Lifespan of solutions to a damped fourth-order wave equation with logarithmic nonlinearity
aa r X i v : . [ m a t h . A P ] J un Lifespan of solutions to a damped fourth-orderwave equation with logarithmic nonlinearity
Yuzhu Han † Qi Li
School of Mathematics, Jilin University, Changchun 130012, P.R. China
Abstract
This paper is devoted to the lifespan of solutions to a damped fourth-order waveequation with logarithmic nonlinearity u tt + ∆ u − ∆ u − ω ∆ u t + α ( t ) u t = | u | p − u ln | u | . Finite time blow-up criteria for solutions at both lower and high initial energy levels are es-tablished, and an upper bound for the blow-up time is given for each case. Moreover, byconstructing a new auxiliary functional and making full use of the strong damping term, alower bound for the blow-up time is also derived.
Keywords
Lifespan; Damped; Fourth-order wave equation; Logarithmic nonlinearity; Ini-tial energy.
AMS Mathematics Subject Classification 2010:
In this paper, we are concerned with the following initial boundary value problem for adamped fourth-order wave equation with logarithmic nonlinearity u tt + ∆ u − ∆ u − ω ∆ u t + α ( t ) u t = | u | p − u ln | u | , ( x, t ) ∈ Ω × (0 , T ) ,u ( x, t ) = ∆ u ( x, t ) = 0 , ( x, t ) ∈ ∂ Ω × (0 , T ) ,u ( x,
0) = u ( x ) , u t ( x,
0) = u ( x ) , x ∈ Ω , (1.1)where Ω ⊂ R n ( n ≥
1) is a bounded domain with smooth boundary ∂ Ω, T ∈ (0 , + ∞ ] is themaximal existence time of the solution u ( x, t ), ω > α ( t ) : [0 , ∞ ) → [0 , ∞ ) is a nonincreasingbounded differentiable function, and the exponent p satisfies( A ) 2 < p < ∗ , where 2 ∗ = + ∞ if n ≤ ∗ = 2 nn − n ≥ † Corresponding author.Email addresses: [email protected](Y. Han). ∗ Supported by NSFC (11401252) and by The Education Department of Jilin Province (JJKH20190018KJ). u t and weak u t ) prevent solutions from blowing up while the nonlinear termsforce solutions to blow up. So it is of great interest to investigate how one dominates the other,and much effort has been devoted to this direction during the past few years. For example,Gazzola et al. [9] investigated the following damped wave equation u tt − ∆ u − ω ∆ u t + µu t = | u | p − u (1.2)in a bounded domain of R n , where ω ≥ µ > − ωλ and p >
2. By using the potentialwell method first proposed by Sattinger et al. [23, 24], they obtained the existence of globaland finite time blow-up solutions to (1.2) for initial data at different energy levels. As forthe damped fourth-order wave equations, Lin et al. [20] considered the following hyperbolicequation with strong damping u tt + ∆ u − ∆ u − ω ∆ u t = f ( u ) (1.3)in a bounded domain of R n with ω >
0. Under certain conditions on the initial data and on thenonlinearity f , they proved the existence of global weak solutions and global strong solutionsby using the classical potential well method. When the nonlinearity f ( u ) grows super-linearlywith respect to u as u tends to infinity, the solutions to (1.3) may blow up in finite time. In2018, Wu [25] considered the following initial boundary value problem u tt + ∆ u − ∆ u − ω ∆ u t + α ( t ) u t = | u | p − u, ( x, t ) ∈ Ω × (0 , T ) ,u ( x, t ) = ∆ u ( x, t ) = 0 , ( x, t ) ∈ ∂ Ω × (0 , T ) ,u ( x,
0) = u ( x ) , u t ( x,
0) = u ( x ) , x ∈ Ω , (1.4)where ω and α ( t ) fulfill the same conditions as that of problem (1.1) and p satisfies the so-calledsubcritical condition, i.e., p ∈ (2 , ∞ ) if n ≤ p ∈ (cid:0) , n − n − (cid:1) if n ≥ . After showing that the unstable set is invariant under the flow of (1.4), he proved a blow-upresult for problem (1.4) with initial energy smaller than the depth of the potential well, byapplying concavity argument. Moreover, a lower bound for the blow-up time is derived. Later,problem (1.4) was reconsidered by Guo et al. [10] and the results of [25] were extended in twoaspects. The first is that they obtained a blow-up result for high initial energy, and the secondis that lower bound for the blow-up time is also derived for some supercritical p , with the helpof inverse H¨older’s inequality and interpolation inequality.On the other hand, evolution equations with logarithmic nonlinearity have also attractedmore and more attention in recent years, due to their wide applications to quantum field theoryand other applied sciences. Among the huge amount of interesting literature, we only referthe interested reader to [5, 6, 7, 8, 11, 12, 14, 15, 16, 19, 22], where qualitative properties ofsolutions to hyperbolic or parabolic equations with logarithmic nonlinearities were studied. Inparticular, Di et al. [8] considered the following initial boundary value problem for a semilinearwave equation with strong damping and logarithmic nonlinearity u tt − ∆ u − ∆ u t = | u | p − u, ( x, t ) ∈ Ω × (0 , T ) ,u ( x, t ) = 0 , ( x, t ) ∈ ∂ Ω × (0 , T ) ,u ( x,
0) = u ( x ) , u t ( x,
0) = u ( x ) , x ∈ Ω , (1.5)when Ω ⊂ R n ( n ≥
1) is a bounded domain with smooth boundary ∂ Ω, 2 < p < + ∞ if n = 1 , < p < nn − if n ≥
3. The existence of global or finite time blow-up solutions to problem21.5) with initial energy less than or equal to the depth of the potential well was investigated byusing the potential well method. Moreover, the decay rate of the energy functional was obtainedfor global solutions and upper and lower bounds for the blow-up time were also derived for blow-up solutions. However, the case that the initial energy is larger than the depth of the potentialwell was not considered in [8], and we do not know whether or not problem (1.5) admits finitetime blow-up solutions for this case. In addition, the lower bound for the blow-up time wasobtained when p is subcritical, i.e., p < n − n − . When p ∈ [ n − n − , nn − ) for n ≥
3, whether a lowerbound for the blow-up time can be obtained is still open.Motivated mainly by [8, 10, 25], we will consider problem (1.1) and investigate how thedamping terms and logarithmic nonlinearity determine the blow-up conditions and blow-uptime of the solutions. More precisely, we shall present some sufficient conditions for the solutionsto problem (1.1) to blow up in finite time with both lower and high initial energy and derive anupper bound for the blow-up time for each case. Moreover, we also estimate a lower bound forthe blow-up time, which, thanks to the strong damping term, also includes some supercriticalcase. For simplicity, we only consider (1.1) for the case ω = 1 and α ( t ) ≡
1. The main resultscan be extended to the general case with little difficulty.The organization of this paper is as follows. In Section 2, as preliminaries, some notations,definitions and lemmas that will be used in the sequel are introduced. Finite time blow-upof solutions and upper bound for the blow-up time with lower and high initial energy will beconsidered in Section 3 and Section 4, respectively. In Section 5 we derive a lower bound forthe blow-up time.
In this section, we introduce some notations and lemmas which will be used in the sequel.In what follows, we denote by k · k r the L r (Ω)-norm (1 ≤ r ≤ ∞ ), by ( · , · ) the L (Ω)-innerproduct and by λ > − ∆ in Ω under homogeneous Dirichlet boundarycondition. Set H = { u ∈ H (Ω) ∩ H (Ω) : u = ∆ u = 0 on ∂ Ω } , and equip it with the norm k u k H = q k ∆ u k + k∇ u k . For simplicity, we also denote the H (Ω)-norm by k u k = q k u k + k∇ u k . Obviously, for any u ∈ H , we have λ k u k ≤ k u k H . (2.1)For any u ∈ H , define J ( u ) = 12 k u k H − p Z Ω | u | p ln | u | d x + 1 p k u k pp , (2.2) I ( u ) = k u k H − Z Ω | u | p ln | u | d x, (2.3) N = { u ∈ H \ { } : I ( u ) = 0 } , (2.4) d = inf u ∈ H \{ } sup λ> J ( λu ) = inf u ∈N J ( u ) , (2.5)3here N is called the Nehari manifold and d is the depth of the potential well (also calledmountain pass level). In what follows, we shall show that N is non-empty and d is positive.The following lemma gives some properties of the so-called fibering map J ( λu ). Since theproof is more or less standard (see [8] for example), we omit it here. Lemma 2.1.
Let p satisfy (A). Then for any u ∈ H \ { } , we have (i) lim λ → + J ( λu ) = 0 , lim λ → + ∞ J ( λu ) = −∞ . (ii) there exists a unique λ ∗ = λ ∗ ( u ) > such that ddλ J ( λu ) | λ = λ ∗ = 0 . J ( λu ) is increasingon < λ < λ ∗ , decreasing on λ ∗ < λ < + ∞ and takes its maximum at λ = λ ∗ . (iii) I ( λu ) > on < λ < λ ∗ , I ( λu ) < on λ ∗ < λ < + ∞ and I ( λ ∗ u ) = 0 . Let σ be any positive number such that p + σ < ∗ . Then it is well known that the embeddingfrom H to L p + σ (Ω) is compact and there is a positive constant B σ such that k u k p + σ ≤ B σ k u k H , ∀ u ∈ H. (2.6) Lemma 2.2.
There is a positive constant C ∗ such that k u k H ≥ C ∗ for any u ∈ N .Proof. First, it follows from Lemma 2.1 (iii) that N is non-empty. For any u ∈ N , using (2.6)and the basic inequality ln s ≤ eσ s σ for s ≥ σ >
0, we have k u k H = Z Ω | u | p ln | u | d x = Z Ω | u | p ln | u | d x + Z Ω | u | p ln | u | d x ≤ Z Ω | u | p ln | u | d x ≤ eσ Z Ω | u | p + σ d x ≤ eσ k u k p + σp + σ ≤ B p + σσ eσ k u k p + σH , (2.7)where Ω = { x ∈ Ω : | u ( x ) | < } and Ω = { x ∈ Ω : | u ( x ) | ≥ } . Recalling that p >
2, weobtain from (2.7) that k u k H ≥ (cid:16) eσB p + σσ (cid:17) / ( p + σ − , C ∗ . The proof is complete. Lemma 2.3.
The depth d of the potential well is positive and there is a nonnegative function v ∈ N such that J ( v ) = d .Proof. By (2.2) and (2.3) we have J ( u ) = p − p k u k H + 1 p I ( u ) + 1 p k u k pp , u ∈ H. (2.8)Therefore, for any u ∈ N , by combining Lemma 2.4 with (2.8) we obtain J ( u ) ≥ p − p k u k H ≥ p − p C ∗ , d > . (2.9)By the definition of d one sees that d ≥ d , i.e., d is positive.To show that d can be attained, let { v k } ∞ k =1 ⊂ N be a minimizing sequence of J . It is easyto check that {| v k |} ∞ k =1 ⊂ N is also a minimizing sequence of J . Therefore, without loss ofgenerality, we may assume that v k ≥ k ∈ N . Then J ( v k ) is bounded, which,together with (2.8), implies that { v k } ∞ k =1 ⊂ N is bounded in H . Noticing that the embedding4rom H to L p + σ (Ω) is compact, we see that there is a subsequence of { v k } ∞ k =1 ⊂ N , which westill denote by { v k } ∞ k =1 ⊂ N , and a v ∈ H such that v k ⇀ v weakly in H as k → ∞ ,v k → v strongly in L p + σ (Ω) as k → ∞ ,v k → v a.e. in Ω as k → ∞ . Hence, v ≥ Z Ω | v | p ln | v | d x = lim k →∞ Z Ω | v k | p ln | v k | d x, (2.10) Z Ω | v | p d x = lim k →∞ Z Ω | v k | p d x. (2.11)Moreover, by the weak lower semicontinuity of k · k H , we have k v k H ≤ lim inf k →∞ k v k k H . (2.12)Therefore, it follows from (2.10)-(2.12) that J ( v ) ≤ lim inf k →∞ J ( v k ) = d, (2.13)and I ( v ) ≤ lim inf k →∞ I ( v k ) = 0 . (2.14)It remains to show that v I ( v ) = 0 to complete the proof. By (2.10) and Lemma4.3 we know Z Ω | v | p ln | v | d x = lim k →∞ Z Ω | v k | p ln | v k | d x = lim k →∞ k v k k H ≥ C ∗ , which implies that v I ( v ) <
0, then by Lemma 2.1 (iii) we know that there exists a λ ∗ ∈ (0 ,
1) such that I ( λ ∗ v ) = 0, i.e., λ ∗ v ∈ N . By the definition of d , we see d ≤ J ( λ ∗ v ) = ( p − λ ∗ p k v k H + λ ∗ p p k v k pp = λ ∗ h p − p k v k H + λ ∗ p − p k v k pp i <λ ∗ h p − p k v k H + 1 p k v k pp i ≤ λ ∗ lim inf k →∞ h p − p k v k k H + 1 p k v k k pp i = λ ∗ lim inf k →∞ J ( v k ) = λ ∗ d, a contradiction. Therefore, I ( v ) = 0 and v
0, which means that v ∈ N . Recalling (2.13)and the definition of d again one sees that J ( v ) = d . The proof is complete.5n this paper, we consider weak solutions to problem (1.1). For completeness, we state,without proof, the local existence theorem which can be established by slightly modifying theargument in [21]. Sometimes u ( x, t ) will be simply written as u ( t ) if no confusion arises. Theorem 2.1. ([10, 25]) Let u ∈ H and u ∈ L (Ω) . Then the problem (1.1) admits aunique weak solution u ∈ L ∞ (0 , T ; H ) , u t ∈ L ∞ (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)) , for T > suitably small. Moreover, the energy functional satisfies E ′ ( t ) = − Z Ω ( u t + |∇ u t | )d x ≤ , (2.15) where E ( t ) = E ( u ( t )) = 12 k u t k + J ( u ( t )) . (2.16)At the end of this section, we present the well-known concavity lemma which will playessential role in proving the blow-up result. Lemma 2.4. (See [13, 17]) Suppose that a positive, twice-differentiable function ψ ( t ) sat-isfies the inequality ψ ′′ ( t ) ψ ( t ) − (1 + θ )( ψ ′ ( t )) ≥ , where θ > . If ψ (0) > , ψ ′ (0) > , then ψ ( t ) → ∞ as t → t ∗ ≤ t ∗ = ψ (0) θψ ′ (0) . In this section, we will investigate the blow-up phenomena of solutions to problem (1.1)with lower initial energy. We first show that the unstable set U is invariant under the flow ofproblem (1.1), where U = { u ∈ H : I ( u ) < , J ( u ) < d } , (3.1)and d is the depth of the potential well defined in (2.5). Lemma 3.1.
Let u ∈ U and u ∈ L (Ω) such that E (0) < d . Then u ( t ) ∈ U for all t ∈ [0 , T ) and p − p k u ( t ) k H + 1 p k u ( t ) k pp > d, ∀ t ∈ [0 , T ) . (3.2) Proof.
First, it follows from (2.2), (2.16) and (2.15) that J ( u ( t )) ≤ E ( t ) ≤ E (0) < d, ∀ t ∈ [0 , T ) . Therefore, in order to prove u ( t ) ∈ U for all t ∈ [0 , T ), it suffices to show that I ( u ( t )) < t ∈ [0 , T ). Assume by contradiction that there exists a t ∈ (0 , T ) such that u ( t ) ∈ N . Thenby the variational definition of d , we obtain d ≤ J ( u ( t )) ≤ E ( t ) ≤ E (0) < d, a contradiction.For any t ∈ [0 , T ), since I ( u ( t )) <
0, it follows from Lemma 2.1 (iii) that there exists a λ ( t ) ∈ (0 ,
1) such that I ( λ ( t ) u ( t )) = 0, i.e., λ ( t ) u ( t ) ∈ N . By the definition of d and (2.8), wehave p − p k u ( t ) k H + 1 p k u ( t ) k pp ≥ ( p − λ ( t )2 p k u ( t ) k H + λ p ( t ) p k u ( t ) k pp = J ( λ ( t ) u ( t )) ≥ d. The proof is complete. 6ith the preliminaries given above, we can show the first blow-up results for problem (1.1)with lower initial energy.
Theorem 3.1.
Let p satisfy (A), u ∈ U and u ∈ L (Ω) such that E (0) < d . Then thesolution u ( x, t ) to problem (1.1) blows up at a finite time T in the sense that lim t → T − (cid:16) k u ( t ) k + Z t k u ( s ) k d s (cid:17) = ∞ . (3.3) Moreover, the blow-up time T can be estimated from above as follows T ≤ h(cid:0) a + ( p − b k u k (cid:1) / + a i ( p − b , (3.4) where a, b are constants that will be fixed in the proof.Proof. Assume by contradiction that the solution u exists globally. As was done in [25], fix T ∗ > G ( t ) = k u ( t ) k + Z t k u ( s ) k d s + ( T ∗ − t ) k u k + b ( t + τ ) , t ∈ [0 , T ∗ ] , (3.5)where T ∗ , b and τ are positive constants to be fixed later. Taking derivative we have G ′ ( t ) = 2( u, u t ) + 2 Z t Z Ω ( uu s + ∇ u · ∇ u s )d x d s + 2 b ( t + τ ) . (3.6)Taking derivative again and using (2.15), (2.16) and Lemma 3.1, we obtain G ′′ ( t ) =2 k u t k + 2( u, u tt ) + 2 Z Ω ( uu t + ∇ u · ∇ u t )d x + 2 b =( p + 2) k u t k + ( p − k u k H − pE ( t ) + 2 b + 2 p k u k pp ≥ ( p + 2) k u t k + 2 pd − pE ( t ) + 2 b =( p + 2) k u t k + 2 p ( d − E (0)) + 2 p Z t k u s ( s ) k d s + 2 b. (3.7)Choosing b = d − E (0) > p > G ′′ ( t ) ≥ ( p + 2)[ k u t k + Z t k u s ( s ) k d s + b ] > . (3.8)Combining (4.15), (3.6) with (3.8) we know, for any t ∈ [0 , T ∗ ], that G ( t ) G ′′ ( t ) − p + 24 ( G ′ ( t )) ≥ ( p + 2) h k u ( t ) k + Z t k u ( s ) k d s + b ( t + τ ) i · h k u t k + Z t k u s ( s ) k d s + b i − ( p + 2) h ( u, u t ) + Z t Z Ω ( uu s + ∇ u · ∇ u s )d x d s + b ( t + τ ) i . Since ( u, u t ) ≤ k u ( t ) k k u t k , (3.9)7 t Z Ω uu s d x d s ≤ (cid:16) Z t Z Ω u d x d s (cid:17) / (cid:16) Z t Z Ω u s d x d s (cid:17) / , (3.10) Z t Z Ω ∇ u · ∇ u s d x d s ≤ (cid:16) Z t Z Ω |∇ u | d x d s (cid:17) / (cid:16) Z t Z Ω |∇ u s | d x d s (cid:17) / , (3.11)it can be directly verified by using Cauchy-Schwarz inequality that G ( t ) G ′′ ( t ) − p + 24 ( G ′ ( t )) ≥ , t ∈ [0 , T ∗ ] . Take τ = max n , k u k − ( p − u , u )( p − b o , (3.12)then G (0) = k u k + T ∗ k u k + bτ > ,G ′ (0) = 2( u , u ) + 2 bτ > , and 4 G (0)( p − G ′ (0) = 2[ k u k + T ∗ k u k + bτ ]( p − u , u ) + bτ ] ≤ T ∗ , (3.13)for suitably large T ∗ . According to Lemma 2.4, there exists a T ∗ > T ∗ ≤ G (0)( p − G ′ (0) (3.14)such that G ( t ) → ∞ as t → T −∗ . This contradicts with the assumption that G ( t ) is well defined on the closed [0 , T ∗ ] for any T ∗ > T be the maximalexistence time of u ( x, t ) (which is finite by the above argument) and let G ( t ) be given in (4.15),with the exception that T ∗ is replaced by T and t ∈ [0 , T ], where T ∈ (0 , T ). Similarly to theforegoing arguments, one can show that T ≤ k u k + T k u k + bτ ]( p − u , u ) + bτ ] , where we require that τ , which is independent of T , still satisfies (3.12). It then follows fromthe arbitrariness of T < T that T ≤ k u k + T k u k + bτ ]( p − u , u ) + bτ ] , (3.15)which guarantees T ≤ T ( τ ) , k u k + bτ )( p − u , u ) + bτ ] − k u k . (3.16)Set a = 2 k u k − ( p − u , u ) and τ = [ a + ( p − b k u k ] / + a ( p − b . Then it is an easymatter to verify that τ satisfies (3.12), T ( τ ) attains its minimum at τ and T ( τ ) = 4 h(cid:0) a + ( p − b k u k (cid:1) / + a i ( p − b . T ≤ h(cid:0) a + ( p − b k u k (cid:1) / + a i ( p − b . The proof is complete.
Remark 3.1. By (2.16) and (2.8) and recalling that p > one sees E (0) < implies I ( u ) < . Therefore, Theorem 3.1 implies that the solution u ( x, t ) to problem (1.1) blows upin finite time for negative initial energy. In this section we shall build a blow-up criterion for problem (1.1) at high initial energylevel. Some ideas used in this section are borrowed from [10] and [18]. As a preliminary, wefirst establish a lemma that will play a fundamental role.
Lemma 4.1.
Let p satisfy ( A ) . Assume that u ∈ H and u ∈ L (Ω) such that < E (0) < C p ( u , u ) . (4.1) Then the solution u ( x, t ) to problem (1.1) satisfies ( u, u t ) − pC E ( t ) ≥ h ( u , u ) − pC E (0) i e C t , t ∈ [0 , T ) . (4.2) Here C = min n p + 2 , p ( p − λ , ( p − λ + λ )2 o > . (4.3) Proof.
Set F ( t ) = ( u, u t ). By direct calculations and recalling (2.16) we have F ′ ( t ) = k u t k + ( u, u tt )= k u t k + ( u, − ∆ u + ∆ u + ∆ u t − u t + | u | p − u ln | u | )= k u t k − k u k H − Z Ω ∇ u · ∇ u t d x − ( u, u t ) + Z Ω | u | p ln | u | d x = p + 22 k u t k + p − k u k H − Z Ω ∇ u · ∇ u t d x − ( u, u t ) − pE ( t ) + 1 p k u k pp ≥ p + 22 k u t k + p − k u k H − Z Ω ∇ u · ∇ u t d x − ( u, u t ) − pE ( t ) . (4.4)By using Cauchy inequality, we can estimate the third and fourth terms in the last inequalityas follows | Z Ω ∇ u · ∇ u t d x | ≤ C p k∇ u k + pC k∇ u t k , (4.5) | ( u, u t ) | ≤ C p k u k + pC k u t k . (4.6)Substituting (4.5) and (4.6) into (4.4) we arrive at F ′ ( t ) ≥ p + 22 k u t k + p − k u k H − C p k u k − pC k u t k − pE ( t ) . (4.7)9et H ( t ) = F ( t ) − pC E ( t ). Then in view of (2.1), (2.15), (4.3) and (4.7) we obtain H ′ ( t ) = F ′ ( t ) − pC E ′ ( t ) = F ′ ( t ) + pC k u t k ≥ p + 22 k u t k + p − k u k H − C p k u k − pE ( t ) ≥ p + 22 k u t k + h ( p − λ − p ( p − λ p i k u k − pE ( t )= p + 22 k u t k + ( p − λ k u k − pE ( t ) ≥ p + 22 k u t k + ( p − λ + λ )4 k u k − pE ( t ) ≥ C h k u t k + 12 k u k − pC E ( t ) i ≥ C H ( t ) . (4.8)Since H (0) > H ( t ).The proof is complete.With Lemma 4.1 at hand, we are now in the position to prove high initial energy blow-upand estimate an upper bound for the blow-up time for problem (1.1). Theorem 4.1.
Let all the assumptions in Lemma 4.1 hold. Then the solution u ( x, t ) toproblem (1.1) blows up at some finite time T in the sense of (3.3) . Moreover, if E (0) < C p k u k , (4.9) then T ≤ k u k + βt )( p − u , u ) + βt ] − k u k . (4.10) Here C is the positive constant given in (4.3) , β = 2 h C p k u k − E (0) i > , (4.11) t is suitably large such that ( p − h ( u , u ) + βt i > k u k . (4.12) Proof.
We divide the proof into two steps.
Step I: Finite time blow-up.
Suppose by contradiction that (3.3) will not happen forany finite T . Then k u ( · , t ) k is well-defined for all t ≥
0. Without loss of generality, we mayassume that E ( t ) ≥ t ≥
0. Otherwise by Remark 3.1 we know that u ( x, t ) blows up infinite time.On one hand, it follows from Lemma 4.1 that ddt k u ( t ) k = 2( u, u t ) ≥ H (0) e C t + 2 pC E ( t ) ≥ H (0) e C t . (4.13)Integration of (4.13) over [0 , t ] yields k u ( t ) k = k u k + 2 Z t Z Ω uu τ d x d τ ≥ k u k + 2 Z t H (0) e C τ d τ = k u k + 2 H (0) C ( e C t − . (4.14)10n the other hand, by virtue of Minkowski inequality, H¨older inequality, (2.15), the defini-tion of λ and the fact E ( t ) ≥ k u ( t ) k ≤k u k + k u ( t ) − u k = k u k + k Z t u τ d τ k ≤k u k + Z t k u τ k d τ ≤ k u k + 1 √ λ Z t k u τ k d τ ≤k u k + √ t √ λ (cid:16) Z t k u τ k d τ (cid:17) / = k u k + √ t √ λ ( E (0) − E ( t )) / ≤k u k + s E (0)1 + λ t / , which contradicts (4.14) when t is sufficiently large. Therefore, u ( x, t ) blows up in finite time. Step II: Upper bound for the blow-up time.
From now on, we assume that
T > u ( x, t ), which is finite by Step I. According to Lemma 4.1 and theassumption that E ( t ) ≥
0, we see that ddt k u ( t ) k = 2( u, u t ) ≥ H (0) e C t + 2 pC E ( t ) > , t ∈ [0 , T ) , which implies k u ( t ) k is increasing with respect to t . To estimate T from above, as was donein the proof of Theorem 3.1, we define K ( t ) = k u ( t ) k + Z t k u ( s ) k d s + ( T − t ) k u k + β ( t + t ) , t ∈ [0 , T ) , (4.15)where β and t are given in (4.11) and (4.12), respectively. By applying similar argument tothat in the proof of the first part of Theorem 3.1, we obtain K ( t ) K ′′ ( t ) − p + 24 ( K ′ ( t )) =2 K ( t ) h k u t k − k u k H + Z Ω | u | p ln | u | d x + β i − ( p + 2) h ( u, u t ) + Z t Z Ω ( uu s + ∇ u · ∇ u s )d x d s + β ( t + t ) i =2 K ( t ) h k u t k − k u k H + Z Ω | u | p ln | u | d x + β i + ( p − h η ( t ) − (cid:0) K ( t ) − ( T − t ) k u k (cid:1)(cid:0) k u t k + Z t k u s k d s + β (cid:1)i , (4.16)where η ( t ) = h k u ( t ) k + Z t k u ( s ) k d s + β ( t + t ) ih k u t k + Z t k u s k d s + β i − h ( u, u t ) + Z t Z Ω ( uu s + ∇ u · ∇ u s )d x d s + β ( t + t ) i . (4.17)Using (3.9)-(3.11) and Cauchy-Schwarz inequality we can show that η ( t ) ≥ , T ). There-11ore, recalling (2.15), (2.16), (4.11) and the monotonicity of k u ( t ) k , we have K ( t ) K ′′ ( t ) − p + 24 ( K ′ ( t )) ≥ K ( t ) h k u t k − k u k H + Z Ω | u | p ln | u | d x + β i − ( p + 2) K ( t ) h k u t k + Z t k u s k d s + β i = K ( t ) h ( p − k u k H − pE (0) + ( p − Z t k u s k d s + 2 p k u k pp − pβ i ≥ K ( t ) h ( p − λ + λ ) k u k − pE (0) − pβ i ≥ K ( t ) h ( p − λ + λ ) k u k − pE (0) − pβ i =2 pK ( t ) h C p k u k − E (0) − β/ i ≥ , t ∈ [0 , T ) . (4.18)Besides, K (0) = k u k + T k u k + βt > K ′ (0) = 2( u , u )+2 βt > K ( t ) yields T ≤ K (0)( p − K ′ (0) = 2( k u k + T k u k + βt )( p − u , u ) + βt ] . Since 2 k u k ( p − u , u ) + βt ] < T ≤ k u k + βt )( p − u , u ) + βt ] − k u k . The proof of Theorem 4.1 is complete.
Remark 4.1.
As was done in deriving (3.4) , one can also minimize the right-hand sideterm of (4.10) for t satisfying (4.12) to obtain a more accurate upper bound for T . Interestedreader may check it. Since the lower bound for the blow-up time provides a safe time interval for the systemunder consideration, it is more important in practice to estimate T from below. In this section,our aim is to determine a lower bound for the blow-up time of problem (1.1) by constructing anew auxiliary functional. Throughout this section we shall use C, C , C , · · · , to denote genericpositive constants which may depend on Ω , p, n , but are independent of the solution u ( x, t ). Theorem 5.1.
Assume that p satisfies nn + 2 < n ( p − n + 2 < ∗ , i.e., p ∈ (2 , ∞ ) if n ≤ p ∈ (cid:0) , n − n − (cid:1) if n ≥ . (5.1) Let u ( x, t ) be a weak solution to problem (1.1) that blows up at T in the sense of (3.3) . Then T ≥ Z ∞ N (0) d sC + C s p − µ , where N (0) = k u k + k u k H . roof. For simplicity, we only prove this theorem for n ≥
3. The case for n = 1 , , T ) on which the quantity k u ( t ) k H isbounded. Clearly T is a lower bound for T since both k u ( t ) k and k u ( t ) k can be bounded by k u ( t ) k H .Define N ( t ) = k u t ( t ) k + k u ( t ) k H , t ∈ [0 , T ) . (5.2)Then lim t → T − N ( t ) = + ∞ . (5.3)Differentiating (5.2) and making use of Green’s second identity, we obtain N ′ ( t ) = 2[( u t , u tt ) + (∆ u, ∆ u t ) + ( ∇ u, ∇ u t )]= 2( u t , u tt + ∆ u − ∆ u )= 2( u t , ∆ u t − u t + | u | p − ln | u | )= − k u t k + 2 Z Ω u t | u | p − u ln | u | d x. (5.4)Set Ω = Ω ( t ) = { x ∈ Ω : | u ( x, t ) | < } and Ω = Ω ( t ) = { x ∈ Ω : | u ( x, t ) | ≥ } . Since p satisfies (5.1), we can choose µ > n ( p − µ ) n +2 < ∗ , which impliesthat H can be embedded into L n ( p − µ ) n +2 (Ω) continuously. We use B µ to denote the embeddingconstant from H to L n ( p − µ ) n +2 (Ω), i.e., k v k n ( p − µ ) n +2 ≤ B µ k v k H , ∀ v ∈ H. (5.5)Using H¨older’s inequality, Cauchy inequality, (5.5) and the basic inequalities | s p − ln s | ≤ ( e ( p − − for 0 < s < s ≤ eµ s µ for s ≥
1, we can estimate the second term on the right-hand side of (5.4) as follows Z Ω u t | u | p − u ln | u | d x = Z Ω u t | u | p − u ln | u | d x + Z Ω u t | u | p − u ln | u | d x ≤ (cid:16) Z Ω | u t | nn − d x (cid:17) n − n (cid:16) Z Ω (cid:12)(cid:12) | u | p − u ln | u | (cid:12)(cid:12) nn +2 d x (cid:17) n +22 n + (cid:16) Z Ω | u t | nn − d x (cid:17) n − n (cid:16) Z Ω (cid:12)(cid:12) | u | p − u ln | u | (cid:12)(cid:12) nn +2 d x (cid:17) n +22 n ≤k u t k nn − h ( e ( p − − | Ω | n +22 n + ( eµ ) − (cid:16) Z Ω | u | n ( p − µ ) n +2 d x (cid:17) n +22 n i ≤ C k u t k h ( e ( p − − | Ω | n +22 n + ( eµ ) − B p − µµ k u k p − µH i ≤ ε k u t k + C ( ε ) h C + C k u k p − µ ) H i ≤ ε k u t k + C ( ε ) h C + C N p − µ ( t ) i . (5.6)Therefore, it follows by taking ε ≤ N ′ ( t ) ≤ C + C N p − µ ( t ) . (5.7)13ntegrating (5.7) over [0 , t ], we have Z t N ′ ( τ ) C + C N p − µ ( τ ) d τ ≤ t. (5.8)Letting t → T − and recalling (5.3), we obtain Z ∞ N (0) d sC + C s p − µ ≤ T ≤ T. (5.9)Recalling that p − µ >
1, the left-hand side term in (5.9) is finite. The proof is complete.
Remark 5.1.
By making full use of the damping term, we obtain the lower bound for theblow-up time not only for subcritical exponent p , but also for some supercritical ones. We pointout that this observation can also be applied to problem (1.5) considered in [8]. Acknowledgement
The authors would like to express their sincere gratitude to Professor Wenjie Gao for his en-thusiastic guidance and constant encouragement.
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