Regularity and stability analysis for a class of semilinear nonlocal differential equations in Hilbert spaces
aa r X i v : . [ m a t h . A P ] D ec REGULARITY AND STABILITY ANALYSIS FOR A CLASS OFSEMILINEAR NONLOCAL DIFFERENTIAL EQUATIONS INHILBERT SPACES
TRAN DINH KE ♮ , NGUYEN NHU THANG, LAM TRAN PHUONG THUY Abstract.
We deal with a class of semilinear nonlocal differential equationsin Hilbert spaces which is a general model for some anomalous diffusion equa-tions. By using the theory of integral equations with completely positive kerneltogether with local estimates, some existence, regularity and stability resultsare established. An application to nonlocal partial differential equations isshown to demonstrate our abstract results. Introduction
Let H be a separable Hilbert space. Consider the following problem ddt [ k ∗ ( u − u )]( t ) + Au ( t ) = f ( u ( t )) , t > , (1.1) e1 u (0) = u , (1.2) e2 where the unknown function u takes values in H , the kernel k ∈ L loc ( R + ), A is anunbounded linear operator, and f : H → H is a given function. Here ∗ denotes theLaplace convolution, i.e., ( k ∗ v )( t ) = R t k ( t − s ) v ( s ) ds .It should be mentioned that, nonlocal equations have been employed to modeldifferent problems related to processes in materials with memory (see, e.g., [3,4, 5, 13]). In particular, when the kernel k ( t ) = g − α ( t ) := t − α / Γ(1 − α ) , α ∈ (0 , ddt [ k ∗ ( u − u )] represents the Caputo fractional derivative of order α , and thisequation has been a subject of an extensive study. In a specific setting, for example,when H = L (Ω) , Ω ⊂ R N , and A = − ∆ is the Laplace operator associated witha boundary condition of Dirichlet/Neumann type, equation (1.1) with a class ofkernel functions is utilized to describe anomalous diffusion phenomena includingslow/ultraslow diffusions, which were remarked in [14].Our motivation for the present work is that, up to our knowledge, no attempthas been made to establish regularity results for (1.1)-(1.2). Moreover, the stabilityanalysis in the sense of Lyapunov for (1.1) has been less known. In the special casewhen k = g − α , we refer to some results on stability analysis given in [1, 8, 9]. Inthe recent paper [15], Vergara and Zacher investigated a concrete model of type(1.1), which is a nonlocal semilinear partial differential equation (PDE). Using amaximum principle for the linearized equation, they proved the asymptotic stability Mathematics Subject Classification.
Key words and phrases. nonlocal differential equation; weak solution; mild solution; asymp-totic stability. ♮ Corresponding author. Email: [email protected] (T.D.Ke). for zero solution of this equation. It is worth noting that, the technique used in[15] does not work for the abstract equation (1.1). In this paper, the stabilityof solutions to (1.1) will be analyzed by using a new representation of solutionstogether with a new Gronwall type inequality. In order to deal with (1.1), we makethe following standing hypotheses.( A ) The operator A : D ( A ) → H is densely defined, self-adjoint, and positivelydefinite with compact resolvent. ( K ) The kernel function k ∈ L loc ( R + ) is nonnegative and nonincreasing, andthere exists a function l ∈ L loc ( R + ) such that k ∗ l = 1 on (0 , ∞ ) . ( F ) The nonlinear function f : H → H is locally Lipschitzian, i.e., for each ρ > there is a nonnegative number κ ( ρ ) such that k f ( v ) − f ( v ) k ≤ κ ( ρ ) k v − v k , ∀ v , v ∈ B ρ , where B ρ is the closed ball in H with center at origin and radius ρ . Noting that, the hypothesis ( K ) was used in a lot of works, e.g. [6, 10, 12, 14,15, 18]. This enables us to transform equations of type (1.1) to a Volterra integralequation with completely positive kernel, which is a main subject discussed in [13].In this case, one writes ( k, l ) ∈ PC . Some typical examples of ( k, l ) were given in[14], e.g., • k ( t ) = g − α ( t ) and l ( t ) = g α ( t ) , t >
0: slow diffusion (fractional order) case. • k ( t ) = Z g β ( t ) dβ and l ( t ) = Z ∞ e − pt p dp, t >
0: ultra-slow diffusion(distributed order) case. • k ( t ) = g − α ( t ) e − γt , γ ≥
0, and l ( t ) = g α ( t ) e − γt + γ Z t g α ( s ) e − γs ds, t > K ), we refer the reader to [12].Owing these hypotheses, we are able to derive, in the next section, a variation-of-parameter formula as well as the concept of mild solution for inhomogeneousequations. We show that a mild solution is also a weak solution, and it is classicalif the external force function is H¨older continuous and the kernel function l issmooth enough. Section 3 is devoted to the semilinear equations, in which weprove the local/global solvability and asymptotic stability for (1.1). In addition,we show that, the mild solution of semilinear problem is also H¨older continuous.Consequently, we present in the last section an application of the abstract results.2. Preliminaries
For µ ∈ R + , consider the following scalar integral equations s ( t ) + µ ( l ∗ s )( t ) = 1 , t ≥ , (2.1) eq-s r ( t ) + µ ( l ∗ r )( t ) = l ( t ) , t > . (2.2) eq-r In the sequel, we assume, in addition, that l is continuous on (0 , ∞ ). Under thisassumption, the existence and uniqueness of s and r were examined in [11]. In thecase l ( t ) = g α ( t ), following from the Laplace transform of s ( · ) and r ( · ), we knowthat s ( t ) = E α, ( − µt α ) and r ( t ) = t α − E α,α ( − µt α ), here E α,β is the Mittag-Leffler EMILINEAR NONLOCAL DIFFERENTIAL EQUATIONS 3 function defined by E α,β ( z ) = ∞ X n =0 z n Γ( αn + β ) , z ∈ C . Recall that the kernel function l is said to be completely positive iff s ( · ) and r ( · )take nonnegative values for every µ >
0. The complete positivity of l is equivalentto that (see [3]), there exist α ≥ k ∈ L loc ( R + ) nonnegative and nonincreasingwhich satisfy αl + l ∗ k = 1. In particular, the hypothesis ( K ) ensures that l iscompletely positive.Denote by s ( · , µ ) and r ( · , µ ) the solutions of (2.1) and (2.2), respectively. Wecollect some properties of these functions. pp-sr Proposition 2.1.
Let the hypothesis ( K ) hold. Then for every µ > , s ( · , µ ) , r ( · , µ ) ∈ L loc ( R + ) . In addition, we have:(1) The function s ( · , µ ) is nonnegative and nonincreasing. Moreover, s ( t, µ ) (cid:20) µ Z t l ( τ ) dτ (cid:21) ≤ , ∀ t ≥ . (2.3) eq-s1 (2) The function r ( · , µ ) is nonnegative and the following relations hold s ( t, µ ) = 1 − µ Z t r ( τ, µ ) dτ = k ∗ r ( · , µ )( t ) , t ≥ . (2.4) eq-sr1 (3) For each t > , the functions µ s ( t, µ ) and µ r ( t, µ ) are nonincreasing.Proof. The justification for (2.3) and (2.4) can be found in [3]. We prove the laststatement. For β ∈ L loc ( R + ), we denote by ˆ β the Laplace transform of β . It followsfrom (2.1)-(2.2) that ˆ s ( λ, µ ) = λ − (1 + µ ˆ l ) − , ˆ r ( λ, µ ) = ˆ l (1 + µ ˆ l ) − , λ > . Then ∂∂µ ˆ s ( λ, µ ) = − λ − ˆ l (1 + µ ˆ l ) − = − ˆ s ( λ, µ )ˆ r ( λ, µ ) ,∂∂µ ˆ r ( λ, µ ) = − ˆ l (1 + µ ˆ l ) − = − ˆ r ( λ, µ )ˆ r ( λ, µ ) . Taking the inverse transform and using the convolution rule, we get ∂∂µ s ( t, µ ) = − s ( · , µ ) ∗ r ( · , µ )( t ) ≤ ,∂∂µ r ( t, µ ) = − r ( · , µ ) ∗ r ( · , µ )( t ) ≤ , t > . The proof is complete. (cid:3) rm-sr
Remark 2.1. (1) As mentioned in [15] , the functions s ( · , µ ) and r ( · , µ ) takenonnegative values even in the case µ ≤ .(2) Equation (2.1) is equivalent to the problem ddt [ k ∗ ( s − µs = 0 , s (0) = 1 . T.D. KE, N.N. THANG, L.T.P. THUY
This can be seen by convoluting both side of equation [ s −
1] + µl ∗ s = 0 with k and using k ∗ l = 1 .(3) Let v ( t ) = s ( t, µ ) v + ( r ( · , µ ) ∗ g )( t ) , here g ∈ L loc ( R + ) . Then v solves theproblem ddt [ k ∗ ( v − v )]( t ) + µv ( t ) = g ( t ) , v (0) = v . Indeed, by formulation and the relation k ∗ r = s , we have k ∗ ( v − v ) = k ∗ ( s − v + k ∗ r ∗ g = k ∗ ( s − v + s ∗ g. So ddt [ k ∗ ( v − v )] = ddt [ k ∗ ( s − v + s (0 , µ ) g + s ′ ( · , µ ) ∗ g = − µs ( · , µ ) v + g − µr ( · , µ ) ∗ g = − µ [ s ( · , µ ) v + r ( · , µ ) ∗ g ] + g = − µv + g, thanks to the fact that s (0 , µ ) = 1 and s ′ ( t, µ ) = − µr ( t, µ ) , t > .(4) We deduce from (2.3) that, if l L ( R + ) then lim t →∞ s ( t, µ ) = 0 for every µ > .(5) It follows from (2.4) that R t r ( τ, µ ) dτ ≤ µ − , ∀ t > . If l L ( R + ) then R ∞ r ( τ, µ ) dτ = µ − for every µ > . We are now in a position to prove a Gronwall type inequality, which play animportant role in our analysis. pp-gronwall
Proposition 2.2.
Let v be a nonnegative function satisfying v ( t ) ≤ s ( t, µ ) v + Z t r ( t − τ, µ )[ αv ( τ ) + β ( τ )] dτ, t ≥ , (2.5) pp-gronwall-0 for µ > , v ≥ , α > and β ∈ L loc ( R + ) . Then v ( t ) ≤ s ( t, µ − α ) v + Z t r ( t − τ, µ − α ) β ( τ ) dτ. Particularly, if β is constant then v ( t ) ≤ s ( t, µ − α ) v + βµ − α (1 − s ( t, µ − α )) . Proof.
Let w ( t ) be the expression in the right hand side of (2.5). Then v ( t ) ≤ w ( t )for t ≥
0, and w solves the problem ddt [ k ∗ ( w − v )]( t ) + µw ( t ) = αv ( t ) + β ( t ) ,w (0) = v , thanks to Remark 2.1 (2). This is equivalent to ddt [ k ∗ ( w − v )]( t ) + ( µ − α ) w ( t ) = α ( v ( t ) − w ( t )) + β ( t ) ,w (0) = v , EMILINEAR NONLOCAL DIFFERENTIAL EQUATIONS 5 which implies w ( t ) = s ( t, µ − α ) v + Z t r ( t − τ, µ − α )[ α ( v ( τ ) − w ( τ )) + β ( τ )] dτ ≤ s ( t, µ − α ) v + Z t r ( t − τ, µ − α ) β ( τ ) dτ, in accordance with v ( τ ) − w ( τ ) ≤ τ ≥ r .Finally, if β is constant, we employ (2.4) to get Z t r ( t − τ, µ − α ) βdτ = β Z t r ( τ, µ − α ) dτ = βµ − α (1 − s ( t, µ − α )) , which completes the proof. (cid:3) Let us mention that, the hypothesis ( A ) ensures the existence of an orthonormalbasis of H consisting of eigenfunctions { e n } ∞ n =1 of the operator A and we have Av = ∞ X n =1 λ n v n e n , where λ n > e n of A , D ( A ) = { v = ∞ X n =1 v n e n : ∞ X n =1 λ n v n < ∞} . We can assume that 0 < λ ≤ λ ≤ ... ≤ λ n → ∞ as n → ∞ .For γ ∈ R , one can define the fractional power of A as follows D ( A γ ) = ( v = ∞ X n =1 v n e n : ∞ X n =1 λ γn v n < ∞ ) ,A γ v = ∞ X n =1 λ γn v n e n . Let V γ = D ( A γ ). Then V γ is a Banach space endowed with the norm k v k γ = k A γ v k = ∞ X n =1 λ γn v n ! . Furthermore, for γ >
0, we can identify the dual space V ∗ γ of V γ with V − γ .We now define the following operators S ( t ) v = ∞ X n =1 s ( t, λ n ) v n e n , t ≥ , v ∈ H, (2.6) op-S R ( t ) v = ∞ X n =1 r ( t, λ n ) v n e n , t > , v ∈ H. (2.7) op-R It is easily seen that S ( t ) and R ( t ) are linear. We show some basic properties ofthese operators in the following lemma. lm-SR Lemma 2.3.
Let { S ( t ) } t ≥ and { R ( t ) } t> , be the families of linear operators de-fined by (2.6) and (2.7) , respectively. Then T.D. KE, N.N. THANG, L.T.P. THUY (1) For each v ∈ H and T > , S ( · ) v ∈ C ([0 , T ]; H ) and AS ( · ) v ∈ C ((0 , T ]; H ) .Moreover, k S ( t ) v k ≤ s ( t, λ ) k v k , t ∈ [0 , T ] , (2.8) lm-SR1 k AS ( t ) v k ≤ k v k (1 ∗ l )( t ) , t ∈ (0 , T ] . (2.9) lm-SR2 (2) Let v ∈ H, T > and g ∈ C ([0 , T ]; H ) . Then R ( · ) v ∈ C ((0 , T ]; H ) , R ∗ g ∈ C ([0 , T ]; H ) and A ( R ∗ g ) ∈ C ([0 , T ]; V − ) . Furthermore, k R ( t ) v k ≤ r ( t, λ ) k v k , t ∈ (0 , T ] , (2.10) lm-SR3a k ( R ∗ g )( t ) k ≤ Z t r ( t − τ, λ ) k g ( τ ) k dτ, t ∈ [0 , T ] , (2.11) lm-SR3b k A ( R ∗ g )( t ) k − ≤ (cid:18)Z t r ( t − τ, λ ) k g ( τ ) k dτ (cid:19) , t ∈ [0 , T ] . (2.12) lm-SR3c Proof. (1) For the first statement, we observe that k S ( t ) v k = ∞ X n =1 s ( t, λ n ) v n . (2.13) lm-SR4 Since s ( t, λ n ) ≤ t ≥ , n ∈ N , this series is uniformly convergent on[0 , T ]. So is series (2.6). Due to the fact that s ( · , λ n ) is continuous, we get S ( · ) v ∈ C ([0 , T ]; H ). Estimate (2.8) is deduced from (2.13) by using s ( t, λ n ) ≤ s ( t, λ ) forall n > AS ( t ) v = ∞ X n =1 λ n s ( t, λ n ) v n e n , (2.14) lm-SR5 we have k AS ( t ) v k = ∞ X n =1 λ n s ( t, λ n ) v n . (2.15) lm-SR5a In view of (2.3), we get λ n s ( t, λ n ) ≤ λ n λ n (1 ∗ l )( t ) ≤ ∗ l )( t ) , ∀ t > . Substituting into (2.15), we have k AS ( t ) v k ≤ k v k (1 ∗ l )( t ) , for every t >
0. In addition,for any δ such that 0 < δ < T , one has λ n s ( t, λ n ) ≤ ∗ l )( δ ) , which impliesthat the convergence of (2.15) as well as (2.14) is uniform on [ δ, T ]. That is, AS ( · ) v ∈ C ([ δ, T ]; H ).(2) Recall that r ( · , µ ) is continuous on (0 , ∞ ) (see, e.g. [11]). So for any δ ∈ (0 , T )and µ > r ( · , µ ) ∈ C ([ δ, T ]). This ensures that the series k R ( t ) v k = ∞ X n =1 r ( t, λ n ) v n (2.16) lm-SR6 is uniformly convergent on [ δ, T ]. So is series (2.7). Inequality (2.10) follows from(2.16) since r ( t, · ) is nonincreasing. EMILINEAR NONLOCAL DIFFERENTIAL EQUATIONS 7
We now prove that R ∗ g ∈ C ([0 , T ]; H ). Denoting g n ( t ) = ( g ( t ) , e n ), we firstcheck that ( R ∗ g )( t ) = ∞ X n =1 [ r ( · , λ n ) ∗ g n ]( t ) e n . (2.17) lm-SR7 Indeed, since g ∈ C ([0 , T ]; H ), the series k g ( t ) k = ∞ P n =1 | g n ( t ) | is uniformly conver-gent on [ δ, T ]. Given ǫ >
0, for δ ≤ τ ≤ t ≤ T and p ∈ N , we have k n + p X k = n r ( τ, λ k ) g k ( t − τ ) e k k ≤ r ( τ, λ ) n + p X k = n | g k ( t − τ ) | ! < ǫ, provided that n is large enough. So the series ∞ P n =1 r ( τ, λ n ) g n ( t − τ ) e n convergesuniformly on [ δ, T ] and one can take integration term by term on [ δ, t ], i.e. Z tδ R ( t − τ ) g ( τ ) dτ = ∞ X n =1 (cid:18)Z tδ r ( τ, λ n ) g n ( t − τ ) dτ (cid:19) e n . Fix t > h n ( δ ) = t R δ r ( τ, λ n ) g n ( t − τ ) dτ . Arguing as above for the uniformconvergence of the series ∞ P n =1 h n ( δ ) e n on [0 , t ], we can pass to the limit as δ → | [ r ( · , λ n ) ∗ g n ]( t ) | ≤ Z t p r ( t − τ, λ n ) p r ( t − τ, λ n ) | g n ( τ ) | dτ ≤ (cid:18)Z t r ( t − τ, λ n ) dτ (cid:19) (cid:18)Z t r ( t − τ, λ n ) | g n ( τ ) | dτ (cid:19) ≤ (cid:18) λ n (1 − s ( t, λ n )) (cid:19) (cid:18)Z t r ( t − τ, λ ) | g n ( τ ) | dτ (cid:19) ≤ λ n (cid:18)Z t r ( t − τ, λ ) | g n ( τ ) | dτ (cid:19) , (2.18) lm-SR8 thanks to (2.4) and the monotonicity of r ( t, · ). Then it follows n + p X k = n | [ r ( · , λ k ) ∗ g k ]( t ) | ≤ λ Z t r ( t − τ, λ ) n + p X k = n | g k ( τ ) | ! dτ ≤ ǫλ Z t r ( t − τ, λ ) dτ ≤ ǫλ , for n large, thanks to the uniform convergence of ∞ P n =1 | g n ( t ) | on [0 , T ] and relation(2.4). Hence (2.17) is uniformly convergent on [0 , T ] and then R ∗ g ∈ C ([0 , T ]; H ).Estimate (2.11) takes place by employing (2.10).Finally, we show that A ( R ∗ g ) ∈ C ([0 , T ]; V − ). Noticing that A ( R ∗ g )( t ) = ∞ X n =1 λ n [ r ( · , λ n ) ∗ g n ]( t ) e n , T.D. KE, N.N. THANG, L.T.P. THUY we obtain k A ( R ∗ g )( t ) k − = k A ( R ∗ g )( t ) k = ∞ X n =1 (cid:16) λ n [ r ( · , λ n ) ∗ g n ]( t ) (cid:17) . (2.19) lm-SR9 Using estimate (2.18), one can claim the uniform convergence of (2.19) on [0 , T ]and estimate (2.12) follows. Thus A ( R ∗ g ) ∈ C ([0 , T ]; V − ) as desired.The proof is complete. (cid:3) rm-SR Remark 2.2. (1) Obviously, S (0) v = v for every v ∈ H .(2) We have ( R ∗ g )(0) = 0 . Indeed, it follows from (2.11) that k ( R ∗ g )( t ) k ≤ sup τ ∈ [0 ,T ] k g ( τ ) k Z t r ( t − τ, λ ) dτ = sup τ ∈ [0 ,T ] k g ( τ ) k λ − (1 − s ( t, λ )) → as t → . (3) Lemma 2.3 implies that A S ( · ) v, A ( R ∗ g ) ∈ C ((0 , T ]; H ) for every v ∈ H and g ∈ C ([0 , T ]; H ) . Equivalently, S ( · ) v, R ∗ g ∈ C ((0 , T ]; V ) . Given g ∈ C ([0 , T ]; H ), consider the linear problem ddt [ k ∗ ( u − u )]( t ) + Au ( t ) = g ( t ) , t ∈ (0 , T ] , (2.20) le1 u (0) = u . (2.21) le2 Based on the operators S ( t ) and R ( t ), we introduce the following definition of mildsolutions to (2.20)-(2.21). def-mild-sol Definition 2.1.
A function u ∈ C ([0 , T ]; H ) is called a mild solution to the problem (2.20) - (2.21) on [0 , T ] iff u ( t ) = S ( t ) u + Z t R ( t − s ) g ( s ) ds, t ∈ [0 , T ] . (2.22) def-mild-sol-0 Weak solution and regularity
Existence and uniqueness.
In the sequel, we will define weak solution for(2.20)-(2.21) and show that a mild solution is also a weak solution.
Definition 3.1.
A function u ∈ C ([0 , T ]; H ) ∩ C ((0 , T ]; V ) is said to be a weaksolution to (2.20) - (2.21) on [0 , T ] iff u (0) = u and equation (2.20) holds in V − . th-wsol Theorem 3.1. If u is a mild solution to the problem (2.20) - (2.21) , then it is aweak solution.Proof. Let u be defined by (2.22). Then Lemma 2.3 ensures that S ( · ) u and R ∗ g belong to C ([0 , T ]; H ), so u = S ( · ) u + R ∗ g ∈ C ([0 , T ]; H ). By Remark 2.2, weget u (0) = u and u ∈ C ((0 , T ]; V ). EMILINEAR NONLOCAL DIFFERENTIAL EQUATIONS 9
By formulation, we have k ( τ )( u ( t − τ ) − u ) = ∞ X n =1 k ( τ )[ s ( t − τ, λ n ) − u n e n + ∞ X n =1 k ( τ )[ r ( · , λ n ) ∗ g n ]( t − τ ) e n for δ ≤ τ ≤ t ≤ T , where δ ∈ (0 , T ), and these series are uniformly convergent on[ δ, t ]. So one has Z tδ k ( τ )( u ( t − τ ) − u ) dτ = ∞ X n =1 Z tδ k ( τ )[ s ( t − τ, λ m ) − dτ u n e n + ∞ X n =1 Z tδ k ( τ )[ r ( · , λ n ) ∗ g n ]( t − τ ) dτ e n . (3.1) th-wsol1 For fixed t ∈ (0 , T ], put h n ( δ ) = Z tδ k ( τ )[ s ( t − τ, λ m ) − dτ u n + Z tδ k ( τ )[ r ( · , λ n ) ∗ g n ]( t − τ ) dτ. Obviously, h n is continuous on [0 , t ] for all n , and the function δ h ( δ ) = R tδ k ( τ )( u ( t − τ ) − u ) dτ is also continuous on [0 , t ]. Then the series P ∞ n =1 h n ( δ ) e n converges uniformly on [0 , t ], which enables us to pass to the limit in (3.1) to obtain k ∗ ( u − u )( t ) = ∞ X n =1 k ∗ ( s ( · , λ n ) − t ) u n e n + ∞ X n =1 k ∗ [ r ( · , λ n ) ∗ g n ]( t ) e n = ∞ X n =1 k ∗ ( s ( · , λ n ) − t ) u n e n + ∞ X n =1 [ s ( · , λ n ) ∗ g n ]( t ) e n , (3.2) th-wsol2 thanks to (2.4). We testify that, it is possible to take differentiation term by termin (3.2). It suffices to prove that the series ∞ X n =1 ddt [ k ∗ ( s ( · , λ n ) − t ) u n e n + ∞ X n =1 ddt [ s ( · , λ n ) ∗ g n ]( t ) e n (3.3) th-wsol3 is uniformly convergent on [ δ, T ] for any δ ∈ (0 , T ). Indeed, by Remark 2.1 we have ddt [ k ∗ ( s ( · , λ n ) − t ) = − λ n s ( t, λ n ) ,ddt [ s ( · , λ n ) ∗ g n ]( t ) = g n ( t ) − λ n [ r ( · , λ n ) ∗ g ]( t ) . Therefore, (3.3) becomes − ∞ X n =1 λ n s ( t, λ n ) u n e n − ∞ X n =1 λ n [ r ( · , λ n ) ∗ g ]( t ) e n + ∞ X n =1 g n ( t ) e n = − AS ( t ) u − A ( R ∗ g )( t ) + g ( t ) , which are uniformly convergent on [ δ, T ] as shown in Lemma 2.3. Hence, we cantake differentiation in (3.2) and get the equation ddt [ k ∗ ( u − u )]( t ) = − AS ( t ) u − A ( R ∗ g )( t ) + g ( t ) = − Au ( t ) + g ( t ) , t ∈ (0 , T ] , which holds in V − . The proof is complete. (cid:3) We are in a position to prove the uniqueness of weak solution. th-uniq
Theorem 3.2.
Problem (1.1) - (1.2) has a unique weak solution.Proof. It remains to show the uniqueness. Let h µ = − s ′ µ = µr , then h µ is nonneg-ative and solves the equation h µ ( t ) + µ ( h µ ∗ l )( t ) = µl ( t ) , t > , µ > . In addition, for 1 ≤ p < ∞ , f ∈ L p (0 , T ), one has h n ∗ f → f in L p (0 , T ) as n → ∞ ([17]). Put k µ = k ∗ h µ , then k µ = µk ∗ r = µs µ , thanks to (2.4). Hence k µ ∈ W , (0 , T ). This enable us to employ the fundamental identity ([14, Lemma2.3])( v ( t ) , ( k µ ∗ v ) ′ ( t )) = 12 ( k µ ∗ k v ( · ) k ) ′ ( t ) + 12 k µ ( t ) k v ( t ) k + 12 Z t k v ( t ) − v ( t − s ) k [ − k ′ µ ( s )] ds, t ∈ [0 , T ] , v ∈ C ([0 , T ]; H ) . Therefore( v ( t ) , ( k µ ∗ v ) ′ ( t )) ≥
12 ( k µ ∗ k v ( · ) k ) ′ ( t ) , t ∈ [0 , T ] , v ∈ C ([0 , T ]; H ) , (3.4) th-uniq1 thanks to the fact that k µ is nonincreasing.Let u and u be weak solutions of (1.1)-(1.2). Put v = u − u , then we have(( k ∗ v ) ′ ( t ) , w ) + ( Av ( t ) , w ) = 0 , ∀ t ∈ (0 , T ] , w ∈ V ,v (0) = 0 . Then (( h n ∗ k ∗ v )( t ) , w ) + ( h n ∗ ∗ Av ( t ) , w ) = 0 , ∀ t ∈ (0 , T ] , w ∈ V . which is equivalent to(( k n ∗ v ) ′ ( t ) , w ) + ( h n ∗ Av ( t ) , w ) = 0 , ∀ t ∈ (0 , T ] , w ∈ V . Taking w = v ( t ) and using (3.4) yields12 ( k n ∗ k v ( · ) k ) ′ ( t ) + ( h n ∗ Av ( t ) , v ( t )) ≤ , ∀ t ∈ (0 , T ] . Let g ( t ) = ( k n ∗ k v ( · ) k ) ′ ( t ) + ( h n ∗ Av ( t ) , v ( t )), then g ( t ) ≤ , ∀ t ∈ (0 , T ]. Notingthat, the relation12 ( k n ∗ k v ( · ) k ) ′ ( t ) = g n ( t ) := g ( t ) − ( h n ∗ Av ( t ) , v ( t ))is equivalent to (see [14, Lemma 2.4])12 k v ( t ) k = 1 n g n ( t ) + l ∗ g n ( t ) , t ∈ (0 , T ] , (3.5) th-uniq2 and the fact that g n ( t ) → g ( t ) − ( Av ( t ) , v ( t )) as n → ∞ , for t ∈ (0 , T ], we obtain12 k v ( t ) k = l ∗ [ g ( · ) − k A v ( · ) k ]( t ) ≤ , t ∈ (0 , T ] . Thus v = 0 and the proof is complete. (cid:3) EMILINEAR NONLOCAL DIFFERENTIAL EQUATIONS 11
Regularity.
By using ( K ), the problem (2.20)-(2.21) can be transformed tothe integral equation u ( t ) + l ∗ Au ( t ) = u + l ∗ g ( t ) , t ∈ [0 , T ] . This allows us to employ the resolvent theory in [13] for regularity analysis. Notingthat the solution operator for the equation u ( t ) + l ∗ Au ( t ) = u , t ∈ [0 , T ] , (3.6) eq-base is given by S ( t ) u = u ( t ), where S ( t ) is defined by (2.6). We refer to S ( · ) as theresolvent family.We recall some notions and fact stated in [13]. Definition 3.2.
Let l ∈ L loc ( R + ) be a function of subexponential growth, i.e. Z ∞ | l ( t ) | e − ǫt dt < ∞ for every ǫ > . • l is said to be of positive type if Re Z T ( l ∗ ϕ )( t ) ¯ ϕ ( t ) dt ≥ for each ϕ ∈ C ( R + ; C ) and T > . • For given m ∈ N , l is called m -regular if there exists a constant c > suchthat | λ n ˆ l ( n ) ( λ ) | ≤ c | ˆ l ( λ ) | for all Re λ > , ≤ n ≤ m, here ˆ l is the Laplace transform of l . It is easily seen that, if l is nonnegative and nonincreasing on (0 , ∞ ), then l is ofpositive type. Indeed, let ϕ ( t ) = p ( t ) + iq ( t ), thenRe Z T ( l ∗ ϕ )( t ) ¯ ϕ ( t ) dt = Z T dt Z t l ( t − s )[ p ( s ) p ( t ) + q ( s ) q ( t )] ds ≥ l ( T ) Z T [ P ( t ) p ( t ) + Q ( t ) q ( t )] dt = 12 l ( T )[ P ( T ) + Q ( T )] , where P = 1 ∗ p and Q = 1 ∗ q , the primitive of p and q , respectively. Definition 3.3.
Equation (3.6) is called parabolic if the following conditions hold:(1) ˆ l ( λ ) = 0 , / ˆ l ( λ ) ∈ ρ ( − A ) for all Re λ ≥ .(2) There is a constant M ≥ such that U ( λ ) = λ − ( I + ˆ l ( λ ) A ) − satisfies k U ( λ ) k ≤ M | λ | for all Re λ > . We have the following sufficient condition for (3.6) to be parabolic.
Proposition 3.3. [13, Corollary 3.1]
Assume that l ∈ L loc ( R + ) is of subexponential pp-parabolic growth and of positive type. If − A generates a bounded analytic semigroup in H ,then (3.6) is parabolic. Let us mention that, by the assumption ( A ), − A generates a contraction C -semigroup in H , which is given by e − tA v = ∞ X n =1 e − tλ n ( v, e n ) e n , v ∈ H. So the semigroup { e − tA } t ≥ is analytic due to [16, Corollary 7.1.1].The following result on the resolvent family for (3.6) plays an important role inour analysis. Proposition 3.4. [13, Theorem 3.1]
Assume that (3.6) is parabolic and the kernel pp-reg function l is m -regular for some m ≥ . Then there is a resolvent family S ( · ) ∈ C ( m − ((0 , ∞ ); L ( H )) for (3.6) , and a constant M ≥ such that k t n S ( n ) ( t ) k ≤ M, for all t > , n ≤ m − . In order to obtain the differentiability of the resolvent family, we replace ( K ) bya stronger assumption.( K *) The assumption ( K ) is satisfied with l being 2-regular, nonincreasing andof subexponential growth. rm-K Remark 3.1.
As mentioned in [3] , the assumption ( K ) does not guarantee that l is nonincreasing. However if k is positive, decreasing, log-convex ( ln k is a convexfunction), and k (0+) = ∞ , then l is nonincreasing. Employing Proposition 3.4, we have the following statement. lm-reg
Lemma 3.5.
Let ( A ) and ( K *) hold. Then the resolvent family S ( · ) defined by (2.6) is differentiable on (0 , ∞ ) , the relation S ′ ( t ) = − AR ( t ) , t ∈ (0 , ∞ ) , (3.7) lm-reg1 and the estimate k S ′ ( t ) k ≤ Mt , t ∈ (0 , ∞ ) , (3.8) lm-reg2 hold for some M ≥ .Proof. Since the kernel function l is nonnegative and nonincreasing, it is of positivetype. In addition, the assumption ( A ) ensures that − A generates a bounded analyticsemigroup. So (3.6) is parabolic, according to Proposition 3.3. Therefore, it followsfrom Proposition 3.4 that S ( · ) is differentiable on (0 , ∞ ) and estimate (3.8) takesplace. Finally, it is deduced from the formulation of S and R given by (2.6)-(2.7)that S ′ ( t ) v = ∞ X n =1 ∂ t s ( t, λ n )( v, e n ) e n = ∞ X n =1 − λ n r ( t, λ n )( v, e n ) e n = − AR ( t ) v, t > , v ∈ H, thanks to (2.4), which proves (3.7). (cid:3) Denote by C γ ([ a, b ]; H ), γ ∈ (0 , a, b ], that is, f ∈ C γ ([ a, b ]; H ) iff k f k C γ = sup t ,t ∈ [ a,b ] k f ( t ) − f ( t ) k| t − t | γ < ∞ . th-reg Theorem 3.6.
Let the hypotheses of Lemma 3.5 hold. Assume that the function g in (2.20) belongs to C γ ([0 , T ]; H ) , and u is the weak solution of (2.20) - (2.21) . Then u ∈ C ([0 , T ]; H ) ∩ C γ ([ δ, T ]; H ) for any < δ < T , and u is a classical solution. EMILINEAR NONLOCAL DIFFERENTIAL EQUATIONS 13
Proof.
Recall that the unique weak solution of (2.20)-(2.21) is given by u ( t ) = S ( t ) u + ( R ∗ g )( t ) = u ( t ) + u ( t ) , t ∈ [0 , T ] . (3.9) th-reg0 We first show that u is H¨older continuous on [ δ, T ]. Indeed, for t ∈ [ δ, T ) , h > k u ( t + h ) − u ( t ) k ≤ Z t k R ( τ ) kk g ( t + h − τ ) − g ( t − τ ) k dτ + Z t + ht k R ( τ ) kk g ( t + h − τ ) k dτ = I + I . Considering I , one gets I ≤ Z t r ( τ, λ ) k g k C γ h γ dτ = k g k C γ h γ λ − (1 − s ( t, λ )) ≤ k g k C γ λ − h γ . Concerning I , the relation S ′ ( t ) = − AR ( t ) for t > I ≤ k ( − A ) − k Z t + ht k S ′ ( τ ) kk g ( t + h − τ ) k dτ ≤ k A − kk g k ∞ M Z t + ht dττ = k A − kk g k ∞ M ln (cid:18) ht (cid:19) ≤ k A − kk g k ∞ M γ − (cid:18) ht (cid:19) γ ≤ k A − kk g k ∞ M γ − δ − γ h γ , here we utilize the inequalityln(1 + r ) ≤ r γ γ for r > , γ ∈ (0 , . So we have proved that k u ( t + h ) − u ( t ) k ≤ Ch γ with C = k g k C γ λ − + k A − kk g k ∞ M γ − δ − γ . It remains to show that u ∈ C γ ([ δ, T ]; H ). Let 0 < δ ≤ t < T and h >
0. Usingthe mean value formula S ( t + h ) v − S ( t ) v = h Z S ′ ( t + θh ) vdθ, v ∈ H, we have k u ( t + h ) − u ( t ) k = k S ( t + h ) u − S ( t ) u k≤ h Z k S ′ ( t + θh kk v k dθ ≤ M k v k h Z dθt + θh = M k v k ln (cid:18) ht (cid:19) ≤ M k v k γ − δ − γ h γ . Finally, we have to show that u is classical, that is, u given by (3.9) obeys thesystem (2.20)-(2.21) in H . In the proof of Theorem 3.1, we have testified that(2.20) holds in V − by reasoning that A ( R ∗ g )( t ) ∈ V − for t >
0. So it suffices to prove A ( R ∗ g )( t ) ∈ H for t > g is H¨older continuous.Indeed, using the relation S ′ ( t ) = − AR ( t ) for t > A ( R ∗ g )( t ) = Z t AR ( t − τ ) g ( τ ) dτ = − Z t S ′ ( t − τ ) g ( τ ) dτ = − Z t S ′ ( t − τ )[ g ( τ ) − g ( t )] dτ + [ I − S ( t )] g ( t ) . Then k A ( R ∗ g )( t ) k ≤ Z t k S ′ ( t − τ ) kk g ( τ ) − g ( t ) k dτ + k [ I − S ( t )] g ( t ) k≤ M k g k C γ Z t ( t − τ ) γ − dτ + k [ I − S ( t )] g ( t ) k≤ M k g k C γ γ − T γ + 2 k g k ∞ , for 0 < t ≤ T, which completes the proof. (cid:3) Stability and regularity for semilinear equations
Definition 4.1.
A function u ∈ C ([0 , T ]; H ) is called a mild solution of the problem (1.1) - (1.2) on [0 , T ] iff u ( t ) = S ( t ) u + Z t R ( t − s ) f ( u ( s )) ds, for every t ∈ [0 , T ] , where S ( · ) and R ( · ) are given by (2.6) - (2.7) . In the next theorem, we prove a local solvability result. th-locsol
Theorem 4.1.
Let ( A ), ( K ) and ( F ) be satisfied. Then there exists t ∗ > such thatthe problem (1.1) - (1.2) has a mild solution defined on [0 , t ∗ ] . Moreover, u ( t ) ∈ V for all t ∈ (0 , t ∗ ] .Proof. We make use of the contraction mapping principle. For given ζ ∈ (0 , T ], letΦ : C ([0 , ζ ]; H ) → C ([0 , ζ ]; H ) be the mapping defined byΦ( u )( t ) = S ( t ) u + Z t R ( t − τ ) f ( u ( τ )) dτ, t ∈ [0 , ζ ] . (4.1) sol-op Taking ρ > k u k and assuming that u ∈ B ρ , the closed ball in C ([0 , ζ ]; H ) withcenter at origin and radius ρ , we have k Φ( u )( t ) k ≤ k S ( t ) kk u k + Z t k R ( t − τ ) kk f ( u ( τ )) k dτ ≤ s ( t, λ ) k u k + Z t r ( t − τ, λ )[ κ ( ρ ) k u ( τ ) k + k f (0) k ] dτ ≤ k u k + [ κ ( ρ ) ρ + k f (0) k ] λ − (1 − s ( t, λ )) , t ∈ [0 , ζ ] , here we employ the hypothesis ( F ), Proposition 2.1 and Lemma 2.3. Since s ( · , λ ) ∈ AC ([0 , ζ ]) and s (0 , λ ) = 1, one can choose ζ such that the last expression is smallerthan ρ as long as t ∈ [0 , ζ ]. That is, Φ( B ρ ) ⊂ B ρ . EMILINEAR NONLOCAL DIFFERENTIAL EQUATIONS 15
Using ( F ) again, one gets k Φ( u )( t ) − Φ( u )( t ) k ≤ Z t r ( t − τ, λ ) k f ( u ( τ ) − f ( u ( τ )) k dτ ≤ Z t r ( t − τ, λ ) κ ( ρ ) k u ( τ ) − u ( τ ) k dτ ≤ κ ( ρ ) k u − u k ∞ λ − (1 − s ( t, λ )) , t ∈ [0 , ζ ] , where k · k ∞ is the sup norm in C ([0 , ζ ]; H ). Taking t ∗ ≤ ζ such that κ ( ρ )(1 − s ( t ∗ , λ )) < λ , we see that Φ is a contraction as a map from B ρ into itself, with B ρ now in C ([0 , t ∗ ]; H ). So the problem (1.1)-(1.2) has a solution defined on [0 , t ∗ ].In addition, since t g ( t ) = f ( u ( t )) is a continuous function, Φ( u )( t ) ∈ D ( A ) for t > u ( t ) ∈ V for t >
0. The proof is complete. (cid:3)
We now discuss some circumstances, in which solutions exist globally. th-glosol1
Theorem 4.2.
Let ( A ) and ( K ) hold. If the nonlinear function f is globally Lips-chitzian, that is, κ ( ρ ) in ( F ) is constant, then the problem (1.1) - (1.2) has a uniqueglobal mild solution u ∈ C ([0 , T ]; H ) ∩ C ((0 , T ]; V ) . If, in addition, that κ < λ and l L ( R + ) , then every mild solution to (1.1) is globally bounded and asymptoticallystable.Proof. Fixed
T >
0, let β > k u k β = sup t ∈ [0 ,T ] e − βt k u ( t ) k . Then k · k β isequivalent to the sup norm in C ([0 , T ]; H ). From the estimate k Φ( u )( t ) − Φ( u )( t ) k ≤ Z t r ( t − τ, λ ) κ k u ( τ ) − u ( τ ) k dτ, we get e − βt k Φ( u )( t ) − Φ( u )( t ) k ≤ (cid:18) κ Z t r ( t − τ, λ ) e − β ( t − τ ) dτ (cid:19) k u − u k β ≤ κ Z T r ( t, λ ) e − βt dt ! k u − u k β . Choosing β > κ Z T r ( t, λ ) e − βt dt < , we obtain Φ is a contraction map from C ([0 , T ]; H ) endowed with the norm k · k β into itself, which ensures the existence and uniqueness of solution to (1.1)-(1.2). Inaddition, we have u ( t ) ∈ V for t ∈ (0 , T ], by the same reasoning as in the proof ofTheorem 4.1.Now assume that κ < λ . Let u be a solution of (1.1)-(1.2), then we have k u ( t ) k ≤ s ( t, λ ) k u k + Z t r ( t − τ, λ )[ κ k u ( τ ) k + k f (0) k ] dτ, ∀ t ≥ . Using the Gronwall type inequality given in Proposition 2.2, we get k u ( t ) k ≤ s ( t, λ − κ ) k u k + 1 λ − κ k f (0) k (1 − s ( t, λ − κ )) ≤ k u k + 1 λ − κ k f (0) k , ∀ t ≥ , which yields the global boundedness of u .Let u and v be solutions of (1.1), then we have k u ( t ) − v ( t ) k ≤ s ( t, λ ) k u (0) − v (0) k + Z t r ( t − τ, λ ) κ k u ( τ ) − v ( τ ) k dτ, thanks to ( F ) and Lemma 2.8. Employing Proposition 2.2 again, we obtain k u ( t ) − v ( t ) k ≤ s ( t, λ − κ ) k u (0) − v (0) k , ∀ t ≥ . Since l L ( R + ), it follows from Proposition 2.1(1) that s ( t, λ − κ ) → t → ∞ ,which completes the proof. (cid:3) The following theorems show the main results of this section. th-glosol2
Theorem 4.3.
Let ( A ), ( K ) and ( F ) hold. If lim v → k f ( v ) kk v k = α with α ∈ [0 , λ ) ,then there exists δ > such that the problem (1.1) - (1.2) admits a unique globalmild solution u ∈ C ([0 , T ]; H ) ∩ C ((0 , T ]; V ) , provided that k u k ≤ δ .Proof. In order to prove the global existence, we make use of the Schauder fixedpoint theorem. By assumption on the behaviour of f , for θ ∈ (0 , λ − α ), thereexists η > k f ( v ) k ≤ ( α + θ ) k v k as long as k v k ≤ η . Now we considerthe solution map Φ : B η → C ([0 , T ]; H ) defined by (4.1). We see that k Φ( u )( t ) k ≤ s ( t, λ ) k u k + Z t r ( t − τ, λ )( α + θ ) k u ( τ ) k dτ ≤ s ( t, λ ) k u k + ( α + θ ) ηλ − (1 − s ( t, λ )) ≤ s ( t, λ )[ k u k − ( α + θ ) λ − η ] + ( α + θ ) λ − η ≤ η, t ∈ [0 , T ] , provided that k u k ≤ αλ − η , thanks to the fact that ( α + θ ) λ − <
1. Fixing an θ and η mentioned above, for δ = αλ − η , we have shown that Φ( B η ) ⊂ B η as k u k ≤ δ .In the next step, we construct a compact convex subset D ⊂ B η which isstill invariant under Φ by using the same routine as in [2]. Put M = B η and M k +1 = co Φ( M k ) , k ∈ N , where co denotes the closure of convex hull of subsets in C ([0 , T ]; H ). Obviously, M k is closed, convex, and M k +1 ⊂ M k for all k ∈ N . Let M = ∞ T k =1 M k , then M is also a closed convex set and Φ( M ) ⊂ M . We verify that M ( t ) is relatively compact for each t ≥
0. Indeed, let χ be the Hausdorff measureof noncompactness on H . Then it suffices to testify that lim k →∞ χ ( M k ( t )) = 0 foreach t ≥
0. For D ⊂ B η , we get χ ( f ( D ( t ))) ≤ κ ( η ) χ ( D ( t )) , ∀ t ≥ , EMILINEAR NONLOCAL DIFFERENTIAL EQUATIONS 17 thanks to the Lipschitz property of f (see [7]), here D ( t ) = { v ( t ) : v ∈ D } . There-fore, χ ( M k +1 ( t )) = χ (Φ( M k )( t )) ≤ χ (cid:18)Z t R ( t − τ ) f ( M k ( τ )) dτ (cid:19) ≤ Z t k R ( t − τ ) k χ ( f ( M k ( τ ))) dτ ≤ Z t r ( t − τ, λ ) κ ( η ) χ ( M k ( τ )) dτ. Let µ k ( t ) = χ ( M k ( t )) , t ≥
0, then µ k is nonincreasing and the last estimate reads µ k +1 ( t ) ≤ Z t r ( t − τ, λ ) κ ( η ) µ k ( τ ) dτ. Passing to the limit in the last relation, one gets µ ∞ ( t ) ≤ Z t r ( t − τ, λ ) κ ( η ) µ ∞ ( τ ) dτ, where µ ∞ ( t ) = lim k →∞ µ ( t ) , t ≥
0. Now applying Proposition 2.2 yields µ ∞ ( t ) = 0 forevery t ≥
0. We have proved that M ( t ) is relatively compact for each t ≥ D = co Φ( M ). Since Φ( M ) ⊂ M and M is closed and convex, we see that D ⊂ M , and thenΦ( D ) ⊂ Φ( M ) ⊂ coΦ( M ) = D . Moreover, since D ( t ) ⊂ M ( t ), we have D ( t ) relatively compact for every t ≥ D is a equicontinuous subset, which is equivalent to thatΦ( M ) has this property. Observe that, if Y ⊂ H is a relatively compact set, thenthe set R ( · ) Y = { R ( · ) y : y ∈ Y } is equicontinuous on (0 , T ]. The same reasoningensures the set R ( · − τ ) f ( M ( τ )) is equicontinuous on ( τ, T ]. This implies theequicontinuity of the setΦ( M ) = { v : v ( t ) ∈ S ( t ) u + Z t R ( t − τ ) f ( M ( τ )) dτ } . Thus D is compact due to the Arzela-Ascoli theorem. This enables us to utilizethe Schauder fixed point theorem for Φ : D → D to obtain a mild solution for(1.1)-(1.2).Finally, if u and u is two mild solutions for (1.1)-(1.2), v = u − u , and ρ = max {k u k ∞ , k u k ∞ } then k v ( t ) k ≤ Z t k R ( t − τ ) kk f ( u ( τ )) − f ( u ( τ )) k dτ ≤ Z t r ( t − τ, λ ) κ ( ρ ) k v ( τ ) k dτ, t ∈ [0 , T ] . Applying Proposition 2.2 yields v = 0. The proof is complete. (cid:3) th-stab Theorem 4.4.
Let the hypotheses of Theorem 4.3 hold. If l L ( R + ) , then thezero solution of (1.1) - (1.2) is asymptotically stable. Proof.
Taken θ and δ from the proof of Theorem 4.3, for k u k ≤ δ and a corre-sponding solution u of (1.1)-(1.2), we have k u ( t ) k ≤ s ( t, λ ) k u k + Z t r ( t − τ, λ )( α + θ ) k u ( τ ) k dτ. Using Proposition 2.2, we get k u ( t ) k ≤ s ( t, λ − α − θ ) k u k , ∀ t ≥ . Since l L ( R + ) and λ − α − θ >
0, we have s ( t, λ − α − θ ) → t → ∞ , andthe last inequality ensures the stability and attractivity of the zero solution. Theproof is complete. (cid:3) We now present a linearized stability result as a consequence of Theorem 4.4.
Corollary 4.5.
Let ( A ) and ( K ) hold. Assume that the nonlinearity f is contin-uously differentiable such that f (0) = 0 and A − f ′ (0) remains positively definite.Then the zero solution of (1.1) is asymptotically stable.Proof. Denote ˜ A = A − f ′ (0), ˜ f ( v ) = f ( v ) − f ′ (0) v . Then equation (1.1) is equiv-alent to ddt [ k ∗ ( u − u )]( t ) + ˜ Au ( t ) = ˜ f ( u ( t )) , t > . (4.2) ee By assumption, ˜ A fulfills ( A ). Furthermore, ˜ f is also continuously differentiable, soit is locally Lipschitzian and, therefore, ˜ f satisfies ( F ). Observing that k ˜ f ( v ) k = o ( k v k ) as k v k →
0, one can apply Theorem 4.4 for (4.2) (with α = 0) to get theconclusion. (cid:3) To end this section, we prove the H¨older continuity of the mild solution to (1.1)-(1.2). th-reg-sm
Theorem 4.6.
Let ( A ), ( K *) and ( F ) hold. Then the mild solution to (1.1) - (1.2) is H¨older continuous on [ δ, T ] for every < δ < T .Proof. Let u be the mild solution to (1.1)-(1.2). Then u ( t ) = S ( t ) u + Z t R ( t − τ ) f ( u ( τ )) dτ = u ( t ) + u ( t ) . By the same reasoning as in the proof of Theorem 3.6, we have u ∈ C γ ([ δ, T ]; H )for every 0 < δ < T and γ ∈ (0 , EMILINEAR NONLOCAL DIFFERENTIAL EQUATIONS 19
Regarding u , let ρ = k u k ∞ and 0 < δ ≤ t ≤ T , then we see that k u ( t + h ) − u ( t ) k ≤ Z t k R ( τ ) kk f ( u ( t + h − τ )) − f ( u ( t − τ )) k dτ + Z t + ht k R ( τ ) kk f ( u ( t + h − τ )) k dτ ≤ Z t r ( τ, λ ) κ ( ρ ) k u ( t + h − τ ) − u ( t − τ ) k dτ + k A − k Z t + ht k S ′ ( τ ) k ( k f (0) k + κ ( ρ ) ρ ) dτ ≤ Z t r ( t − τ, λ ) κ ( ρ ) k u ( τ + h ) − u ( τ ) k dτ + k A − k M ( k f (0) k + κ ( ρ ) ρ ) γ − δ − γ h γ , here we use ( F ) and the arguments as in the proof of Theorem 3.6 for estimatingthe second integral.Applying Proposition 2.2 for v ( t ) = k u ( t + h ) − u ( t ) k , one gets k u ( t + h ) − u ( t ) k ≤ k A − k M ( k f (0) k + κ ( ρ ) ρ ) γ − δ − γ s ( t, λ − κ ( ρ )) h γ , which implies u ∈ C γ ([ δ, T ]; H ). (cid:3) Application
Let Ω ⊂ R N be a bounded domain with smooth boundary ∂ Ω. We apply theobtained results to the following two-term fractional-in-time PDE: ∂ αt u ( t, x ) + µ ∂ βt u ( t, x ) + ( − Λ) γ u ( t, x ) = F (cid:18)Z Ω u ( t, x ) dx (cid:19) G ( x, u ( t, x )) , (5.1) ap1 for t > , x ∈ Ω ,u ( t, x ) = 0 , for t ≥ , x ∈ ∂ Ω , (5.2) ap2 u (0 , x ) = u ( x ) , for x ∈ Ω , (5.3) ap3 where 0 < α < β < µ ≥ , γ > ∂ αt and ∂ βt stand for the Caputo fractionalderivatives of order α and β in t , respectively. The operator Λ is defined by D (Λ) = H (Ω) ∩ H (Ω) , Λ u = N X i,j =1 ∂ x i ( a ij ( x ) ∂ x j u ) , where a ij ∈ L ∞ (Ω) , a ij = a ji , ≤ i, j ≤ N , subject to the condition N P i,j =1 a ij ( x ) ξ i ξ j ≥ θ | ξ | , for some θ >
0. Let H = L (Ω) with the inner product ( u, v ) = Z Ω u ( x ) v ( x ) dx .Put k ( t ) = g − α ( t ) + µ g − β ( t ) , (5.4) ap-k A = ( − Λ) γ ,f ( v )( x ) = F (cid:18)Z Ω v ( x ) dx (cid:19) G ( x, v ( x )) , v ∈ L (Ω) . Then the problem (5.1)-(5.3) is in the form of (1.1)-(1.2). Observe that, the kernelfunction k is completely monotonic, i.e. ( − n k ( n ) ( t ) ≥ t ∈ (0 , ∞ ). Asmentioned in [14], k admits a resolvent function l such that k ∗ l = 1 on (0 , ∞ ) andin this case, (1 ∗ l )( t ) ∼ g α ( t ) as t → ∞ . Thus s ( t, λ ) ≤
11 + λ (1 ∗ l )( t ) → t → ∞ . Noting that, the nonlinearity in (5.1) can be seen as a perturbation depending notonly on the state but also on the energy of the system. We assume that • F ∈ C ( R ) obeys the estimate | F ( r ) | ≤ a + b | r | ν , for some nonnegativenumbers a, b and ν . • G : Ω × R → R is a Carath´eodory function and satisfies the Lipschitzcondition in the second variable, i.e. | G ( x, y ) − G ( x, y ) | ≤ h ( x ) | y − y | , ∀ x ∈ Ω , y , y ∈ R , here h ∈ L ∞ (Ω) is a nonnegative function. In addition, assume that G ( x,
0) = 0 for a.e. x ∈ Ω.Then one can verify that f maps L (Ω) into itself. More precisely, we get k f ( v ) k = F (cid:18)Z Ω v ( x ) dx (cid:19) (cid:18)Z Ω | G ( x, v ( x )) | dx (cid:19) ≤ F (cid:18)Z Ω v ( x ) dx (cid:19) (cid:18)Z Ω h ( x ) v ( x ) dx (cid:19) . Hence k f ( v ) k ≤ ( a + b k v k ν ) k h k ∞ k v k , (5.5) ap4 where k h k ∞ = ess sup x ∈ Ω | h ( x ) | . In addition, we can check that f is locally Lips-chitzian due to the assumption that F ′ is continuous and G is Lipschitzian. Specif-ically, for v , v ∈ L (Ω) , k v k , k v k ≤ ρ , we see that k f ( v ) − f ( v ) k ≤ | F ( k v k ) − F ( k v k ) | (cid:18)Z Ω | G ( x, v ( x )) | dx (cid:19) + | F ( k v k ) | (cid:18)Z Ω | G ( x, v ( x )) − G ( x, v ( x )) | dx (cid:19) ≤ | F ′ (cid:0) θ k v k + (1 − θ ) k v k (cid:1) | · |k v k − k v k | · k h k ∞ k v k + ( a + b k v k ν ) k h k ∞ k v − v k≤ κ ( ρ ) k v − v k , where κ ( ρ ) = 2 ρ k h k ∞ sup r ∈ [0 ,ρ ] | F ′ ( r ) | + ( a + bρ ν ) k h k ∞ . Let λ △ be the first eigenvalue of − ∆, that is λ △ = inf {k∇ u k : k u k = 1 } . Denoteby µ the first eigenvalue of − Λ associated with the eigenfunction ϕ , then µ k ϕ k = ( − Λ ϕ, ϕ ) ≥ θ k∇ ϕ k ≥ θλ △ k ϕ k , thanks to the Poincar´e inequality. This implies that the first eigenvalue λ of A = ( − Λ) γ is given by λ = µ γ ≥ θ γ λ γ △ . EMILINEAR NONLOCAL DIFFERENTIAL EQUATIONS 21
On the other hand, it follows from (5.5) that lim v → k f ( v ) kk v k ≤ a k h k ∞ if ν > v → k f ( v ) kk v k ≤ ( a + b ) k h k ∞ as ν = 0. Employing Theorem 4.4, we have theconclusion that, the zero solution of (5.1) is asymptotically stable in the followingcases:(1) ν > a k h k ∞ < θ γ λ γ △ ,(2) ν = 0 and ( a + b ) k h k ∞ < θ γ λ γ △ .Let us mention that, the mild solution for (5.1)-(5.5) is H¨older continuous on [ δ, T ]for every 0 < δ < T . Indeed, since k given by (5.4) is positive, decreasing andlog-convex on (0 , ∞ ), by Remark 3.1, the kernel function l is nonincreasing. More-over, the Laplace transform ˆ l is given by ˆ l ( λ ) = ( λ − α + µλ − β ) − . By a directcomputation, one has λ ˆ l ′ ( λ ) = − ( λ − α + µλ − β ) − [(1 − α ) λ − α + µ (1 − β ) λ − β ] λ ˆ l ′′ ( λ ) = 2( λ − α + µλ − β ) − [(1 − α ) λ − α + µ (1 − β ) λ − β ] + ( λ − α + µλ − β ) − [ α (1 − α ) λ − α + µβ (1 − β ) λ − β ] . Observing that, for every η , η ∈ (0 ,
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Tran Dinh KeDepartment of Mathematics, Hanoi National University of Education136 Xuan Thuy, Cau Giay, Hanoi, Vietnam
E-mail address : [email protected] Nguyen Nhu ThangDepartment of Mathematics, Hanoi National University of Education136 Xuan Thuy, Cau Giay, Hanoi, Vietnam
E-mail address : [email protected] Lam Tran Phuong ThuyDepartment of Mathematics, Electric Power University,235 Hoang Quoc Viet, Hanoi, Vietnam
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