ t ) | ξ | σ χ ( | ξ | ) (cid:17) ( t, · ) , and F − (cid:16) e − c | ξ | δ t | ξ | β cos( c | ξ | σ t ) χ ( | ξ | ) (cid:17) ( t, · ) , where β ≥ c is a positive constant and c = 0 is a real constant. Then, in the second step we estimatethe following more structured oscillating integrals: F − (cid:16) e − c | ξ | δ t | ξ | β sin (cid:0) c | ξ | σ f ( | ξ | ) t (cid:1) | ξ | σ f ( | ξ | ) χ ( | ξ | ) (cid:17) ( t, · ) , and F − (cid:16) e − c | ξ | δ t | ξ | β cos (cid:0) c | ξ | σ f ( | ξ | ) t (cid:1) χ ( | ξ | ) (cid:17) ( t, · ) , where f ( | ξ | ) = r − | ξ | δ − σ . Lemma 3.1.
The following estimate holds in R n for any n ≥ : (cid:13)(cid:13)(cid:13) F − (cid:16) e − c | ξ | δ t | ξ | β sin( c | ξ | σ t ) | ξ | σ χ ( | ξ | ) (cid:17) ( t, · ) (cid:13)(cid:13)(cid:13) L . ( t for t ∈ (0 , ,t (2+[ n ])(1 − σ δ )+ σ − β δ for t ∈ [1 , ∞ ) , with β ≥ . Here c is a positive and c = 0 is a real constant. roof. We follow ideas from the proofs to Proposition 4 in [18] and Lemma 3 . t ∈ (0 ,
1] and t ∈ [1 , ∞ ). First, in order to treat the firstcase t ∈ (0 , | x | ≤
1. Then, we derive immediately for small values of | ξ | the estimate (cid:13)(cid:13)(cid:13) F − (cid:16) e − c | ξ | δ t | ξ | β − σ sin( c | ξ | σ t ) χ ( | ξ | ) (cid:17) ( t, · ) (cid:13)(cid:13)(cid:13) L ( | x |≤ . t. (31)For this reason we assume now | x | ≥
1. We introduce the function I = I ( t, x ) := F − (cid:16) e − c | ξ | δ t | ξ | β − σ sin( c | ξ | σ t ) χ ( | ξ | ) (cid:17) ( t, x ) . Because the functions in the parenthesis are radial symmetric with respect to ξ , the inverse Fourier transformis radial symmetric with respect to x , too. Applying modified Bessel functions leads to I ( t, x ) = c Z ∞ e − c r δ t r β − σ sin( c r σ t ) χ ( r ) r n − ˜ J n − ( r | x | ) dr. (32)Let us consider odd spatial dimension n = 2 m + 1 , m ≥
1. We introduce the vector field Xf ( r ) := ddr (cid:0) r f ( r ) (cid:1) as in the proof of Proposition 4 in [18]. Then carrying out m + 1 steps of partial integration we have I ( t, x ) = − c | x | n Z ∞ ∂ r (cid:0) X m (cid:0) e − c r δ t sin( c r σ t ) χ ( r ) r β − σ +2 m (cid:1)(cid:1) sin( r | x | ) dr. (33)A standard calculation leads to the following presentation of the right-hand side of (33): I ( t, x ) = m X j =0 j +1 X k =0 c jk | x | n Z ∞ ∂ j +1 − kr e − c r δ t ∂ kr (cid:0) sin( c r σ t ) χ ( r ) (cid:1) r β − σ + j sin( r | x | ) dr + m X j =0 j X k =0 c jk | x | n Z ∞ ∂ j − kr e − c r δ t ∂ k +1 r (cid:0) sin( c r σ t ) χ ( r ) (cid:1) r β − σ + j sin( r | x | ) dr + m X j =1 j X k =0 c jk | x | n Z ∞ ∂ j − kr e − c r δ t ∂ kr (cid:0) sin( c r σ t ) χ ( r ) (cid:1) r β − σ + j − sin( r | x | ) dr with some constants c jk . Now, we control the integrals I j,k ( t, x ) := Z ∞ ∂ j +1 − kr e − c r δ t ∂ kr (cid:0) sin( c r σ t ) χ ( r ) (cid:1) r β − σ + j sin( r | x | ) dr. (34)Because of small values of r , we notice that the following estimates hold on the support of χ and on thesupport of its derivatives: (cid:12)(cid:12) ∂ lr e − c r δ t (cid:12)(cid:12) . ( l = 0 ,r δ − l t if l = 1 , · · · , m, (cid:12)(cid:12) ∂ lr (cid:0) sin( c r σ t ) χ ( r ) (cid:1)(cid:12)(cid:12) . r σ − l t for all l = 0 , · · · , m. As a result, we obtain for small r , j = 0 , · · · , m and k = 0 , · · · , j (cid:12)(cid:12) ∂ j +1 − kr e − c r δ t ∂ kr (cid:0) sin( c r σ t ) χ ( r ) (cid:1) r β − σ + j (cid:12)(cid:12) . r δ +2 β − t on the support of χ and on the support of its derivatives. We divide the integral (34) into two parts toderive on the one hand (cid:12)(cid:12)(cid:12) Z π | x | ∂ j +1 − kr e − c r δ t ∂ kr (cid:0) sin( c r σ t ) χ ( r ) (cid:1) r β − σ + j sin( r | x | ) dr (cid:12)(cid:12)(cid:12) . t | x | δ . (35)8n the other hand, we can carry out one more step of partial integration in estimating the remaining integralas follows: (cid:12)(cid:12)(cid:12) Z ∞ π | x | ∂ j +1 − kr e − c r δ t ∂ kr (cid:0) sin( c r σ t ) χ ( r ) (cid:1) r β − σ + j sin( r | x | ) dr (cid:12)(cid:12)(cid:12) . | x | (cid:12)(cid:12)(cid:12) ∂ j +1 − kr e − c r δ t ∂ kr (cid:0) sin( c r σ t ) χ ( r ) (cid:1) r β − σ + j cos( r | x | ) (cid:12)(cid:12)(cid:12) ∞ r = π | x | + 1 | x | Z ∞ π | x | (cid:12)(cid:12)(cid:12) ∂ r (cid:16) ∂ j +1 − kr e − c r δ t ∂ kr (cid:0) sin( c r σ t ) χ ( r ) (cid:1) r β − σ + j (cid:17) cos( r | x | ) (cid:12)(cid:12)(cid:12) dr . t | x | , (36)since δ + 2 β > σ ≥
1. Here we also note that for all j = 0 , · · · , m and k = 0 , · · · , j we have (cid:12)(cid:12)(cid:12) ∂ r (cid:16) ∂ j +1 − kr e − c r δ t ∂ kr (cid:0) sin( c r σ t ) χ ( r ) (cid:1) r β − σ + j (cid:17)(cid:12)(cid:12)(cid:12) . r δ +2 β − t . Hence, from (33) to (36) we have produced terms | x | − ( n +2 δ ) and | x | − ( n +1) which guarantee the L propertyin x to prove that for all t ∈ (0 ,
1] and n = 2 m + 1 the following estimate holds: (cid:13)(cid:13) F − (cid:0) e − c | ξ | δ t | ξ | β − σ sin( c | ξ | σ t ) χ ( | ξ | ) (cid:1) ( t, · ) (cid:13)(cid:13) L ( | x |≥ . t . (37)Let us consider even spatial dimension n = 2 m, m ≥
1, in the first case t ∈ (0 , µ = 1 and the fifth rule for µ = 0, and repeating the above calculationsas we did to get (37) we can conclude the following estimate: (cid:13)(cid:13) F − (cid:0) e − c | ξ | δ t | ξ | β − σ sin( c | ξ | σ t ) χ ( | ξ | ) (cid:1) ( t, · ) (cid:13)(cid:13) L ( | x |≥ . t , for n = 2 m, m ≥ . (38)Let us turn to the second case t ∈ [1 , ∞ ). Then, by the change of variables ξ = t − δ η as we did in the proofof the case t ∈ (0 ,
1] to Lemma 3 . (cid:13)(cid:13) F − (cid:0) e − c | ξ | δ t | ξ | β − σ sin( c | ξ | σ t ) χ ( | ξ | ) (cid:1) ( t, · ) (cid:13)(cid:13) L ( | x |≤ . t − βδ , (39)and (cid:13)(cid:13) F − (cid:0) e − c | ξ | δ t | ξ | β − σ sin( c | ξ | σ t ) χ ( | ξ | ) (cid:1) ( t, · ) (cid:13)(cid:13) L ( | x |≥ . ( t ( m +2)(1 − σ δ )+ σ − β δ if n = 2 m + 1 ,t ( m +1)(1 − σ δ )+ σ − β δ if n = 2 m. (40)Here we also note that | ξ | ∈ (0 , r ∈ (0 , t δ ] and rt − δ ≤ Remark 3.1.
Let us explain the result for the case n = 1. We explained the proofs to Lemma 3.1 for n ≥ n = 1 we only carry out partial integration with no need of the supportof the vector field Xf ( r ) as we did in (33). Then, following the steps of our considerations for odd spatialdimensions we may conclude that the statements of this lemma also hold for n = 1.Following the proof of Lemma 3.1 we may prove the following L estimate, too. Lemma 3.2.
The following estimate holds in R n for any n ≥ : (cid:13)(cid:13)(cid:13) F − (cid:16) e − c | ξ | δ t | ξ | β cos( c | ξ | σ t ) χ ( | ξ | ) (cid:17) ( t, · ) (cid:13)(cid:13)(cid:13) L . ( for t ∈ (0 , ,t (2+[ n ])(1 − σ δ ) − βδ for t ∈ [1 , ∞ ) , with β ≥ . Here c is a positive and c = 0 is a real constant. Lemma 3.3.
The following estimate holds in R n for any n ≥ : (cid:13)(cid:13)(cid:13) F − (cid:16) e − c | ξ | δ t | ξ | β sin( c | ξ | σ f ( | ξ | ) t ) | ξ | σ f ( | ξ | ) χ ( | ξ | ) (cid:17) ( t, · ) (cid:13)(cid:13)(cid:13) L . ( t for t ∈ (0 , ,t (2+[ n ])(1 − σ δ )+ σ − β δ for t ∈ [1 , ∞ ) , where f ( | ξ | ) = r − | ξ | δ − σ and β ≥ . Here c is a positive and c = 0 is a real constant.Proof. We will follow the proof of Lemma 3 . | x | ≥ t ∈ (0 , n = 2 m + 1 and even spatial dimensions n = 2 m with m ≥
1, we assert the following estimateson the support of χ ( r ) and on the support of its derivatives: (cid:12)(cid:12)(cid:12) ∂ kt (cid:16) sin (cid:0) c r σ f ( r ) t (cid:1) f ( r ) χ ( r ) (cid:17)(cid:12)(cid:12)(cid:12) . r σ − k t for all k = 1 , · · · , m, where f ( r ) = r − r δ − σ . Here Fa`a di Bruno’s formula comes into play for all our estimates. We split the proof of the above estimateinto several sub-steps as follows:Step 1: Applying Proposition 5.9 with h ( s ) = √ s and g ( r ) = 1 − r δ − σ ) we have (cid:12)(cid:12) ∂ kr f ( r ) (cid:12)(cid:12) . (cid:12)(cid:12)(cid:12) X · m + ··· + k · m k = k, m i ≥ g ( r ) − ( m + ··· + m k ) k Y j =1 (cid:16) − r δ − σ ) − j (cid:17) m j (cid:12)(cid:12)(cid:12) . X · m + ··· + k · m k = k, m i ≥ r δ − σ )( m + ··· + m k ) − k . r − k (cid:16) since 34 ≤ g ( r ) ≤ r ≤ (cid:17) . In the same way we derive (cid:12)(cid:12)(cid:12) ∂ kr (cid:16) f ( r ) (cid:17)(cid:12)(cid:12)(cid:12) . r − k for k = 1 , · · · , m. (41)Step 2: Applying Proposition 5.9 with h ( s ) = sin( c s ) and g ( r ) = r σ f ( r ) t we get (cid:12)(cid:12) ∂ kr sin (cid:0) c r σ f ( r ) t (cid:1)(cid:12)(cid:12) . (cid:12)(cid:12)(cid:12) X · m + ··· + k · m k = k, m i ≥ sin (cid:0) c r σ f ( r ) t (cid:1) ( m + ··· + m k ) k Y j =1 (cid:16) ∂ jr (cid:0) r σ f ( r ) t (cid:1)(cid:17) m j (cid:12)(cid:12)(cid:12) . (cid:12)(cid:12)(cid:12) X · m + ··· + k · m k = k, m i ≥ k Y j =1 (cid:16) t j X l =0 C lj r σ − j + l f ( r ) ( l ) (cid:17) m j (cid:12)(cid:12)(cid:12) . X · m + ··· + k · m k = k, m i ≥ k Y j =1 ( t r σ − j ) m j . X · m + ··· + k · m k = k, m i ≥ r − k ( t r σ ) m + ··· + m k . r σ − k t. (42)10herefore, from (41) and (42) using the product rule for higher derivatives we may conclude (cid:12)(cid:12)(cid:12) ∂ kr (cid:16) sin (cid:0) c r σ f ( r ) t (cid:1) f ( r ) (cid:17)(cid:12)(cid:12)(cid:12) . r σ − k t for k = 1 , · · · , m. Next, let us turn to consider | x | ≥ t ∈ [1 , ∞ ). To derive the desired estimates by using similar ideasas in the proof to Lemma 3.1, we shall prove the following auxiliary estimates on the support of χ ( t − δ r )and on the support of its partial derivatives: (cid:12)(cid:12)(cid:12) ∂ kr (cid:16) sin (cid:0) c r σ f ( r ) t − σ δ (cid:1) f ( r ) (cid:17)(cid:12)(cid:12)(cid:12) . t − σ δ r σ − k (1 + r σ t − σ δ ) k − if k = 1 , · · · , m, where f ( r ) = r − t σ − δδ r δ − σ ) . Step 1: Applying Proposition 5.9 with h ( s ) = √ s and g ( r ) = 1 − t σ − δδ r δ − σ ) we get (cid:12)(cid:12) ∂ kr f ( r ) (cid:12)(cid:12) . (cid:12)(cid:12)(cid:12) X · m + ··· + k · m k = k, m i ≥ g ( r ) − ( m + ··· + m k ) k Y j =1 (cid:16) − t σ − δδ r δ − σ ) − j (cid:17) m j (cid:12)(cid:12)(cid:12) . X · m + ··· + k · m k = k, m i ≥ (cid:0) t σ − δδ r δ − σ ) (cid:1) m + ··· + m k r − k (cid:16) since 34 ≤ g ( r ) ≤ r ≤ t δ (cid:17) . r − k X · m + ··· + k · m k = k, m i ≥ ( t − δ r ) δ − σ )( m + ··· + m k ) . r − k (cid:16) since t − δ r ≤ r ≤ t δ (cid:17) . An analogous treatment leads to (cid:12)(cid:12)(cid:12) ∂ kr (cid:16) f ( r ) (cid:17)(cid:12)(cid:12)(cid:12) . r − k for k = 1 , · · · , m. (43)Step 2: Repeating the proof as we did in Lemma 3 . (cid:12)(cid:12) ∂ kr sin (cid:0) c r σ f ( r ) t − σ δ (cid:1)(cid:12)(cid:12) . t − σ δ r σ − k (cid:0) t − σ δ r σ (cid:1) k − . (44)Therefore, from (43) and (44) using the product rule for higher derivatives we may conclude (cid:12)(cid:12)(cid:12) ∂ kr (cid:16) sin (cid:0) c r σ f ( r ) t − σ δ (cid:1) f ( r ) (cid:17)(cid:12)(cid:12)(cid:12) . t − σ δ r σ − k (cid:0) t − σ δ r σ (cid:1) k − for k = 1 , · · · , m. Summarizing, Lemma 3.3 is proved.Following the steps of the proof to Lemma 3.3 we may conclude the following statement, too.
Lemma 3.4.
The following estimate holds in R n for any n ≥ : (cid:13)(cid:13)(cid:13) F − (cid:16) e − c | ξ | δ t | ξ | β cos (cid:0) c | ξ | σ f ( | ξ | ) t (cid:1) χ ( | ξ | ) (cid:17) ( t, · ) (cid:13)(cid:13)(cid:13) L . ( for t ∈ (0 , ,t (2+[ n ])(1 − σ δ ) − βδ for t ∈ [1 , ∞ ) , where f ( | ξ | ) = r − | ξ | δ − σ and β ≥ . Here c is a positive and c = 0 is a real constant. emark 3.2. Following the proof of Lemmas from 3.1 to 3.4 we can conclude that all the desired statementsstill hold in the case δ = σ . Proof of Proposition 3.1.
In order to prove the first statement, by the relation (29) we replace 2 β = a + 2 δ and 2 β = a , respectively, in Lemmas 3.3 and 3.4. For the sake of the relation (30), plugging 2 β = a inLemma 3.3 we may conclude the second statement. Remark 3.3.
Repeating the same arguments as in the proof of Proposition 3.1 we may see that thesestatements still hold in the case δ = σ . L estimates for large frequencies Our approach is based on the paper [18]. According to the considerations in Section 5 . L estimates for large frequencies. Proposition 3.2.
The following estimates hold in R n for any n ≥ : (cid:13)(cid:13) F − (cid:0) | ξ | a ˆ K (cid:0) − χ ( | ξ | ) (cid:1)(cid:1) ( t, · ) (cid:13)(cid:13) L . ( t − a σ − δ ) for t ∈ (0 , ,t − a δ for t ∈ [1 , ∞ ) , (cid:13)(cid:13) F − (cid:0) | ξ | a ˆ K (cid:0) − χ ( | ξ | ) (cid:1)(cid:1) ( t, · ) (cid:13)(cid:13) L . ( t − a δ for t ∈ (0 , ,t − a σ − δ ) for t ∈ [1 , ∞ ) , for any non-negative number a . Finally, from the statements of Propositions 3.1 and 3.2, we may conclude the following L estimates. Proposition 3.3.
The following estimates hold in R n for any n ≥ : (cid:13)(cid:13) F − (cid:0) | ξ | a ˆ K (cid:1) ( t, · ) (cid:13)(cid:13) L . ( t − a σ − δ ) for t ∈ (0 , ,t (2+[ n ])(1 − σ δ ) − a δ for t ∈ [1 , ∞ ) , (cid:13)(cid:13) F − (cid:0) | ξ | a ˆ K (cid:1) ( t, · ) (cid:13)(cid:13) L . ( t − a δ for t ∈ (0 , ,t n ])(1 − σ δ ) − a δ for t ∈ [1 , ∞ ) , for any non-negative number a .3.1.3. L ∞ estimates Proposition 3.4.
The following estimates hold in R n for any n ≥ : (cid:13)(cid:13) F − (cid:0) | ξ | a ˆ K χ ( | ξ | ) (cid:1) ( t, · ) (cid:13)(cid:13) L ∞ . ( for t ∈ (0 , ,t − n + a δ for t ∈ [1 , ∞ ) , (45) (cid:13)(cid:13) F − (cid:0) | ξ | a ˆ K (cid:0) − χ ( | ξ | ) (cid:1)(cid:1) ( t, · ) (cid:13)(cid:13) L ∞ . t − n + a σ − δ ) for t ∈ (0 , ∞ ) , (46) (cid:13)(cid:13) F − (cid:0) | ξ | a ˆ K χ ( | ξ | ) (cid:1) ( t, · ) (cid:13)(cid:13) L ∞ . ( t for t ∈ (0 , ,t − n + a δ for t ∈ [1 , ∞ ) , (47) (cid:13)(cid:13) F − (cid:0) | ξ | a ˆ K (cid:0) − χ ( | ξ | ) (cid:1)(cid:1) ( t, · ) (cid:13)(cid:13) L ∞ . t − n + a σ − δ ) for t ∈ (0 , ∞ ) , (48) for any non-negative number a .Proof. Taking account of the representation for ˆ K we can re-write it as follows:ˆ K ( t, ξ ) = e λ t − e ( λ − λ ) t λ − λ = ( te λ t R e − θi √ | ξ | σ −| ξ | δ t dθ for small | ξ | ,te λ t R e − θ √ | ξ | δ − | ξ | σ t dθ for large | ξ | . (cid:12)(cid:12) ˆ K ( t, ξ ) (cid:12)(cid:12) . te −| ξ | δ t , (cid:12)(cid:12) ˆ K ( t, ξ ) (cid:12)(cid:12) . e −| ξ | δ t for small | ξ | , (cid:12)(cid:12) ˆ K ( t, ξ ) (cid:12)(cid:12) . te −| ξ | σ − δ ) t , (cid:12)(cid:12) ˆ K ( t, ξ ) (cid:12)(cid:12) . e − c | ξ | σ − δ ) t for large | ξ | , where c is a suitable positive constant. Therefore, we may conclude all the statements that we wanted toprove. Remark 3.4.
Following the approach to prove Proposition 3.4 we may notice that the statements (45) and(47) still hold in the case δ = σ .From Proposition 3.4 the following statement follows immediately. Proposition 3.5.
The following estimates hold in R n for any n ≥ : (cid:13)(cid:13) F − (cid:0) | ξ | a ˆ K (cid:1) ( t, · ) (cid:13)(cid:13) L ∞ . ( t − n + a σ − δ ) for t ∈ (0 , ,t − n + a δ for t ∈ [1 , ∞ ) , and (cid:13)(cid:13) F − (cid:0) | ξ | a ˆ K (cid:1) ( t, · ) (cid:13)(cid:13) L ∞ . ( t − n + a σ − δ ) for t ∈ (0 , ,t − n + a δ for t ∈ [1 , ∞ ) , for any non-negative number a .3.1.4. L r estimates From the statements of Propositions 3.3 and 3.5, by employing an interpolation argument we may concludethe following L r estimates. Proposition 3.6.
The following estimates hold in R n for n ≥ : (cid:13)(cid:13) F − (cid:0) | ξ | a ˆ K (cid:1) ( t, · ) (cid:13)(cid:13) L r . ( t − n σ − δ ) (1 − r ) − a σ − δ ) if t ∈ (0 , ,t (2+[ n ])(1 − σ δ ) r − n δ (1 − r ) − a δ if t ∈ [1 , ∞ ) , (cid:13)(cid:13) F − (cid:0) | ξ | a ˆ K (cid:1) ( t, · ) (cid:13)(cid:13) L r . ( t − n σ − δ ) (1 − r ) − a σ − δ ) if t ∈ (0 , ,t n ])(1 − σ δ ) r − n δ (1 − r ) − a δ if t ∈ [1 , ∞ ) , for all r ∈ [1 , ∞ ] and any non-negative number a . Corollary 3.1. L p − L q estimates not necessarily on the conjugate lineLet δ ∈ (cid:0) σ , σ (cid:1) in (3) and ≤ p ≤ q ≤ ∞ . Then, the solutions to (3) satisfy the L p − L q estimates (cid:13)(cid:13) | D | a u ( t, · ) (cid:13)(cid:13) L q . t − n σ − δ ) (1 − r ) − a σ − δ ) k u k L p + t − n σ − δ ) (1 − r ) − a σ − δ ) k u k L p if t ∈ (0 , ,t (2+[ n ])(1 − σ δ ) r − n δ (1 − r ) − a δ k u k L p + t n ])(1 − σ δ ) r − n δ (1 − r ) − a δ k u k L p if t ∈ [1 , ∞ ) , (cid:13)(cid:13) | D | a u t ( t, · ) (cid:13)(cid:13) L q . t − n σ − δ ) (1 − r ) − a +2 δ σ − δ ) k u k L p + t − n σ − δ ) (1 − r ) − a +2 δ σ − δ ) k u k L p if t ∈ (0 , ,t n ])(1 − σ δ ) r − n δ (1 − r ) − a +2 σ − δ δ k u k L p + t (2+[ n ])(1 − σ δ ) r − n δ (1 − r ) − a δ k u k L p if t ∈ [1 , ∞ ) , where q = r + p , for any non-negative number a and for all n ≥ . roof. For the sake of the statements in Proposition 3.6, applying Young’s convolution inequality we mayconclude the first statement. In order to prove some estimates for the time derivative of the solution wenotice that the following relations hold: ∂ t ˆ K = −| ξ | σ ˆ K and ∂ t ˆ K = ˆ K − | ξ | δ ˆ K . Hence, employing again Young’s convolution inequality and using Proposition 3.6 we may conclude thesecond statement. Summarizing, Corollary 3.1 is proved.
Remark 3.5.
Let us compare our results with some known results in [18]. In the special case σ = 1 onemay observe that the decay estimates for the solution itself appearing in Corollary 3.1 are asymptoticallythe same as the corresponding ones obtained in the cited paper if we consider the case of sufficiently largespace dimensions n . Remark 3.6.
We can see that there appears the singular behavior of the time-dependent coefficients for t → +0 in Corollary 3.1. This causes some difficulties to treat the semi-linear models. Hence, in order toovercome this, we state the following corollary by assuming additional regularity for the data. Corollary 3.2. L q − L q estimates with additional L m regularity for the dataLet δ ∈ (cid:0) σ , σ (cid:1) in (3), q ∈ (1 , ∞ ) and m ∈ [1 , q ) . Then the Sobolev solutions to (3) satisfy the following ( L m ∩ L q ) − L q estimates: (cid:13)(cid:13) | D | a u ( t, · ) (cid:13)(cid:13) L q . (1 + t ) (2+[ n ])(1 − σ δ ) r − n δ (1 − r ) − a δ k u k L m ∩ H a,q + (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) − a δ k u k L m ∩ H [ a − δ ]+ ,q , (cid:13)(cid:13) | D | a u t ( t, · ) (cid:13)(cid:13) L q . (1 + t ) (1+[ n ])(1 − σ δ ) r − n δ (1 − r ) − a +2( σ − δ )2 δ k u k L m ∩ H a +2( σ − δ ) ,q + (1 + t ) (2+[ n ])(1 − σ δ ) r − n δ (1 − r ) − a δ k u k L m ∩ H a,q . Moreover, the following L q − L q estimates are satisfied: (cid:13)(cid:13) | D | a u ( t, · ) (cid:13)(cid:13) L q . (1 + t ) (2+[ n ])(1 − σ δ ) − a δ k u k H a,q + (1 + t ) n ])(1 − σ δ ) − a δ k u k H [ a − δ ]+ ,q , (cid:13)(cid:13) | D | a u t ( t, · ) (cid:13)(cid:13) L q . (1 + t ) (1+[ n ])(1 − σ δ ) − a +2( σ − δ )2 δ k u k H a +2( σ − δ ) ,q + (1 + t ) (2+[ n ])(1 − σ δ ) − a δ k u k H a,q . Here q = r + m , a is a non-negative number and the dimension n ≥ .Proof. To derive the ( L m ∩ L q ) − L q estimates, on the one hand we control the L q norm of the small frequencypart of the solutions by the L m norm of the data. On the other hand, its high-frequency part is estimatedby using the L q − L q estimates with a suitable regularity of the data depending on the order a of derivatives.Finally, applying Young’s convolution inequality we may conclude all the statements what we wanted toprove. δ = σ First, let us explain our strategies to deal with estimates in the case δ = σ . We recall that in the case δ ∈ ( σ , σ ) our goal is to obtain L p − L q estimates not necessarily on the conjugate line with 1 ≤ p ≤ q ≤ ∞ .For this reason, we need to develop some techniques from the paper [18] in order to conclude L estimates, L ∞ estimates and L r estimates, with r ∈ [1 , ∞ ], as well. Moreover, this strategy is also applied with anextension to the case δ = σ to get these estimates for small frequencies (see later, Section 3.2.1). Meanwhile,for large frequencies in the case δ = σ this strategy fails to derive L estimates as in Proposition 3.2, and L ∞ estimates as in Proposition 3.4 for (46) and (48). Hence, it is reasonable to apply the Mikhlin- H¨omandermultiplier theorem for large frequencies in the case δ = σ . It is clear that this approach is only to conclude L q − L q estimates for large frequencies with q ∈ (1 , ∞ ). Therefore, in the case δ = σ the aim to obtain L p − L q estimates not necessarily on the conjugate line with 1 ≤ p ≤ q ≤ ∞ is beyond the scope of ourpaper. 14 .2.1. L p − L q estimates not necessarily on the conjugate line for small frequencies By interpolation argument, from Remarks 3.3 and 3.4 we can conclude the following L r estimates for smallfrequencies. Proposition 3.7.
The following estimates hold in R n for any n ≥ : (cid:13)(cid:13) F − (cid:0) | ξ | a ˆ K χ ( | ξ | ) (cid:1) ( t, · ) (cid:13)(cid:13) L r . ( if t ∈ (0 , ,t (2+[ n ]) r − n σ (1 − r ) − a σ if t ∈ [1 , ∞ ) , (cid:13)(cid:13) F − (cid:0) | ξ | a ˆ K χ ( | ξ | ) (cid:1) ( t, · ) (cid:13)(cid:13) L r . ( t if t ∈ (0 , ,t (1+[ n ]) r − n σ (1 − r ) − a σ if t ∈ [1 , ∞ ) , for all r ∈ [1 , ∞ ] and any non-negative number a . Repeating the proof of Corollary 3.1 we obtain the following estimate by using Proposition 3.7.
Corollary 3.3.
Let δ = σ in (3) and ≤ p ≤ q ≤ ∞ . Then, the solutions to (3) satisfy the L p − L q estimates (cid:13)(cid:13) | D | a u χ ( t, · ) (cid:13)(cid:13) L q . (1 + t ) (2+[ n ]) r − n σ (1 − r ) − a σ k u k L p + t (1 + t ) (1+[ n ]) r − n σ (1 − r ) − a σ k u k L p , (cid:13)(cid:13) | D | a ∂ t u χ ( t, · ) (cid:13)(cid:13) L q . (1 + t ) (1+[ n ]) r − n σ (1 − r ) − a σ k u k L p + (1 + t ) (2+[ n ]) r − n σ (1 − r ) − a σ k u k L p , where q = r + p , for any non-negative number a and for all n ≥ . Remark 3.7.
Let us compare our results with some known results in [25]. In the special case σ = δ = 1 wemay observe that the time-dependent coefficients in the L − L estimate for the solution itself appearingin Corollary 3.3 are asymptotically the same as the corresponding ones obtained in the cited paper if weconsider the case of sufficiently large space dimensions n . L q − L q estimates for large frequencies First, we can re-write the characteristic roots as follows: λ ( ξ ) = − − µ ( ξ ) and λ ( ξ ) = −| ξ | σ + 1 + µ ( ξ ) , (49)where µ ( ξ ) = − g (cid:16) | ξ | σ (cid:17) and g ( s ) = Z (1 − θs ) − dθ. (50)Now, we introduce the following abbreviations: K = K ( t, x ) := F − (cid:16) λ ( ξ ) e λ ( ξ ) t λ ( ξ ) − λ ( ξ ) v ( ξ ) (cid:0) − χ ( ξ ) (cid:1)(cid:17) ( t, x ) ,K = K ( t, x ) := F − (cid:16) λ ( ξ ) e λ ( ξ ) t λ ( ξ ) − λ ( ξ ) v ( ξ ) (cid:0) − χ ( ξ ) (cid:1)(cid:17) ( t, x ) ,K = K ( t, x ) := F − (cid:16) e λ ( ξ ) t λ ( ξ ) − λ ( ξ ) v ( ξ ) (cid:0) − χ ( ξ ) (cid:1)(cid:17) ( t, x ) ,K = K ( t, x ) := F − (cid:16) e λ ( ξ ) t λ ( ξ ) − λ ( ξ ) v ( ξ ) (cid:0) − χ ( ξ ) (cid:1)(cid:17) ( t, x ) . We shall prove the following results. 15 roposition 3.8.
Let q ∈ (1 , ∞ ) . Then, the following estimates hold: (cid:13)(cid:13) ∂ jt | D | a K ( t, · ) (cid:13)(cid:13) L q . e − ct k u k H a,q , (cid:13)(cid:13) ∂ jt | D | a K ( t, · ) (cid:13)(cid:13) L q . e − ct k u k H [2 σj − σ + a ]+ ,q , (cid:13)(cid:13) ∂ jt | D | a K ( t, · ) (cid:13)(cid:13) L q . e − ct k u k H [ a − σ ]+ ,q , (cid:13)(cid:13) ∂ jt | D | a K ( t, · ) (cid:13)(cid:13) L q . e − ct k u k H [2 σj − σ + a ]+ ,q , for any t > , a ≥ , integer j ≥ and a suitable positive constant c . According to application of the Mikhlin- H¨omander multiplier theorem (see also [2, 14]) for Fouriermultipliers from Proposition 5.6, in order to prove Proposition 3.8 we shall show the following auxiliaryestimates.
Lemma 3.5.
The following estimates hold in R n for sufficiently large | ξ | : (cid:12)(cid:12) ∂ αξ | ξ | − σ (cid:12)(cid:12) . | ξ | − σ −| α | for all α, (51) (cid:12)(cid:12) ∂ αξ | ξ | pσ (cid:12)(cid:12) . | ξ | pσ −| α | for all α and p ∈ R , (52) (cid:12)(cid:12)(cid:12) ∂ αξ g (cid:16) | ξ | σ (cid:17)(cid:12)(cid:12)(cid:12) . | ξ | − σ −| α | for all | α | ≥ , and (cid:12)(cid:12)(cid:12) g (cid:16) | ξ | σ (cid:17)(cid:12)(cid:12)(cid:12) . , (53) (cid:12)(cid:12) ∂ αξ µ ( ξ ) (cid:12)(cid:12) . | ξ | − σ −| α | for all α, (54) (cid:12)(cid:12) ∂ αξ λ ( ξ ) (cid:12)(cid:12) . | ξ | σ −| α | for all α, (55) (cid:12)(cid:12) ∂ αξ λ ( ξ ) (cid:12)(cid:12) . | ξ | − σ −| α | for all | α | ≥ , and (cid:12)(cid:12) λ ( ξ ) (cid:12)(cid:12) . , (56) (cid:12)(cid:12)(cid:12) ∂ αξ (cid:0) λ ( ξ ) − λ ( ξ ) (cid:1) − (cid:12)(cid:12)(cid:12) . | ξ | − σ −| α | for all α, (57) (cid:12)(cid:12) ∂ αξ λ j ( ξ ) (cid:12)(cid:12) . | ξ | σj −| α | for all α and j ≥ , (58) (cid:12)(cid:12) ∂ αξ λ j ( ξ ) (cid:12)(cid:12) . | ξ | −| α | for all α and j ≥ , (59) (cid:12)(cid:12) ∂ αξ (cid:0) | ξ | b λ j ( ξ ) (cid:1)(cid:12)(cid:12) . | ξ | σj + b −| α | for all α, for any b ∈ R and j ≥ , (60) (cid:12)(cid:12) ∂ αξ (cid:0) | ξ | b λ j ( ξ ) (cid:1)(cid:12)(cid:12) . | ξ | b −| α | for all α, for any b ∈ R and j ≥ , (61) (cid:12)(cid:12) ∂ αξ (cid:0) e λ ( ξ ) t (cid:1)(cid:12)(cid:12) . e − ct | ξ | −| α | , (62) for all α and t > , where c is a suitable positive constant , (cid:12)(cid:12) ∂ αξ (cid:0) e λ ( ξ ) t (cid:1)(cid:12)(cid:12) . e − ct | ξ | −| α | , (63) for all α and t > , where c is a suitable positive constant , (cid:12)(cid:12)(cid:12) ∂ αξ (cid:16) λ ( ξ ) e λ ( ξ ) t λ j ( ξ ) | ξ | b λ ( ξ ) − λ ( ξ ) (cid:17)(cid:12)(cid:12)(cid:12) . e − ct | ξ | σj + b − σ −| α | , (64) for all α, for any b ∈ R , j ≥ and t > , where c is a suitable positive constant , (cid:12)(cid:12)(cid:12) ∂ αξ (cid:16) e λ ( ξ ) t λ j ( ξ ) | ξ | b λ ( ξ ) − λ ( ξ ) (cid:17)(cid:12)(cid:12)(cid:12) . e − ct | ξ | σj + b − σ −| α | , (65) for all α, for any b ∈ R , j ≥ and t > , where c is a suitable positive constant , (cid:12)(cid:12)(cid:12) ∂ αξ (cid:16) λ ( ξ ) e λ ( ξ ) t λ j ( ξ ) | ξ | b λ ( ξ ) − λ ( ξ ) (cid:17)(cid:12)(cid:12)(cid:12) . e − ct | ξ | b −| α | , (66) for all α, for any b ∈ R , j ≥ and t > , where c is a suitable positive constant , (cid:12)(cid:12)(cid:12) ∂ αξ (cid:16) e λ ( ξ ) t λ j ( ξ ) | ξ | b λ ( ξ ) − λ ( ξ ) (cid:17)(cid:12)(cid:12)(cid:12) . e − ct | ξ | b − σ −| α | , (67) for all α, for any b ∈ R , j ≥ and t > , where c is a suitable positive constant , roof. In order to prove all statements in Lemma 3.5, we shall apply Lemma 5.1 and Leibniz rule of themultivariable calculus. Indeed, we will indicate the proof of the above estimates as follows:First, we can that the proof of (51) and (52) is trivial. By (51) applying Lemma 5.1 with h ( s ) = g ( s ) and f ( ξ ) = 4 | ξ | − σ we can conclude (53). By (49) and (50), the statements from (54) to (56) are immediatelyfollow from (52) to (53). In the analogous way, by (55) and (56) we get (57) with h ( s ) = s − and f ( ξ ) = λ ( ξ ) − λ ( ξ ). By (55) we obtain (58) with h ( s ) = s j and f ( ξ ) = λ ( ξ ). By (56) we derive (59) with h ( s ) = s j and f ( ξ ) = λ ( ξ ). Using the Leibniz rule we conclude (60) after using (52) and (58). Analogously,we obtain (61) by using (52) and (59). Applying Lemma 5.1 with h ( s ) = e st and f ( ξ ) = λ ( ξ ) we have (62)by taking account of (55), where we note that λ ( ξ ) ≤ − | ξ | σ . In the same way, by (56) and λ ( ξ ) ≤ − h ( s ) = e st and f ( ξ ) = λ ( ξ ). Combining (56), (57), (60) and (62) we may conclude (64)and (65) by using the Leibniz rule. Finally, combining (55), (57), (61) and (63) we arrive at (66) and (67)by using again the Leibniz rule. Proof of Proposition 3.8.
First, taking account of estimates for K and some its derivatives we will divideour consideration into two cases. In the first case, if 2 σj − σ + a ≥
0, then we can write ∂ jt | D | a K ( t, x ) = F − (cid:16) λ ( ξ ) e λ ( ξ ) t λ j ( ξ ) | ξ | σ − σj λ ( ξ ) − λ ( ξ ) (cid:0) − χ ( ξ ) (cid:1) | ξ | σj − σ + a v ( ξ ) (cid:17) ( t, x ) . By choosing b = 2 σ − σj in (64), we get for all α the estimates (cid:12)(cid:12)(cid:12) ∂ αξ (cid:16) λ ( ξ ) e λ ( ξ ) t λ j ( ξ ) | ξ | σ − σj λ ( ξ ) − λ ( ξ ) (cid:17)(cid:12)(cid:12)(cid:12) . e − ct | ξ | −| α | , where c is a suitable positive constant. By Proposition 5.6, we can conclude (cid:13)(cid:13) ∂ jt | D | a K ( t, · ) (cid:13)(cid:13) L q . e − ct k u k H σj − σ + a,q . (68)In the second case, if 2 σj − σ + a <
0, then we can write ∂ jt | D | a K ( t, x ) = F − (cid:16) λ ( ξ ) e λ ( ξ ) t λ j ( ξ ) | ξ | a λ ( ξ ) − λ ( ξ ) (cid:0) − χ ( ξ ) (cid:1) v ( ξ ) (cid:17) ( t, x ) . By choosing b = a in (64), we derive for all α the estimates (cid:12)(cid:12)(cid:12) ∂ αξ (cid:16) λ ( ξ ) e λ ( ξ ) t λ j ( ξ ) | ξ | a λ ( ξ ) − λ ( ξ ) (cid:17)(cid:12)(cid:12)(cid:12) . e − ct | ξ | σj + a − σ −| α | . e − ct | ξ | −| α | , where c is a suitable positive constant. By Proposition 5.6, we may conclude (cid:13)(cid:13) ∂ jt | D | a K ( t, · ) (cid:13)(cid:13) L q . e − ct k u k L q . (69)Hence, from (68) and (69) we have proved the second statement in Proposition 3.8. By the same argumentswe may also conclude the last statement in Proposition 3.8 by using (65). Analogously, in order to estimatethe third statement we will apply b = 2 σ and b = a in (67), respectively, if a ≥ σ and a < σ . Finally,using (66) with b = 0 immediately leads to the remaining statement. Summarizing, the proof to Proposition3.8 is completed. Remark 3.8.
The exponential decay e − ct appearing in Proposition 3.8 for large frequencies is better thanthe potential decay in Proposition 3.7 for small frequencies. Since we have in mind that the real partof the characteristic roots λ , is negative in the middle zone | ξ | ∈ { ε, ε } with sufficiently small ε , thecorresponding estimates yield in this zone an exponential decay, too.From Proposition 3.8 we conclude the following estimates for large frequencies.17 orollary 3.4. Let δ = σ in (3) and q ∈ (1 , ∞ ) . Then, the solutions to (3) satisfy the L q − L q estimates (cid:13)(cid:13) ∂ jt | D | a u − χ ( t, · ) (cid:13)(cid:13) L q . e − ct (cid:0) k ( u , u ) k H [2 σj − σ + a ]+ ,q + k u k H a,q + k u k H [ a − σ ]+ ,q (cid:1) , for any t > , a ≥ , integer j ≥ and a suitable positive constant c . Remark 3.9.
Let us compare our results with some known results in [25]. In the special case σ = δ = 1we may observe that the decay rates for L q − L q estimate for the solution itself appearing in Corollary 3.4are exactly the same as the corresponding ones obtained in the cited paper.From Corollaries 3.3 and 3.4 we conclude the following estimates. Corollary 3.5. L q − L q estimates with additional L m regularity for the dataLet δ = σ in (3), q ∈ (1 , ∞ ) and m ∈ [1 , q ) . Then the Sobolev solutions to (3) satisfy the following ( L m ∩ L q ) − L q estimates: (cid:13)(cid:13) | D | a u ( t, · ) (cid:13)(cid:13) L q . (1 + t ) (2+[ n ]) r − n σ (1 − r ) − a σ k u k L m ∩ H a,q + (1 + t ) (1+[ n ]) r − n σ (1 − r ) − a σ k u k L m ∩ H [ a − σ ]+ ,q , (cid:13)(cid:13) | D | a u t ( t, · ) (cid:13)(cid:13) L q . (1 + t ) (1+[ n ]) r − n σ (1 − r ) − a σ k u k L m ∩ H a,q + (1 + t ) (2+[ n ]) r − n σ (1 − r ) − a σ k u k L m ∩ H a,q . Moreover, the following L q − L q estimates are satisfied: (cid:13)(cid:13) | D | a u ( t, · ) (cid:13)(cid:13) L q . (1 + t ) (2+[ n ]) − a σ k u k H a,q + (1 + t ) (1+[ n ]) − a σ k u k H [ a − σ ]+ ,q , (cid:13)(cid:13) | D | a u t ( t, · ) (cid:13)(cid:13) L q . (1 + t ) (1+[ n ]) − a σ k u k H a,q + (1 + t ) (2+[ n ]) − a σ k u k H a,q . Here q = r + m , a is a non-negative number and the dimension n ≥ .3.3. L q − L q estimates with additional L m regularity for the data From Corollaries 3.2 and 3.5 we obtain the following result.
Proposition 3.9.
Let δ ∈ (cid:0) σ , σ (cid:3) in (3), q ∈ (1 , ∞ ) and m ∈ [1 , q ) . Then the Sobolev solutions to (3) satisfythe following ( L m ∩ L q ) − L q estimates: (cid:13)(cid:13) | D | a u ( t, · ) (cid:13)(cid:13) L q . (1 + t ) (2+[ n ])(1 − σ δ ) r − n δ (1 − r ) − a δ k u k L m ∩ H a,q + (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) − a δ k u k L m ∩ H [ a − δ ]+ ,q , (70) (cid:13)(cid:13) | D | a u t ( t, · ) (cid:13)(cid:13) L q . (1 + t ) (1+[ n ])(1 − σ δ ) r − n δ (1 − r ) − a +2( σ − δ )2 δ k u k L m ∩ H a +2( σ − δ ) ,q + (1 + t ) (2+[ n ])(1 − σ δ ) r − n δ (1 − r ) − a δ k u k L m ∩ H a,q . (71) Moreover, the following L q − L q estimates are satisfied: (cid:13)(cid:13) | D | a u ( t, · ) (cid:13)(cid:13) L q . (1 + t ) (2+[ n ])(1 − σ δ ) − a δ k u k H a,q + (1 + t ) n ])(1 − σ δ ) − a δ k u k H [ a − δ ]+ ,q , (cid:13)(cid:13) | D | a u t ( t, · ) (cid:13)(cid:13) L q . (1 + t ) (1+[ n ])(1 − σ δ ) − a +2( σ − δ )2 δ k u k H a +2( σ − δ ) ,q + (1 + t ) (2+[ n ])(1 − σ δ ) − a δ k u k H a,q . Here q = r + m , a is a non-negative number and the dimension n ≥ . Remark 3.10.
The statements in Proposition 3.9 are key tools to prove global (in time) existence resultsfor the semi-linear models (1) and (2). Let us compare the results from Proposition 3.9 to those fromProposition 3 . t −→ +0 in the above estimates. On the one hand, it is important tonotice that in (70) we derived a decay estimate for the fractional derivative of order a = 2 δ of the solution18ith respect to the spatial variables by assuming a suitable higher regularity on u , that is, u ∈ L m ∩ H δ,q ,whereas the second data u only belongs to L m ∩ L q . This effect does not appear in the case δ ∈ (cid:0) , σ ). If weassume u ∈ L m ∩ H a,q for a > σ , then we choose the second data u from the function space L m ∩ H a − s,q .That property brings some benefit in the treatment of the semi-linear models (1) and (2) in the next section.On the other hand, with a = 0 in (71) the estimate for the first derivative in time requires less regularityfor the data comparing with respect to the estimate for the derivative in space of order a = σ in (70). Thisproperty is new in comparison with the case δ ∈ (0 , σ ).
4. Treatment of the corresponding semi-linear models
In this section, our goal is to apply the estimates from Proposition 3.9 to prove the global (in time)existence of small data Sobolev solutions to the semi-linear models (1) and (2). Some developed tools fromHarmonic Analysis come into play such as fractional Gargliardo-Nirenberg inequality from Proposition 5.1,fractional Leibniz rule from Proposition 5.2, fractional chain rule from Proposition 5.3 and fractional Sobolevembedding from Corollary 5.2. By recalling the fundamental solutions K and K defined in Section 3 wewrite the solutions to (3) in the following form: u ln ( t, x ) = K ( t, x ) ∗ x u ( x ) + K ( t, x ) ∗ x u ( x ) . We apply Duhamel’s principle to get the following implicit representation of the solutions to (1) and (2): u ( t, x ) = K ( t, x ) ∗ x u ( x ) + K ( t, x ) ∗ x u ( x ) + Z t K ( t − τ, x ) ∗ x f ( u, u t ) dτ =: u ln ( t, x ) + u nl ( t, x ) , where f ( u, u t ) = | u | p or | u t | p . We choose the data space A = A sm,q and introduce the family { X ( t ) } t> ofsolution spaces X ( t ) with the norm k u k X ( t ) := sup ≤ τ ≤ t (cid:16) f ( τ ) − k u ( τ, · ) k L q + f σ ( τ ) − (cid:13)(cid:13) | D | σ u ( τ, · ) (cid:13)(cid:13) L q + f ,s ( τ ) − (cid:13)(cid:13) | D | s u ( τ, · ) (cid:13)(cid:13) L q + f ( τ ) − k u t ( τ, · ) k L q + f ,s ( τ ) − (cid:13)(cid:13) | D | s − δ u t ( τ, · ) (cid:13)(cid:13) L q (cid:17) . Furthermore, we introduce the family { X ( t ) } t> of space X ( t ) := C ([0 , t ] , H s,q ) with the norm k w k X ( t ) := sup ≤ τ ≤ t (cid:16) f ( τ ) − k u ( τ, · ) k L q + f σ ( τ ) − (cid:13)(cid:13) | D | σ u ( τ, · ) (cid:13)(cid:13) L q + f ,s ( τ ) − (cid:13)(cid:13) | D | s u ( τ, · ) (cid:13)(cid:13) L q (cid:17) . In both families of spaces we choose the weights f ( τ ) = (1 + τ ) n ])(1 − σ δ ) r − n δ (1 − r ) , f ,s ( τ ) = (1 + τ ) n ])(1 − σ δ ) r − n δ (1 − r ) − s δ ,f ( τ ) = (1 + τ ) (2+[ n ])(1 − σ δ ) r − n δ (1 − r ) , f ,s ( τ ) = (1 + τ ) n ])(1 − σ δ ) r − n δ (1 − r ) − s δ , and f σ ( τ ) = (1 + τ ) n ])(1 − σ δ ) r − n δ (1 − r ) − σ δ . We define for all t > N : u ∈ X ( t ) −→ N u ∈ X ( t ) by the formula N u ( t, x ) = K ( t, x ) ∗ x u ( x ) + K ( t, x ) ∗ x u ( x ) + Z t K ( t − τ, x ) ∗ x f ( u, u t ) dτ. We will prove that the operator N satisfies the following two inequalities: k N u k X ( t ) . k ( u , u ) k A + k u k pX ( t ) , (72) k N u − N v k X ( t ) . k u − v k X ( t ) (cid:0) k u k p − X ( t ) + k v k p − X ( t ) (cid:1) . (73)Then, employing Banach’s fixed point theorem leads to local (in time) existence results of large data solutionsand global (in time) existence results of small data solutions as well.19 emark 4.1. Replacing a = s and a = σ in the statements from Proposition 3.9 and in the definition ofthe norm of X ( t ) we conclude (cid:13)(cid:13) u ln (cid:13)(cid:13) X ( t ) . k ( u , u ) k A sm,q for s ≥ . Hence, in order to prove (72) it is reasonable to indicate the following inequality: (cid:13)(cid:13) u nl (cid:13)(cid:13) X ( t ) . k u k pX ( t ) . (74)Now we are going to prove our main results from Section 2. s = 2 δ We introduce the data space A := A δm,q and the solution space X ( t ) := C ([0 , t ] , H δ,q ) ∩ C ([0 , t ] , L q ) , where the weight f ,s ( τ ) ≡
0. First, let us prove the inequality (74). In order to control u nl , we use the( L m ∩ L q ) − L q estimates in Proposition 3.9. Hence, we obtain for k = 0 , (cid:13)(cid:13) | D | δk u nl ( t, · ) (cid:13)(cid:13) L q . Z t (1 + t − τ ) n ])(1 − σ δ ) r − n δ (1 − r ) − k (cid:13)(cid:13) | u ( τ, · ) | p (cid:13)(cid:13) L m ∩ L q dτ. Therefore, it is necessary to estimate | u ( τ, x ) | p in L m ∩ L q . We proceed as follows: (cid:13)(cid:13) | u ( τ, · ) | p (cid:13)(cid:13) L m ∩ L q . k u ( τ, · ) k pL mp + k u ( τ, · ) k pL qp . Employing the fractional Gagliardo-Nirenberg inequality from Proposition 5.1 leads to (cid:13)(cid:13) | u ( τ, · ) | p (cid:13)(cid:13) L m ∩ L q . (1 + τ ) p (cid:0) n ])(1 − σ δ ) r − n δ ( m − mp ) (cid:1) k u k pX ( τ ) provided that the condition (6) is fulfilled. From the above estimate we get (cid:13)(cid:13) | D | δk u nl ( t, · ) (cid:13)(cid:13) L q . k u k pX ( t ) Z t (1+ t − τ ) n ])(1 − σ δ ) r − n δ (1 − r ) − k (1+ τ ) p (cid:0) n ])(1 − σ δ ) r − n δ ( m − mp ) (cid:1) dτ. The key tool relies now in Lemma 5.2. Because of the condition (5), applying Lemma 5.2 by choosing α = − − (1 + [ n ])(1 − σ δ ) r + n δ (1 − r ) + k and β = p (cid:0) − − (1 + [ n ])(1 − σ δ ) r + n δ ( m − mp ) (cid:1) we derive Z t (1 + t − τ ) n ])(1 − σ δ ) r − n δ (1 − r ) − k (1 + τ ) p (cid:0) n ])(1 − σ δ ) r − n δ ( m − mp ) (cid:1) dτ . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) − k . As a result, we may conclude for k = 0 , (cid:13)(cid:13) | D | δk u nl ( t, · ) (cid:13)(cid:13) L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) − k k u k pX ( t ) . (75)Analogously, we also obtain (cid:13)(cid:13) | D | σ u nl ( t, · ) (cid:13)(cid:13) L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) − σ δ k u k pX ( t ) , (76) (cid:13)(cid:13) ∂ t u nl ( t, · ) (cid:13)(cid:13) L q . (1 + t ) (2+[ n ])(1 − σ δ ) r − n δ (1 − r ) k u k pX ( t ) . (77)From (75) to (77) and the definition of the norm in X ( t ), we arrive immediately at the inequality (74).20ext, let us prove the estimate (73). Using again the ( L m ∩ L q ) − L q estimates in Proposition 3.9, we havefor two functions u and v from X ( t ) the following estimate for k = 0 , (cid:13)(cid:13) | D | δk (cid:0) N u ( t, · ) − N v ( t, · ) (cid:1)(cid:13)(cid:13) L q . Z t (1 + t − τ ) n ])(1 − σ δ ) r − n δ (1 − r ) − k (cid:13)(cid:13) | u ( τ, · ) | p − | v ( τ, · ) | p (cid:13)(cid:13) L m ∩ L q dτ. Applying H¨older’s inequality leads to (cid:13)(cid:13) | u ( τ, · ) | p − | v ( τ, · ) | p (cid:13)(cid:13) L q . k u ( τ, · ) − v ( τ, · ) k L qp (cid:0) k u ( τ, · ) k p − L qp + k v ( τ, · ) k p − L qp (cid:1) , (cid:13)(cid:13) | u ( τ, · ) | p − | v ( τ, · ) | p (cid:13)(cid:13) L m . k u ( τ, · ) − v ( τ, · ) k L mp (cid:0) k u ( τ, · ) k p − L mp + k v ( τ, · ) k p − L mp (cid:1) . In the same way as the proof of (72), employing the fractional Gagliardo-Nirenberg inequality from Propo-sition 5.1 to the terms k u ( τ, · ) − v ( τ, · ) k L η , k u ( τ, · ) k L η , k v ( τ, · ) k L η with η = qp and η = mp we have for k = 0 , (cid:13)(cid:13) | D | δk (cid:0) N u ( t, · ) − N v ( t, · ) (cid:1)(cid:13)(cid:13) L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) − k k u − v k X ( t ) (cid:0) k u k p − X ( t ) + k v k p − X ( t ) (cid:1) . Analogously, we also derive (cid:13)(cid:13) | D | σ (cid:0) N u ( t, · ) − N v ( t, · ) (cid:1)(cid:13)(cid:13) L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) − σ δ k u − v k X ( t ) (cid:0) k u k p − X ( t ) + k v k p − X ( t ) (cid:1) , (cid:13)(cid:13) ∂ t (cid:0) N u ( t, · ) − N v ( t, · ) (cid:1)(cid:13)(cid:13) L q . (1 + t ) (2+[ n ])(1 − σ δ ) r − n δ (1 − r ) k u − v k X ( t ) (cid:0) k u k p − X ( t ) + k v k p − X ( t ) (cid:1) . From the definition of the norm in X ( t ), we may conclude the inequality (73). Summarizing, the proof toTheorem 2.1 is complete. < s < δ We introduce the data space A := A sm,q and the solution space X ( t ) := C ([0 , t ] , H s,q ) , where the weights f σ ( τ ) = f ( τ ) = f ,s ( τ ) ≡
0. We can notice that X ( t ) and X ( t ) coincide in (73) and(74). In order to prove these two inequalities, we use the ( L m ∩ L q ) − L q estimates from Proposition 3.9.Hence, we derive for k = 0 , (cid:13)(cid:13) | D | ks u nl ( t, · ) (cid:13)(cid:13) L q . Z t (1 + t − τ ) n ])(1 − σ δ ) r − n δ (1 − r ) − ks δ (cid:13)(cid:13) | u ( τ, · ) | p (cid:13)(cid:13) L m ∩ L q dτ, (cid:13)(cid:13) | D | ks (cid:0) N u ( t, · ) − N v ( t, · ) (cid:1)(cid:13)(cid:13) L q . Z t (1 + t − τ ) n ])(1 − σ δ ) r − n δ (1 − r ) − ks δ (cid:13)(cid:13) | u ( τ, · ) | p − | v ( τ, · ) | p (cid:13)(cid:13) L m ∩ L q dτ. In the same way as we did in the proof of Theorem 2.1 we obtain for k = 0 , (cid:13)(cid:13) | D | ks u nl ( t, · ) (cid:13)(cid:13) L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) − ks δ k u k pX ( t ) , (cid:13)(cid:13) | D | ks (cid:0) N u ( t, · ) − N v ( t, · ) (cid:1)(cid:13)(cid:13) L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) − ks δ k u − v k X ( t ) (cid:0) k u k p − X ( t ) + k v k p − X ( t ) (cid:1) , provided that the conditions (11) and (12) are fulfilled. From the definition of the norm in X ( t ) we canconclude immediately the inequalities (74) and (73). Summarizing, Theorem 2.2 is proved.21 .4. Proof of Theorem 2.3: δ < s ≤ δ + nq We introduce the data space A := A sm,q and the solution space X ( t ) := C ([0 , t ] , H s,q ) ∩ C ([0 , t ] , H s − δ,q ) , where the weight f σ ( τ ) ≡
0. First, let us prove the inequality (74). We need to control all norms k u nl ( t, · ) k L q , k u nlt ( t, · ) k L q , (cid:13)(cid:13) | D | s u nl ( t, · ) (cid:13)(cid:13) L q , (cid:13)(cid:13) | D | s − δ u nlt ( t, · ) (cid:13)(cid:13) L q . In the analogous way as we did in the proof of Theorem 2.1, we derive the following estimates: k u nl ( t, · ) k L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) k u k pX ( t ) , (78) k ∂ t u nl ( t, · ) k L q . (1 + τ ) (2+[ n ])(1 − σ δ ) r − n δ (1 − r ) k u k pX ( t ) , (79)provided that the condition (15) is satisfied and p ∈ h qm , ∞ (cid:17) if n ≤ qs, or p ∈ h qm , nn − qs i if n ∈ (cid:16) qs, q sq − m i . (80)Now, let us turn to estimate the norm (cid:13)(cid:13) | D | s − δ u nlt ( t, · ) (cid:13)(cid:13) L q . We use the ( L m ∩ L q ) − L q estimates fromProposition 3.9 to get (cid:13)(cid:13) | D | s − δ u nlt ( t, · ) (cid:13)(cid:13) L q . Z t (1 + t − τ ) n ])(1 − σ δ ) r − n δ (1 − r ) − s δ (cid:13)(cid:13) | u ( τ, · ) | p (cid:13)(cid:13) L m ∩ L q ∩ ˙ H s − δ,q dτ. The integrals with (cid:13)(cid:13) | u ( τ, · ) | p (cid:13)(cid:13) L m ∩ L q and (cid:13)(cid:13) | u ( τ, · ) | p (cid:13)(cid:13) L q will be handled as we did to obtain (78). In order toestimate (cid:13)(cid:13) | u ( τ, · ) | p (cid:13)(cid:13) ˙ H s − δ,q , we shall apply the fractional chain rule with p > ⌈ s − δ ⌉ from Proposition 5.3and the fractional Gagliardo-Nirenberg inequality from Proposition 5.1. Hence, we get (cid:13)(cid:13) | u ( τ, · ) | p (cid:13)(cid:13) ˙ H s − δ,q . k u ( τ, · ) k p − L q (cid:13)(cid:13) | D | s − δ u ( τ, · ) (cid:13)(cid:13) L q . k u ( τ, · ) k ( p − − θ q ) L q (cid:13)(cid:13) | D | s u ( τ, · ) (cid:13)(cid:13) ( p − θ q L q k u ( τ, · ) k − θ q L q (cid:13)(cid:13) | D | s u ( τ, · ) (cid:13)(cid:13) θ q L q . (1 + τ ) p (1+(1+[ n ])(1 − σ δ ) r − n δ ( m − qp )) − s − δ δ k u k pX ( τ ) , where p − q + 1 q = 1 q , θ q = ns (cid:16) q − q (cid:17) ∈ [0 , , θ q = ns (cid:16) q − q + s − δn (cid:17) ∈ h s − δs , i . These conditions imply the restriction1 < p ≤ q δn − qs if n > qs, or p > n ≤ qs. (81)Therefore, we obtain (cid:13)(cid:13) | D | s − δ u nlt ( t, · ) (cid:13)(cid:13) L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) − s δ k u k pX ( t ) . (82)In the analogous way we also derive (cid:13)(cid:13) | D | s u nl ( t, · ) (cid:13)(cid:13) L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) − s δ k u k pX ( t ) . (83)Summarizing, from (78) to (79), (82) to (83) and the definition of the norm in X ( t ) the inequality (74)follows immediately. 22ext, let us prove the inequality (73). Following the proof of Theorem 2.1, the new difficulty is to controlthe term (cid:13)(cid:13) | u ( τ, · ) | p − | v ( τ, · ) | p (cid:13)(cid:13) ˙ H s − δ,q . The integral representation | u ( τ, x ) | p − | v ( τ, x ) | p = p Z (cid:0) u ( τ, x ) − v ( τ, x ) (cid:1) G (cid:0) ωu ( τ, x ) + (1 − ω ) v ( τ, x ) (cid:1) dω, where G ( u ) = u | u | p − , leads to (cid:13)(cid:13) | u ( τ, · ) | p − | v ( τ, · ) | p (cid:13)(cid:13) ˙ H s − δ,q . Z (cid:13)(cid:13)(cid:13) | D | s − δ (cid:16)(cid:0) u ( τ, · ) − v ( τ, · ) (cid:1) G (cid:0) ωu ( τ, · ) + (1 − ω ) v ( τ, · ) (cid:1)(cid:17)(cid:13)(cid:13)(cid:13) L q dω. Applying the fractional Leibniz formula from Proposition 5.2 we derive the following estimate: (cid:13)(cid:13) | u ( τ, · ) | p − | v ( τ, · ) | p (cid:13)(cid:13) ˙ H s − δ,q . (cid:13)(cid:13) | D | s − δ (cid:0) u ( τ, · ) − v ( τ, · ) (cid:1)(cid:13)(cid:13) L r Z (cid:13)(cid:13) G (cid:0) ωu ( τ, · ) + (1 − ω ) v ( τ, · ) (cid:1)(cid:13)(cid:13) L r dω + k u ( τ, · ) − v ( τ, · ) k L r Z (cid:13)(cid:13) | D | s − δ G (cid:0) ωu ( τ, · ) + (1 − ω ) v ( τ, · ) (cid:1)(cid:13)(cid:13) L r dω . (cid:13)(cid:13) | D | s − δ (cid:0) u ( τ, · ) − v ( τ, · ) (cid:1)(cid:13)(cid:13) L r (cid:16) k u ( τ, · ) k p − L r p − + k v ( τ, · ) k p − L r p − (cid:17) + k u ( τ, · ) − v ( τ, · ) k L r Z (cid:13)(cid:13) | D | s − δ G (cid:0) ωu ( τ, · ) + (1 − ω ) v ( τ, · ) (cid:1)(cid:13)(cid:13) L r dω, where 1 r + 1 r = 1 r + 1 r = 1 q . Taking into consideration the fractional Gargliardo- Nirenberg inequality from Proposition 5.1 we obtain (cid:13)(cid:13) | D | s − δ (cid:0) u ( τ, · ) − v ( τ, · ) (cid:1)(cid:13)(cid:13) L r . k u ( τ, · ) − v ( τ, · ) k θ ˙ H s,q k u ( τ, · ) − v ( τ, · ) k − θ L q , k u ( τ, · ) k L r p − . k u ( τ, · ) k θ ˙ H s,q k u ( τ, · ) k − θ L q , k u ( τ, · ) − v ( τ, · ) k L r . k u ( τ, · ) − v ( τ, · ) k θ ˙ H s,q k u ( τ, · ) − v ( τ, · ) k − θ L q , where θ = ns (cid:16) q − r + s − δn (cid:17) ∈ h s − δs , i , θ = ns (cid:16) q − r ( p − (cid:17) ∈ [0 , , θ = ns (cid:16) q − r (cid:17) ∈ [0 , . Because ω ∈ [0 ,
1] is a parameter, employing again the fractional chain rule with p > ⌈ s − δ ⌉ fromProposition 5.3 and the fractional Gagliardo- Nirenberg inequality we get (cid:13)(cid:13) | D | s − δ G (cid:0) ωu ( τ, · ) + (1 − ω ) v ( τ, · ) (cid:1)(cid:13)(cid:13) L r . k ωu ( τ, · ) + (1 − ω ) v ( τ, · ) k p − L r (cid:13)(cid:13) | D | s − δ (cid:0) ωu ( τ, · ) + (1 − ω ) v ( τ, · ) (cid:1)(cid:13)(cid:13) L r . k ωu ( τ, · ) + (1 − ω ) v ( τ, · ) k ( p − θ + θ ˙ H s,q k ωu ( τ, · ) + (1 − ω ) v ( τ, · ) k ( p − − θ )+1 − θ L q , where p − r + 1 r = 1 r , θ = ns (cid:16) q − r (cid:17) ∈ [0 , , θ = ns (cid:16) q − r + s − δn (cid:17) ∈ h s − δs , i . All together it follows Z (cid:13)(cid:13) | D | s − δ G (cid:0) ωu ( τ, · ) + (1 − ω ) v ( τ, · ) (cid:1)(cid:13)(cid:13) L r dω . (cid:0) k u ( τ, · ) k ˙ H s,q + k v ( τ, · ) k ˙ H s,q (cid:1) ( p − θ + θ (cid:0) k u ( τ, · ) k L q + k v ( τ, · ) k L q (cid:1) ( p − − θ )+1 − θ . (cid:13)(cid:13) | u ( τ, · ) | p − | v ( τ, · ) | p (cid:13)(cid:13) ˙ H s − δ,q . (1 + τ ) p (cid:0) n ])(1 − σ δ ) r − n δ ( m − qp ) (cid:1) − s − δ δ k u − v k X ( τ ) (cid:0) k u k p − X ( τ ) + k v k p − X ( τ ) (cid:1) , where we note that θ + ( p − θ = θ + ( p − θ + θ = ns (cid:16) p − q + s − δn (cid:17) . Therefore, we have proved that (cid:13)(cid:13) | D | s − δ ∂ t (cid:0) N u ( t, · ) − N v ( t, · ) (cid:1)(cid:13)(cid:13) L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) − s δ k u − v k X ( t ) (cid:0) k u k p − X ( t ) + k v k p − X ( t ) (cid:1) , (cid:13)(cid:13) | D | s (cid:0) N u ( t, · ) − N v ( t, · ) (cid:1)(cid:13)(cid:13) L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) − s δ k u − v k X ( t ) (cid:0) k u k p − X ( t ) + k v k p − X ( t ) (cid:1) . From the definition of the norm in X ( t ) the inequality (73) follows. Summarizing, the proof of Theorem 2.3is complete. Remark 4.2.
It is clear that one should explain if one can really choose parameters q , q , r , · · · , r and θ , · · · , θ as required in the proof to Theorem 2.3. Following the explanations as we did in Remark 4 . ≤ p ≤ q δn − qs if n > qs, or p ≥ n ≤ qs. (84)These conditions are sufficient to guarantee the existence of all these parameters satisfying the requiredconditions. s > δ + nq We introduce both spaces of the data and the solutions as in Theorem 2.3, where the weight f σ ( τ ) ≡ | u ( τ, · ) | p and | u ( τ, · ) | p − | v ( τ, · ) | p in L m and L q as we did in the proof to Theorem 2.3. On the other hand, let us control the above terms in ˙ H s − δ,q by applying the fractional powers rule and the fractional Sobolev embedding.First, let us begin to estimate (cid:13)(cid:13) | u ( τ, · ) | p (cid:13)(cid:13) ˙ H s − δ,q . We apply Corollary 5.1 for fractional powers with s − δ ∈ (cid:0) nq , p (cid:1) and Corollary 5.2 with a suitable s ∗ < nq to derive (cid:13)(cid:13) | u ( τ, · ) | p (cid:13)(cid:13) ˙ H s − δ,q . k u ( τ, · ) k ˙ H s − δ,q k u ( τ, · ) k p − L ∞ . k u ( τ, · ) k ˙ H s − δ,q (cid:0) k u ( τ, · ) k ˙ H s ∗ ,q + k u ( τ, · ) k ˙ H s − δ,q (cid:1) p − . Using again the fractional Gagliardo-Nirenberg inequality from Proposition 5.1 gives k u ( τ, · ) k ˙ H s − δ,q . k u ( τ, · ) k − θ L q (cid:13)(cid:13) | D | s u ( τ, · ) (cid:13)(cid:13) θ L q . (1 + τ ) n ])(1 − σ δ ) r − n δ (1 − r ) − s − δ δ k u k X ( τ ) , k u ( τ, · ) k ˙ H s ∗ ,q . k u ( τ, · ) k − θ L q (cid:13)(cid:13) | D | s u ( τ, · ) (cid:13)(cid:13) θ L q . (1 + τ ) n ])(1 − σ δ ) r − n δ (1 − r ) − s ∗ δ k u k X ( τ ) , where θ = 1 − δs and θ = s ∗ s . Therefore, we obtain (cid:13)(cid:13) | u ( τ, · ) | p (cid:13)(cid:13) ˙ H s − δ,q . (1 + τ ) p (1+(1+[ n ])(1 − σ δ ) r − n δ (1 − r )) − s − δ δ − ( p − s ∗ δ k u k X ( τ ) . (1 + τ ) p (1+(1+[ n ])(1 − σ δ ) r − n δ ( m − mp )) k u k pX ( τ ) , if we choose s ∗ = nq − ε with a sufficiently small positive ε .Next, let us turn to estimate the term (cid:13)(cid:13) | u ( τ, · ) | p − | v ( τ, · ) | p (cid:13)(cid:13) ˙ H s − δ,q . Then, repeating the corresponding stepsof the proof of Theorem 2.3 and using analogous arguments as in the first step we conclude (cid:13)(cid:13) | u ( τ, · ) | p − | v ( τ, · ) | p (cid:13)(cid:13) ˙ H s − δ,q . (1 + τ ) p (1+(1+[ n ])(1 − σ δ ) r − n δ ( m − mp )) k u − v k X ( t ) (cid:0) k u k p − X ( t ) + k v k p − X ( t ) (cid:1) , provided that the conditions p > p > s − δ are satisfied. Summarizing, Theorem 2.4 is proved.24 .6. Proof of Theorem 2.5: s > δ + nq We introduce both spaces for the data and the solutions as in Theorem 2.3, where the weight f σ ( τ ) ≡ X ( t ) is replaced by the space X ( t ) in both inequalities (73) and (74). First, let us provethe inequality (74). In order to estimate u nl , we apply the L m ∩ L q − L q estimates from Proposition 3.9.Therefore, we derive (cid:13)(cid:13) u nl ( t, · ) (cid:13)(cid:13) L q . Z t (1 + t − τ ) n ])(1 − σ δ ) r − n δ (1 − r ) (cid:13)(cid:13) | u t ( τ, · ) | p (cid:13)(cid:13) L m ∩ L q dτ. Moreover, we get (cid:13)(cid:13) | u t ( τ, · ) | p (cid:13)(cid:13) L m ∩ L q . k u t ( τ, · ) k pL mp + k u t ( τ, · ) k pL qp . After applying the fractional Gagliardo-Nirenberg inequality from Proposition 5.1 we arrive at (cid:13)(cid:13) | u t ( τ, · ) | p (cid:13)(cid:13) L m ∩ L q . (1 + τ ) p ((2+[ n ])(1 − σ δ ) r − n δ ( m − mp )) k u k pX ( τ ) , provided that the condition p ∈ (cid:2) qm , ∞ (cid:1) is fulfilled due to s > δ + nq . Hence, we obtain k u nl ( t, · ) k L q . k u k pX ( t ) Z t (1 + t − τ ) n ])(1 − σ δ ) r − n δ (1 − r ) (1 + τ ) p ((2+[ n ])(1 − σ δ ) r − n δ ( m − mp )) dτ. The key tool relies now in using Lemma 5.2. Because of condition (23), applying Lemma 5.2 after choosing α = − − (cid:16) h n i(cid:17)(cid:16) − σ δ (cid:17) r + n δ (cid:16) − r (cid:17) and β = p (cid:16) − (cid:16) h n i(cid:17)(cid:16) − σ δ (cid:17) r + n δ (cid:16) m − mp (cid:17)(cid:17) , we have Z t (1 + t − τ ) n ])(1 − σ δ ) r − n δ (1 − r ) (1 + τ ) p ((2+[ n ])(1 − σ δ ) r − n δ ( m − mp )) dτ . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) . As a result, we arrive at the following estimate: k u nl ( t, · ) k L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) k u k pX ( t ) . (85)Analogously, we also derive k ∂ t u nl ( t, · ) k L q . (1 + τ ) (2+[ n ])(1 − σ δ ) r − n δ (1 − r ) k u k pX ( t ) . (86)Now, let us control the norm (cid:13)(cid:13) | D | s − δ u nlt ( t, · ) (cid:13)(cid:13) L q . We get (cid:13)(cid:13) | D | s − δ u nlt ( t, · ) (cid:13)(cid:13) L q . Z t (1 + t − τ ) n ])(1 − σ δ ) r − n δ (1 − r ) − s δ (cid:13)(cid:13) | u t ( τ, · ) | p (cid:13)(cid:13) L m ∩ L q ∩ ˙ H s − δ,q dτ. The integrals with estimates for (cid:13)(cid:13) | u t ( τ, · ) | p (cid:13)(cid:13) L m ∩ L q and (cid:13)(cid:13) | u t ( τ, · ) | p (cid:13)(cid:13) L q will be handled as before to obtain(85). In order to control the integral with (cid:13)(cid:13) | u t ( τ, · ) | p (cid:13)(cid:13) ˙ H s − δ,q , we shall apply Corollary 5.1 for fractionalpowers with s − δ ∈ (cid:0) nq , p (cid:1) and Corollary 5.2 with a suitable s ∗ < nq . Hence, we have (cid:13)(cid:13) | u t ( τ, · ) | p (cid:13)(cid:13) ˙ H s − δ,q . k u t ( τ, · ) k ˙ H s − δ,q k u t ( τ, · ) k p − L ∞ . k u t ( τ, · ) k ˙ H s − δ,q (cid:0) k u t ( τ, · ) k ˙ H s ∗ ,q + k u t ( τ, · ) k ˙ H s − δ,q (cid:1) p − . Applying the fractional Gagliardo-Nirenberg inequality leads to k u t ( τ, · ) k ˙ H s ∗ ,q . k u t ( τ, · ) k − θL q (cid:13)(cid:13) | D | s − δ u t ( τ, · ) (cid:13)(cid:13) θL q . (1 + τ ) (2+[ n ])(1 − σ δ ) r − n δ (1 − r ) − s ∗ δ k u k X ( τ ) , θ = s ∗ s − δ . Therefore, we obtain (cid:13)(cid:13) | u t ( τ, · ) | p (cid:13)(cid:13) ˙ H s − δ,q . (1 + τ ) p (cid:0) (2+[ n ])(1 − σ δ ) r − n δ (1 − r ) (cid:1) − s − δ δ − ( p − s ∗ δ k u k pX ( τ ) . (1 + τ ) p ((2+[ n ])(1 − σ δ ) r − n δ ( m − mp )) k u k pX ( τ ) , if we choose s ∗ = nq − ε , where ε is a sufficiently small positive number. By the same arguments as above itfollows (cid:13)(cid:13) | D | s − δ u nlt ( t, · ) (cid:13)(cid:13) L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) − s δ k u k pX ( t ) . (87)Analogously, we also get (cid:13)(cid:13) | D | s u nl ( t, · ) (cid:13)(cid:13) L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) − s δ k u k pX ( t ) . (88)From (85) to (88) and the definition of the norm in X ( t ) we may conclude immediately the inequality (74).Next, let us prove the inequality (73). The new difficulty is to control the term (cid:13)(cid:13) | u t ( τ, · ) | p −| v t ( τ, · ) | p (cid:13)(cid:13) ˙ H s − δ,q .Then, repeating the proof of Theorem 2.3 and using the analogous treatment as in the first step, we obtain (cid:13)(cid:13) | D | s − δ ∂ t (cid:0) N u ( t, · ) − N v ( t, · ) (cid:1)(cid:13)(cid:13) L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) − s δ k u − v k X ( t ) (cid:0) k u k p − X ( t ) + k v k p − X ( t ) (cid:1) , (cid:13)(cid:13) | D | s (cid:0) N u ( t, · ) − N v ( t, · ) (cid:1)(cid:13)(cid:13) L q . (1 + t ) n ])(1 − σ δ ) r − n δ (1 − r ) − s δ k u − v k X ( t ) (cid:0) k u k p − X ( t ) + k v k p − X ( t ) (cid:1) . From the definition of the norm in X ( t ) we may conclude immediately the inequality (73). This completesthe proof of Theorem 2.5.
5. Concluding remarks and open problemsRemark 5.1. (Time-dependent coefficients in the dissipation term)
A next challenge is to study L esti-mates for oscillating integrals and L p − L q linear estimates away from the conjugate line as well to struc-turally damped σ -evolution models with time-dependent coefficients. These estimates are fundamental toolsto prove global (in time) existence results to semi-linear models. Therefore, it is interesting to investigatethe following Cauchy problem: u tt + ( − ∆) σ u + b ( t )( − ∆) δ u t = 0 , u (0 , x ) = u ( x ) , u t (0 , x ) = u ( x ) (89)with σ ≥ δ ∈ [0 , σ ]. Here the coefficient b = b ( t ) should satisfy some “effectiveness assumptions” as in[12]. Remark 5.2. (Blow-up results)
In this paper, we applied ( L m ∩ L q ) − L q and L q − L q estimates forsolution and its derivatives to (3) to prove the global (in time) existence of small data Sobolev solutionsto the semi-linear models (1) and (2) with δ ∈ ( σ , σ ]. It can be expected to find the critical exponents foreach of the two nonlinearities. The “shape” of these critical exponents can be found in [2] by using the testfunction method, where the assumption for integers σ and δ comes into play. In general, the main difficultyis to deal with fractional Laplacian operators ( − ∆) σ as well-known non-local operators. Remark 5.3. (Gevrey smoothing)
We are interested in another qualitative property of solutions to (3),the so-called Gevrey smoothing. It is reasonable to use our estimates with L norms only. Moreover, wesuppose for the Cauchy data ( u , u ) ∈ H σ × L . The study of regularity properties for the solutions allowsto restrict our considerations to large frequencies in the extended phase space. Recalling the definition theGevrey-Sobolev space regularity Γ s,ρ introduced in [5] we may conclude the following statement. Theorem 6.
Let us consider the Cauchy problem (3) with δ ∈ ( σ , σ ) . The data are supposed to belongto the energy space, that is, ( u , u ) ∈ H σ × L . Then, there is a smoothing effect in the sense, that thesolution belongs to the Gevrey-Sobolev space as follows: u ( t, · ) ∈ Γ σ − δ ) ,σ , and | D | σ u ( t, · ) , u t ( t, · ) ∈ Γ σ − δ ) , for all t > . roof of Proposition 3.8. Let us turn to large values of | ξ | . For the sake of the asymptotic behavior of thecharacteristic roots in (27) and (28) we arrive at | ˆ K | . e − c | ξ | σ − δ ) t , | ˆ K | . | ξ | − δ e − c | ξ | σ − δ ) t and | ∂ t ˆ K | . | ξ | σ − δ ) e − c | ξ | σ − δ ) t , | ∂ t ˆ K | . e − c | ξ | σ − δ ) t , for some positive constants c . Therefore, using the representation of the solutions (26) we derive the followingestimates: Z R n exp (cid:0) c | ξ | σ − δ ) t (cid:1) | ξ | σ | v ( t, ξ ) | dξ . Z R n | ξ | σ | v ( ξ ) | dξ + Z R n | v ( ξ ) | dξ, Z R n exp (cid:0) c | ξ | σ − δ ) t (cid:1) | v t ( t, ξ ) | dξ . Z R n | ξ | σ | v ( ξ ) | dξ + Z R n | v ( ξ ) | dξ. We may conclude immediately all the statements we wanted to prove.
Acknowledgments
The PhD study of MSc. T.A. Dao is supported by Vietnamese Government’s Scholarship.
Appendix A
A.1. Fractional Gagliardo-Nirenberg inequality
Proposition 5.1.
Let < p, p , p < ∞ , σ > and s ∈ [0 , σ ) . Then, it holds the following fractionalGagliardo-Nirenberg inequality for all u ∈ L p ∩ ˙ H σp : k u k ˙ H sp . k u k − θL p k u k θ ˙ H σp , where θ = θ s,σ ( p, p , p ) = p − p + sn p − p + σn and sσ ≤ θ ≤ . For the proof one can see [9].
A.2. Fractional Leibniz rule
Proposition 5.2.
Let us assume s > and ≤ r ≤ ∞ , < p , p , q , q ≤ ∞ satisfying the relation r = 1 p + 1 p = 1 q + 1 q . Then, the following fractional Leibniz rule holds: k | D | s ( u v ) k L r . k | D | s u k L p k v k L p + k u k L q k | D | s v k L q for any u ∈ ˙ H sp ∩ L q and v ∈ ˙ H sq ∩ L p . These results can be found in [7].
A.3. Fractional chain rule
Proposition 5.3.
Let us choose s > , p > ⌈ s ⌉ and < r, r , r < ∞ satisfying r = p − r + r . Let usdenote by F ( u ) one of the functions | u | p , ±| u | p − u . Then, it holds the following fractional chain rule: k | D | s F ( u ) k L r . k u k p − L r k | D | s u k L r for any u ∈ L r ∩ ˙ H sr . The proof can be found in [21].
A.4. Fractional powers roposition 5.4. Let p > , < r < ∞ and u ∈ H sr , where s ∈ (cid:0) nr , p (cid:1) . Let us denote by F ( u ) one of thefunctions | u | p , ±| u | p − u with p > . Then, the following estimate holds : k F ( u ) k H sr . k u k H sr k u k p − L ∞ . Corollary 5.1.
Under the assumptions of Proposition 5.4 it holds: k F ( u ) k ˙ H sr . k u k ˙ H sr k u k p − L ∞ . The proof can be found in [3].
A.5. A fractional Sobolev embedding
Proposition 5.5.
Let n ≥ , < s < n , < q ≤ r < ∞ , α < nq ′ where q ′ denotes conjugate number of q ,and γ > − nr , α ≥ γ satisfying r = q + α − γ − sn . Then, it holds: (cid:13)(cid:13) | x | γ | D | − s u (cid:13)(cid:13) L r . (cid:13)(cid:13) | x | α u (cid:13)(cid:13) L q , that is , (cid:13)(cid:13) | x | γ u (cid:13)(cid:13) L r . (cid:13)(cid:13) | x | α | D | s u (cid:13)(cid:13) L q for any u ∈ ˙ H s,qα , where ˙ H s,qα = { u : | D | s u ∈ L q ( R n , | x | αq ) } is the weighted homogeneous Sobolev space ofpotential type with the norm k u k ˙ H s,qα = (cid:13)(cid:13) | x | α | D | s u (cid:13)(cid:13) L q . The proof can be found in [26].
Corollary 5.2.
Let < q < ∞ and < s < nq < s . Then, for any function u ∈ ˙ H s ,q ∩ ˙ H s ,q we have k u k L ∞ . k u k ˙ H s ,q + k u k ˙ H s ,q . For the proof one can see [5].
A.6. A variant of Mikhlin- H¨omander multiplier theorem
Proposition 5.6.
Let q ∈ (1 , ∞ ) , k = [ n ] + 1 and b ≥ . Suppose that m ∈ C k ( R n ) satisfies m ( ξ ) = 0 if | ξ | ≤ and (cid:12)(cid:12) ∂ αξ m ( ξ ) (cid:12)(cid:12) ≤ C | ξ | − nb | q − | (cid:0) A | ξ | b − (cid:1) | α | for all | α | ≤ k , | ξ | ≥ and with some constants A ≥ . Then, the operator T m = F − (cid:0) m ( ξ ) (cid:1) ∗ ( x ) , definedby the action T m f ( x ) := F − ξ → x (cid:0) m ( ξ ) F y → ξ (cid:0) f ( y ) (cid:1)(cid:1) , is continuously bounded from L q into itself and satisfies the following estimate: k T m f ( · ) k L q ≤ CA n | q − | k f k L q . The proof of this lemma can be found in [2] (Theorem 10) and [14] (Theorem 1).
A.7. Modified Bessel functions
Proposition 5.7.
Let f ∈ L p ( R n ) , p ∈ [1 , , be a radial function. Then, the Fourier transform F ( f ) isalso a radial function and it satisfies F n ( ξ ) := F ( f )( ξ ) = c Z ∞ g ( r ) r n − ˜ J n − ( r | ξ | ) dr, g ( | x | ) := f ( x ) , where ˜ J µ ( s ) := J µ ( s ) s µ is called the modified Bessel function with the Bessel function J µ ( s ) and a non-negativeinteger µ . Proposition 5.8.
The the following properties of the modified Bessel function hold:1. sd s ˜ J µ ( s ) = ˜ J µ − ( s ) − µ ˜ J µ ( s ) , . d s ˜ J µ ( s ) = − s ˜ J µ +1 ( s ) ,3. ˜ J − ( s ) = q π cos s and ˜ J ( s ) = q π sin ss ,4. | ˜ J µ ( s ) | ≤ Ce π | Imµ | if s ≤ , and ˜ J µ ( s ) = Cs − cos (cid:0) s − µ π − π (cid:1) + O ( | s | − ) if | s | ≥ ,5. ˜ J µ +1 ( r | x | ) = − r | x | ∂ r ˜ J µ ( r | x | ) , r = 0 , x = 0 .A.8. Fa`a di Bruno’s formula Proposition 5.9.
Let h (cid:0) g ( x ) (cid:1) = ( h ◦ g )( x ) with x ∈ R . Then, we have d n dx n h (cid:0) g ( x ) (cid:1) = X n ! m !1! m m !2! m · · · m n ! n ! m n h ( m + m + ··· + m n ) (cid:0) g ( x ) (cid:1) n Y j =1 (cid:0) g ( j ) ( x ) (cid:1) m j , where the sum is taken over all n - tuples of non-negative integers ( m , m , · · · , m n ) satisfying the constraintof the Diophantine equation: · m + 2 · m + · · · + n · m n = n. For the proof one can see [23].
A.9. A useful lemma
Lemma 5.1.
The following formula of derivative of composed function holds for any multi-index α : ∂ αξ h (cid:0) f ( ξ ) (cid:1) = | α | X k =1 h ( k ) (cid:0) f ( ξ ) (cid:1)(cid:16) X γ + ··· + γ k ≤ α | γ | + ··· + | γ k | = | α | , | γ i |≥ (cid:0) ∂ γ ξ f ( ξ ) (cid:1) · · · (cid:0) ∂ γ k ξ f ( ξ ) (cid:1)(cid:17) , where h = h ( s ) and h ( k ) ( s ) = d k h ( s ) ds k . The result can be found in [24] at the page 202.
Lemma 5.2.
Let α, β ∈ R . Then: I ( t ) := Z t (1 + t − τ ) − α (1 + τ ) − β dτ . (1 + t ) − min { α,β } if max { α, β } > , (1 + t ) − min { α,β } log(2 + t ) if max { α, β } = 1 , (1 + t ) − α − β if max { α, β } < . For the proof one can see [5].
References [1] M. DAbbicco, M.R. Ebert,
An application of L p − L q decay estimates to the semilinear wave equation with parabolic-likestructural damping , Nonlinear Analysis, 99 (2014), 16-34.[2] M. DAbbicco, M.R. Ebert, A new phenomenon in the critical exponent for structurally damped semi-linear evolutionequations , Nonlinear Analysis, 149 (2017), 1-40.[3] Duong T. P., M. Kainane Mezadek, and M. Reissig,
Global existence for semi-linear structurally damped σ -evolutionmodels , J. Math. Anal. Appl., 431 (2015), 569-596.[4] M. D’Abbicco, M. Reissig, Semilinear structural damped waves , Math. Methods Appl. Sci., 37 (2014), 1570-1592.[5] T.A. Dao, M. Reissig,
An application of L estimates for oscillating integrals to parabolic like semi-linear structurallydamped σ -evolution models , 33A4, submitted.[6] M. R. Ebert, M. Reissig, “Methods for partial differential equations, qualitative properties of solutions, phase spaceanalysis, semilinear models”, Birkh¨auser, 2018.[7] L. Grafakos, “Classical and modern Fourier analysis”, Prentice Hall, 2004.[8] V.A. Galaktionov, E.L. Mitidieri, and S.I. Pohozaev, “Blow-up for higher-order prabolic, hyperbolic, dispersion andSchr¨odinger equations”, in: Monogr. Res. Notes Math., Chapman and Hall/CRC, ISBN: 9781482251722, 2014.
9] H. Hajaiej, L. Molinet, T. Ozawa, and B. Wang, “Necessary and sufficient conditions for the fractional Gagliardo-Nirenberginequalities and applications to Navier-Stokes and generalized boson equations, Harmonic analysis and nonlinear partialdifferential equations”, 159-175, RIMS Kokyuroku Bessatsu, B26, Res.Inst.Math.Sci. (RIMS), Kyoto, 2011.[10] R. Ikehata,
Asymptotic profiles for wave equations with strong damping , J. Differential Equations, 257 (2014) 2159-2177.[11] R. Ikehata, G. Todorova, and B. Yordanov,
Wave equations with strong damping in Hilbert spaces, Journal DifferentialEquations , 254 (2013), 3352-3368.[12] M. Kainane, “Structural damped σ -evolution operators”, PhD thesis, TU Bergakademie Freiberg, Germany, 2013.[13] J. Marcinkiewicz, Sur les multiplicateurs des s´eries de Fourier , Studia Math, 8 (1939), 78-91.[14] A. Miyachi,
On some Fourier multipliers for H p ( R n ), J. Fac. Sci. Univ. Tokyo IA, 27 (1980), 157-179.[15] A. Miyachi, On some estimates for the wave equation in L p and H p , J. Fac. Sci. Univ. Tokyo IA, 27 (1980), 331-354.[16] E. Mitidieri, S.I. Pohozaev, Non-existence of weak solutions for some degenerate elliptic and parabolic problems on R n ,J. Evol. Equ., 1 (2001), 189-220.[17] Bui Tang Bao Ngoc, “Semi-linear waves with time-pendent speed and dissipation”, PhD thesis, TU Bergakademie Freiberg,Germany, 2014.[18] T. Narazaki, M. Reissig, L estimates for oscillating integrals related to structural damped wave models , in: M. Cicognani,F. Colombini, D. Del Santo (Eds.), Studies in Phase Space Analysis with Applications to PDEs, in: Progr. NonlinearDifferential Equations Appl., Birkh¨auser, 2013, 215-258.[19] J.C. Peral, L p estimates for the wave equation , J. Funct. Anal., 36 (1980), 114-145.[20] F. Pizichillo, “Linear and non-linear damped wave equations”, Master thesis, 62pp., University of Bari, 2014.[21] A. Palmieri, M. Reissig, Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass anddissipation, II , Mathematische Nachrichten., (2018), 1-34, https://doi.org/10.1002/mana.201700144.[22] T. Runst, W. Sickel, “Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations,De Gruyter series in nonlinear analysis and applications”, Walter de Gruyter & Co., Berlin, 1996.[23] Cav. Francesco Fa`a di Bruno,
Note sur une nouvelle formule de calcul differentiel , Quarterly J. Pure Appl. Math., 1(1857), 359-360.[24] C. G. Simander, “On Dirichlet boundary value problem”, An L p -Theory based on a generalization of G˚arding’s inequality,Lecture Notes in Mathematics, 268, Springer, Berlin, 1972.[25] Y. Shibata, On the rate of decay of solutions to linear viscoelastic equation , Math. Methods Appl. Sci., 23 (2000), 203-226.[26] E. Stein, G. Weiss,
Fractional integrals on n -dimensional Euclidean space , J. Math. Mech., 7 (1958), 503-514.[27] F. Weisz, Marcinkiewicz multiplier theorem and the Sunouchi operator for CiesielskiFourier series , Journal of Approxi-mation Theory, 133 (2005), 195-220., Journal of Approxi-mation Theory, 133 (2005), 195-220.