Research Article On Existence, Uniform Decay Rates, and Blow-Up for Solutions of a Nonlinear Wave Equation with Dissipative and Source

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1 Abstract and Applied Analysis Volume, Article ID 65345, 7 pages doi:.55//65345 Research Article On Existence, Uniform Decay Rates, and Blow-Up for Solutions of a Nonlinear Wave Equation with Dissipative and Source Xiaopan Liu Department of Mathematics, Southeast University, Nanjing 8, China Correspondence should be addressed to Xiaopan Liu, liuxiaopan@6.com Received 4 May ; Revised 7 July ; Accepted 7 July Academic Editor: Narcisa C. Apreutesei Copyright q Xiaopan Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper studies the blow-up and existence, and asymptotic behaviors of the solution of a nonlinear hyperbolic equation with dissipative and source terms. By using Galerkin procedure and the perturbed energy method, the local and global existence of solution is established. In addition, by the concave method, the blow-up of solutions can be obtained.. Introduction In this paper, we investigate the following nonlinear wave equation: u t ρ u tt Δ u u t m u t γδ u t u u, n N σ i u xi u p u, x i x, t,t, u x, u x,u t x, u x, x, x, t,t,. where is a bounded domain in R n with smooth boundary, Δ is a Laplace operator, and u/ n indicates derivative of u in outward normal direction of. In addition to, if n>3, <p< n / n and if n,, then p>. γ is a constant, σ i s i,...,n are given in A later.

2 Abstract and Applied Analysis In 968, Greenberg et al. first suggested and studied the following equation: u tt u xxt σ u x x.. Under the condition σ s > and higher smooth conditions on σ s and initial data, they claimed the global existence of classical solutions for the initial boundary value problem of.. The multidimensional form of the following: u tt Δu t N a i x, t, u xi f x, t x i.3 was first studied by Clement, 3. Exploiting the monotone operator method, he obtained the global existence of weak solutions for the initial boundary value problem of.. Our model comes from 4. In 4, Yang has studied the global existence, asymptotic behavior, and blow-up of solutions for a nonlinear wave equations with dissipative term: u tt Δ u λu t N σ i u xi, x i.4 with the same initial and boundary conditions as that of.. In our model, we add damping and source terms which enhance the difficulty of proving the existence and decay of solution of.. More related studies of the damped hyperbolic equation with dissipative term or damping term can be found in papers 5 5. The paper is organized as follows. In Section, we present some notations, and results needed later and main results. Section 3 contains the statement and the proofs of the decay of solutions. Section 4 gives the statement and the proofs of the blow-up of solutions.. Preliminaries We first introduce the following abbreviations: Q T,T, L p L p, W m,p W m,p,. H m W m,, H m m, W, L,, H. Let, denote the L -inner product. We denote the dual of W,p by W,p,withp p/ p, and,p W,p. Now we make the following assumptions: A σ i C R,σ i s s >and C s r σ i s C s r,.

3 Abstract and Applied Analysis 3 for some r, if n,, else if n 3, then r n/ n, sothat u r B Δu, i,,...,n, where B is the optimal embedding constant. A The initial data u H, u H..3 A3 m<ρ ; If n>3, then <ρ<m<4/ n and if n,, then <ρ<m. A4 If n>3, <p< n / n and if n,, then p>. Without loss of generality, we assume that r<p. And B σ i C R, σ i s s <, and C s r σ i s C s r,.4 for some r, if n,, else if n 3, then r n/ n, sothat u r B Δu, i,,...,n. where B is the optimal embedding constant, and C < r / p. B If n>3, then m<r <p < 4/ n and if n,, then <m<r <p. B3 ρ <r,andifn>3, then ρ<4/ n and if n,, then <ρ. Throughout this paper, we use the embedding H Lq which implies u q B Δu when q n n if n 3, q if n,,.5 where B is an optimal embedding constant. We introduce the following functionals: A i s s σ i η dη, E t ρ u t N Δu J t Δu C r u r A i u xi dx u p p p, r u p p p, I t Δu C u r r u p p Because r<p, we have J t I t r r Δu p r u p p r p..

4 4 Abstract and Applied Analysis Theorem.. Assume that A A4 hold, then problem. has a unique solution u satisfying u L,T ; H W,,T ; L ; u t L,T ; H L.,T ; L, where T<. Theorem.. Assume that A A4 hold, u is the local solution of the problem..and r r / C E r p / E I >, <,. where C max{b C,B },thenu is a global bounded solution, moreover, E t M t / m t m ρ / m t /, t t,,.3 lim t Δu lim t t,.4 where M> is a constant. Theorem.3. Assume that A A4 and pb r p / E p / <.5 p hold, u is the local solution of the problem.7. Consider. are satisfied, then E t Ke κt, t,,.6 where K, κ > are constants. Remark.4. When γ >, we will use perturbed energy method, which is different to the method of the proof of Theorem., to prove Theorem.3. Theorem.5. Assume that B B3, A4 hold, and there exist some u,u H L, then the solution of the problem. blows-up at the limited time T >.

5 Abstract and Applied Analysis 5 3. Decay of Solutions In this section, we prove Theorems..3. First, we give the following Lemma. Lemma 3. see. Let be any bounded domain in R N, {ω k } k L. Then for any ε>, there exists a positive number N ε such that be an orthogonal basis in u Nε / u, ω k ε u,p, 3. k for all u W,p p<. Proof of Theorem.. We look for approximate solutions u n t of problem. of the form n u n t : T jn t ω j, j 3. where {ω j } n j is an orthogonal basis in H,andalsoinL, and the coefficients {T jn } n j satisfy T jn t u n t,ω j with u n t t ρ u n tt t,ω j Δ u n t,ω j u n m t u n t t,ω j γ Δ u n t t,ω j i σ i u n x xi t,ω j u n t p u n t,ω j, t >, j,...,n, i 3.3 u n u n, un t un. Since C is dense in H and L, we choose u n,un C such that u n u in H, un u in H as n. 3.4 The above system of o.d.e. has a local solution u n t defined in some interval,t n.the following will prove that the T n can be substituted by some T>. Multiply 3.3 by T jn t and summing up about j,weget [ d dt ρ u n t ] Δun u n m t m γ Δu n t N σ n i u x i, u n xi t u n p u n,u n t. 3.5

6 6 Abstract and Applied Analysis A simple integration of 3.5 over,t lea to u n t ρ t Δun N t C 3 σ n i u x i, u n xi u n s m m γ un s s t u n p u n,u n s, 3.6 where C 3 / ρ u n / Δun >. We now estimate the last two terms at the right-hand side of 3.6. UsingHölder inequality, Young s inequality, and the embedding theorem, we know there exist q, w >, satisfying that q w n, 3.7 n where q n n, n w n n p q n, n w n n n p n n n n, n n. 3.8 Consider the following: N t σ i u n xi, u n xi s C t u n r u n u n s dx t C u n r r q un w u n s n/ n t C C Δu n r Δu n Δu n s t C C Δu n r Δu n s t C C ɛ C C t ɛ Δu n r ɛ t r/p Δu n p t p r /p ɛ ɛ t Δu n s Δu n s

7 Abstract and Applied Analysis 7 C C p t Δu n p ɛ r C C p r ɛ p ɛ t C C Δu n s, ɛ >, 3.9 assuming that t<. Similarly, t u n p u n,u n s t u n p p q un w u n s n/ n t C Δu n p Δu n Δu n s C ɛ t Δu n p ɛ t C Δu n s, ɛ >. 3. Using , we have t t u n s m m γ Δu n s Δun u n ρ t t C 4 C ε Δu n p ɛ t C Δu n s. 3. Choosing ɛ /C γ in 3., we have t u n s m m Δun ρ t C 4 C Δu n p, γ u n t t Δu n s 3. where C 4 C 3 C C /ɛ p r /p. Assuming Y n t C 4 C t Δun p, we have Y n t CY p n. 3.3 A simple integration of 3.3 over,t lea to Y n t [ Y p n C p t] / p ; 3.4

8 8 Abstract and Applied Analysis this implies that Δu n ρ [ Y p n C p / p. t] 3.5 u n t Though Y n t may blow up, there exists <T<min{,T n } satisfying Δu n u n t C, t,t, 3.6 where C is independent of n. Moreover, t t u n s m m Δu n s C. 3.7 By 3.7, t u n s s m u n s s t m / m m / m u n s s m m C, t,t. 3.8 By 3.6, u n s p u n s p /p p /p un s p p Δun s p C. 3.9 From , we have u n L,T ; H ; 3. u n t L,T ; H L,T ; L. So the solution u n t of problem 3.3 exists on,t for each n. On the other hand, we can extract a subsequence from u n, still denoted by u n, such that u n u weak in L,T ; H, 3. u n t u t weak in L,T ; L L,T ; H, 3.

9 Abstract and Applied Analysis 9 as n.by 3., the Sobolev embedding theorem and the continuity of σ i s, fort,t, u n t u t strongly in L, a.e. on, σ i u n xi t σ i u xi t, a.e. on, i,...n, 3.3 as n. By Lemma 3., , for any ε>, there exist positive constant N ε and N ε independent of u n and u n t, respectively, such that, as n, u n t u t [ Nε ] / u n u, ω k ε u n t u t, Mε, k T u n t s u t s Nε k T u n t s u t s,ω k 3.4 ε T u n t s u t s, Mε. By the arbitrariness of ε we get u n u strongly in L,T ; L, a.e. on Q T, u n t u t strongly in L Q T, a.e. on Q T. 3.5 From the continuity of u t m u t and 3.5 we know that u n t m u n t u t m u t a.e. on Q T.With the same metho used above we easily get u n p u n u p u a.e. on Q T. Integrating 3.3 over,t, t<tgets u n ρ t t ρ,ω j t i t x i σ i u n xi s,ω j Δu n t s, Δω j u n s m u n t s s,ω j γ Δu n s s, Δω j t u n s p u n s,ω j, t >, j,...,n. 3.6 Exploiting , we have t Δu n s, Δω j C t Δu n CT, u n t t m u n t t,ω j u n m t ωj / m u n t ωj / m C, / m

10 Abstract and Applied Analysis t u n s s m u n t s s,ω j u n s m m ωj m CT, m t u n s p u n s,ω j CT. 3.7 Let n in 3.6 and we deduce from 3.3, 3.7 and the Lebesgue-dominated convergence theorem that u t t ρ,ω j ρ i t t t Δu s, Δωj us t m t u s s,ω j γ Δu s s, Δω j σ i u xi s,ω j x i t u s p u s,ω j, t >, j,...,n. 3.8 This implies u L,T ; H is a local weak solution of problem.. The proof of Theorem. is completed. Secondly, we prove Theorem.. First we give two lemmas. It is easy to prove what follows. Lemma 3.. The modified energy functional satisfies, along solutions of., E t u t m m γ Δu t. 3.9 Lemma 3.3. Assume that A A3 hold, satisfying r r / C E I >. r p / E <, 3.3 Then I t >. Proof. Since I >, then there exists by continuity T T such that I t >, for all t,t, this gives J t I t r r Δu p r u p p r p r Δu p r u p p r p. 3.3

11 Abstract and Applied Analysis By using.8,.9, 3.3, and Lemma 3., we easily have Δu r J t r r E t E, t,t. 3.3 We then exploit.,and 3.3 to obtain C u r r B C Δu r B C Δu r r Δu B C r / E Δu, u p p B Δu p r B p / E Δu, t,t Using 3.33, we have C u r r u p r p B C r / E Δu B r r / C E r p / E Δu r p / E Δu < Δu, t,t, 3.34 where C max{b,b C }. Therefore, I t Δu C u r r u p p >, 3.35 for all t,t. By repeating this procedure, and using the fact that lim t T C r r / E t r lim C t T r / E r p / E t r p / E <, 3.36 the proof is completed. Lemma 3.4. Assume that A A4 hold,. satisfy. Then the solution is global existence. More, exist positive constant M> has t t Δu u t p p M; u s m m M; Δu s M, t,. 3.37

12 Abstract and Applied Analysis Proof. It suffices to show that Δu u t 3.38 is bounded independently of t. To achieve this, we use.9, 3.3, and Lemma 3. to get t t E E t u s m m γ Δu s J t t t ρ u t u s m m γ Δu s r Δu p r u p p r p 3.39 t t ρ u t u s m m γ Δu s r Δu t t ρ u t u s m m γ Δu s. Since I t,j t are positive. Therefore, Δu u t CE. 3.4 Moreover, t t u s m m E ; γ Δu s E, 3.4 where C is a positive constant, which depen only on r. Lemma 3.5. Assume that A A4 hold,. satisfy. Then exist C> has t u s Ct m ρ / m ; t t Δu C I s

13 Abstract and Applied Analysis 3 Proof. Using Lemma 3.4, we have t t / m t u s u s m m ρ / m t / m C u s m m t m ρ / m 3.44 Ct m ρ / m. Using 3.34, we have r C u r r u p p BC E : η Δu. r / r p / E Δu 3.45 So η Δu I t ; 3.46 the proof is complete. Lemma 3.6. Assume that A A4 hold,. satisfy. Then there exists a C>, having t I s C t / t m ρ / m t / m Proof. By multiplying the differential equation in. by u and integrating over, using integration by parts.9 and assumption A, weobtain d dt ut ρ u t ρ,u u t ρ Δu u t m u t,u γ Δ u t,u N σ i u xi,u xi u p p d dt ut ρ u t ρ,u u t ρ Δu u t m u t,u 3.48 d dt γ Δu t, Δu C u r r u p p ut ρ u t ρ,u u t ρ I t u t m u t,u γ Δu t, Δu.

14 4 Abstract and Applied Analysis So t t I s u t ρ u u u u s ρ t us m u s,u t γ Δu s, Δu Using A3, , we have t u s m u s udx u t ρ u C u t ρ / Δu C, t u s m m u m C t u s m m Δu t t m / m C u s m m C u s m t / m Ct / m, γ t Δu s, Δu γ t Δu s Δu C t /t Δu s / Ct /. 3.5 Therefore, t I s M t / m t m ρ / m t /. 3.5 Proof of Theorem.. First, t t E t is also absolutely continuous, and we have d t E t E t. 3.5 dt A simple integration of 3.5 over,t lea to t E t E t E s E t u s ρ t N t A i u xi dx p Δu t u p p

15 Abstract and Applied Analysis 5 E t u s ρ t Δu N t A i u xi dx t u p p p E ρ C t r t u s u r r p t t Δu u p p Using the upper inequality, 3.45, 3.46,and 3.5, we have t E t E t u s ρ t t Δu C Δu E t t u s ρ C I s Apply 3.4, 3.47, we can get.3. Using 3.3 and Lemma 3.3, wegetj t,i t >..3 implies lim t E t. So when t, we have u t andj t. It is that.4 is satisfied. Theorem. is complete. Following we will prove Theorem.3. For this purpose we set L t : E t εψ t, 3.55 where ε is a positive constant and Ψ t u t ρ u t udx ρ Lemma 3.7. Let ε be small enough. Then there exist two positive constants α and α such that α L t E t α L t Proof. By Lemma 3.4 and Young s inequality, a direct computation gives L t E t E t ε u t dx ε u dx ρ ρ r E ε ρ u t εb ρ ρ/ Δu

16 6 Abstract and Applied Analysis E t εe t εb r / E ρ/ E t ρ α E t Similarly, we have L t E t ε u t dx ε u dx ρ ρ E t ε ρ u t εb ρ r E ρ/ Δu E t εe t εb r / E ρ/ E t ρ 3.59 α E t, provided that ε is small enough. Lemma 3.8. Assume that the conditions of Theorem.3 hold, then the function Ψ t : u t ρ u t udx 3.6 ρ satisfies, along the solution of., Ψ t 4 E t μ u t m m ω Δu. 3.6 Proof. Applying equations of.,wesee Ψ t ρ u t ρ u t u t ρ u tt udx N Δ u u t m u t σ i u xi γδ u t u p u udx x i ρ u t Δu u t m u t,u N ρ u t Δu N A i u xi dx σ i u xi,u xi γ Δu t, Δu u p p N A i u xi σ i u xi,u xi dx

17 Abstract and Applied Analysis 7 u t m u t,u γ Δu t, Δu u p p ρ 3 u t ρ ρ E t u t m u t,u γ Δu t, Δu ρ 3 ρ ρ u t p N p u p p A i u xi σ i u xi u xi dx. 3.6 Exploiting the assumption A and Young s inequality, we have p p u p N A i u xi σ i u xi u xi dx. p pbp p Δu p pbp p r p / E p / E t u t m u t,u m δ u t m δm m / m m m u t m m m δ u t m m / m m δb m m m δ u t m m / m m r E m/ Δu δb m m r m / E m/ E t, δ > η >. γ Δu t, Δu γη Δu t γη Δu, 3.63 Exploiting 3.63 and 3.6,weget Ψ t [ pbp r p / E p / p δb m m r m / E m/ γη ] r E t m δ u t m m / m m γη ρ 3 ρ E ρ/ B ρ ρ Δu. 3.64

18 8 Abstract and Applied Analysis Choosing δ satisfies δb m m r E m/ 4 [ pbp p r p / ] E p / 3.65 and η satisfies r γη 4 [ pbp p r p / ] E p /, 3.66 at the above; the proof of 3.6 is completed. Proof of Theorem.3. Using 3.6 and Lemma 3.7, we have L t E t εψ t u t m m γ Δu t ε 4 E t εμ u t m m εω Δu t ε 4 E t γ εω Δu t εμ u t m m 3.67 εα 4 L t γ εω Δu t εμ u t m m. Choosing ε satisfies ε min{γ/ω,/μ}. So we have L t εα L t, t A simple integration of 3.68 over,t lea to L t L e εα /4 t, t Exploiting Lemma 3.7 again, we have E t α L e εα /4 t : Ke κt, t, 3.7 where K, κ > are constants. The proof of Theorem.3 is complete. 4. Blow-Up of Solutions Proof of Theorem.5. Assuming that the solution of. is global, we have E t u t m m γ Δu t. 4.

19 Abstract and Applied Analysis 9 So Δu C 3; 4. we set Q t : t E s ot ω u dx, 4.3 where o, ω are constants and will be given later. Consider Q t E t o u dx o u dx E. 4.4 Choosing o satisfies the following condition in 4.4 : o u dx E Q >, 4.5 so we have Q t Q >, t,t. 4.6 Moreover, we have Q t Q E E t t E s t u s m m γ Δu s. 4.7 Define K t : Q γ t ε ρ t u s ρ u s udx, 4.8 where ε> will be given later, and { r ρ <γ min, ρ r p m p m, } r m m r. 4.9

20 Abstract and Applied Analysis Multiplying. by u and a direct computation yield K t γ Q γ Q t γ Q γ Q t t ε u ρ u u dx ε ρ ρ ε ρ u ρ u u dx ε ρ ε u s ρ u ss udx γ Q γ Q t ε u ρ u u dx ε ρ ρ t t ε Δ uu dx ε u s m u s udx εγ N ε t x i σ i u xi udx ε t t t t t u p uu dx γ Q γ Q t ε u ρ u u dx ε ρ ρ t t ε Δu dx εγ Δu s Δudx ε N t t ε σ i u xi u xi dx ε u p p. t t us ρ u s u dx s u s dx u s dx Δ u s udx u s dx u s m u s udx 4. Exploiting 4.7 and B, we have γ t Δu s Δudx γ t Δu η s η γ t γ η Q t Q η γc 3 T, Δu 4. t u s m u s udx ςm m ςm m t t u m dx m m ς m / m t u s m m dx u m dx m m ς m / m Q t Q 4. t t σ i u xi u xi dx C 4 u r r. 4.3

21 Abstract and Applied Analysis Using , we have K t γ Q γ Q t ε u ρ u u dx ε ρ ρ t u s dx t t t εc 4 u r r ε Δu ε u p p εη γtc 3 t Q t ε ςm m ε m ς m / m γ η m ε m ς m / m γ η Q. m u m dx 4.4 Choosing ς, η satisfies ς m / m M Q γ t, η M Q γ t. 4.5 Then we have [ ] K m t γ ε m M εγm Q γ Q t ε t m ρ m M εγm ε M m m Qγ m u s dx ε t t ε u p p εc 4 ε u ρ u u dx ρ Q γ Q u m dx εm Qγ γtc 3 ε t u r r. t Δu 4.6 A simple computing implies [ Q γ t ] γ E s ot ω u dx t u p p p C u r r ot ω u [ t γ C 5 ot ω γ u γ u p p γ ] u r. γ 4.7

22 Abstract and Applied Analysis Using 4.7, B, embedding theorem, and Hölder inequality, we can get α t [ t Q γ m [ γ C 5 ot ω γ u γ t γ m C 6 u m dx α t u p p u p p γ m u r ot ω γ m u γ m, m / p u p dx p m / p γ ]] m u r t m / p α p m / p T p m / p u p p t m / p, : C 8 u p p where C 8 α p m / p T p m / p. Consider α t u m dx α B t u m t m / r α B u r t r m / r t m / r. C 7 u r r 4. Using , B, weget t Q γ m u m dx t γ m C 6 u p p u r γ m ot ω γ m u γ m t m / p t m / r C 8 u p p C 7 u r r

23 Abstract and Applied Analysis 3 t γ m m / p γ m C 6 C 8 u p p u r γ m C 6 ot ω γ m u γ m C 8 t C 7 t u p p u p p u r γ m m / p γ m m / r u r ot ω γ m u γ m C 7 t u p p u r γ m m / r, 4. Using 4.9, 4.7, and Young inequality, we have t Q γ m u m dx C 9 t u p p u r. 4. Similarly, we have TC 3 Q γ C t u p p u r. 4.3 Additionally, we choose ε satisfying ε γ m / m M γm. 4.4 Using 4.6, , weget K t ε u ρ u u dx ε ρ ρ t ε t [ m M Δu ε u p p ε min{,c 4 } t m C 9 C [ m M u r ε t u r r u s dx ] γ M m C 9 C u p p ξq t γ M ]

24 4 Abstract and Applied Analysis N ξ t uxi σ i τ dτ dx t ξ ξ t t u p p p ξ t u s ρ Δu ξ ot ω u, uxi p t u p p σ i τ dτ dx p t C r t t u r r u r r u p p p u p p. 4.5 Using 4.5,weget ξ K t ξq t ρ ε ρ ε min{,c 4 } p t u s ξ t ε ε ξ ε M m m C 9 C [ ε u ρ u u dx ξ t ρ ε ot ω u γ M t Δu u p p M m dx m C 9 C u r γ M ], 4.6 choosing ε <ξ< p ε, u,u satisfying min{,c 4 } ξ p ε >, u ρ u u dx ξ 4.7 t ρ ε ot ω u dx >. Then we choose M,M big enough, satisfying min{,c 4 } ξ p ε M m m C γ 9 C > M u ρ u u dx ξ t ρ ε ot ω u M m dx m C γ 9 C >. M 4.8

25 Abstract and Applied Analysis 5 Using the two above inequalities, we have [ K t Cε Q t t t t u s Δu u r r ] u p p ν, 4.9 where ν>. So K t >K γ ω u dx >. 4.3 Using B B3, A4,andHölder and Young inequalities, we have ρ t u s ρ u s udx ρ C ρ t u s ρ u r u s ρ μ u θ r, t 4.3 where /μ /θ. So we have ρ t u s ρ u s udx / γ t / γ t / γ C u s ρ μ u θ r t μ ρ / γ C u s ρ μ / ρ μ C t θ/ r γ u θ r /θ r t μ ρ / γ C u s t θ/ r γ. C u r r 4.3 Using 4.9, 4.3 and choosing μ satisfy μ ρ / γ, then θ/ r γ <.

26 6 Abstract and Applied Analysis So ρ t u s ρ u s udx / γ t t C u s u r r β t t C u s u r r t t Δu u p p β, 4.33 where β> is a constant. Finally, we can easily get K / γ / γ Q t ε / γ ρ t / γ u s ρ u s udx Combining 4.9 and , we have K t CK / γ t, t T, 4.35 for some constant C>. Integrating the above inequality in,t, weget K / γ K γ/ γ ct /γ, t T The above inequality implies K t blows-up on some time T. Since u exists globally, so we have ε ρ And we know that K t, as t T,so t u s ρ u s udx< Q γ t ; 4.38 this implies Q t,thatistosay t E s. Because t E s te t, 4.39 we know E t, as t T. 4.4

27 Abstract and Applied Analysis 7 This contradicts with the assumption that u is a global solution. So the solution of. blowsup on time T. Acknowledgment This work was supported by the National Natural Science Foundation of China 773. References J. M. Greenberg, R. C. MacCamy, and V. J. Mizel, On the existence,uniqueness and stability of solutions of the equation σ u xx λu xxt ρ u tt, Journal of Mathematics and Mechanics, vol. 7, pp , 968. J. Clements, Existence theorems for a quasilinear evolution equation, SIAM Journal on Applied Mathematics, vol. 6, pp , J. C. Clements, On the existence and uniqueness of solutions of the equation u tt / xi σ i u xi Δ N u t f, Canadian Mathematical Bulletin, vol. 8, no., pp. 8 87, Z. Yang, Global existence, asymptotic behavior and blow-up of solutions for a class of nonlinear wave equations with dissipative term, Journal of Differential Equations, vol. 87, no., pp. 5 54, 3. 5 Z. Yang and G. Chen, Global existence of solutions for quasi-linear wave equations with viscous damping, Journal of Mathematical Analysis and Applications, vol. 85, no., pp , 3. 6 F. Sun and M. Wang, Non-existence of global solutions for nonlinear strongly damped hyperbolic systems, Discrete and Continuous Dynamical Systems Series A, vol., no. 5, pp , 5. 7 F. Sun and M. Wang, Global and blow-up solutions for a system of nonlinear hyperbolic equations with dissipative terms, Nonlinear Analysis. Theory, Metho & Applications, vol. 64, no. 4, pp , 6. 8 S. Berrimi and S. A. Messaoudi, Existence and decay of solutions of a viscoelastic equation with a nonlinear source, Nonlinear Analysis. Theory, Metho & Applications, vol. 64, no., pp , 6. 9 M. M. Cavalcanti, V. N. Domingos Cavalcanti, and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electronic Journal of Differential Equations, vol. 44, pp. 4,. X. Han and M. Wang, General decay of energy for a viscoelastic equation with nonlinear damping, Mathematical Metho in the Applied Sciences, vol. 3, no. 3, pp , 9. G. Li, Y. Sun, and W. Liu, Global existence and blow-up of solutions for a strongly damped Petrovsky system with nonlinear damping, Applicable Analysis, vol. 9, no. 3, pp ,. W. Liu, Global existence, asymptotic behavior and blow-up of solutions for a viscoelastic equation with strong damping and nonlinear source, Topological Metho in Nonlinear Analysis, vol. 36, no., pp ,. 3 W. Liu, Global existence and uniform decay of solutions for a system of wave equations with dispersive and dissipative terms, Frontiers of Mathematics in China, vol. 5, no. 3, pp ,. 4 C. Song and Z. Yang, Existence and nonexistence of global solutions to the Cauchy problem for a nonlinear beam equation, Mathematical Metho in the Applied Sciences, vol. 33, no. 5, pp ,. 5 R. Xu, Y. Liu, and T. Yu, Global existence of solution for Cauchy problem of multidimensional generalized double dispersion equations, Nonlinear Analysis. Theory, Metho & Applications, vol. 7, no., pp , 9.

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