GLOBAL EXISTENCE AND ENERGY DECAY OF SOLUTIONS TO THE CAUCHY PROBLEM FOR A WAVE EQUATION WITH A WEAKLY NONLINEAR DISSIPATION

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1 GLOBAL EXITENCE AND ENERGY DECAY OF OLUTION TO THE CAUCHY PROBLEM FOR A WAVE EQUATION WITH A WEAKLY NONLINEAR DIIPATION ABBÈ BENAIA AND OUFIANE MOKEDDEM Received 1 November 3 We prove the global existence and study decay properties of the solutions to the wave equation with a weak nonlinear dissipative term by constructing a stable set in H 1 ( ). 1. Introduction We consider the Cauchy problem for the nonlinear wave equation with a weak nonlinear dissipation and source terms of the type u x u + λ (x)u + σ(t)g(u ) = u p 1 u in [,+ [, u(x,)= u (x), u (x,)= u 1 (x) in, (1.1) where g : R R is a continuous nondecreasing function and λ and σ are positive functions. When we have a bounded domain instead of, and for the case g(x) = δx (δ >) (without the term λ (x)u), Ikehata and uzuki [8] investigated the dynamics, they have shown that for sufficiently small initial data (u,u 1 ), the trajectory (u(t),u (t)) tends to (,) in H 1 (Ω) L (Ω)ast +.Wheng(x) = δ x m 1 x (m 1, λ, σ 1), Georgiev and Todorova [4] introduced a new method and determined suitable relations between m and p, for which there is global existence or alternatively finite-time blow up. Precisely they showed that the solutions continue to exist globally in time if m p and blow up in finite time if m<pand the initial energy is sufficiently negative. This result was later generalized to an abstract setting by Levine and errin [1] and Levine et al. [11]. In these papers, the authors showed that no solution with negative initial energy can be extended on [, [, if the source term dominates over the damping term (p>m). This generalization allowed them also to apply their result to quasilinear wave equations (see [1, 17]). Quite recently, Ikehata [7] proved that a global solution exists with no relation between p and m by the use of a stable set method due to attinger [18]. For the Cauchy problem (1.1) withλ 1andσ 1, when g(x) = δ x m 1 x (m 1) Todorova [1] (see [16]) proved that the energy decay rate is E(t) (1 + t) ( n(m 1))/(m 1) Copyright 4 Hindawi Publishing Corporation Abstract and Applied Analysis 4:11 (4) Mathematics ubject Classification: 35B4, 35L7 URL:

2 936 Cauchy problem for nonlinear wave equation for t. he used a general method introduced by Nakao [14] on condition that the data have compact support. Unfortunately, this method does not seem to be applicable in the case of more general functions λ and σ. Our purpose in this paper is to give a global solvability in the class H 1 and energy decay estimates of the solutions to the Cauchy problem (1.1) for a weak linear perturbation and a weak nonlinear dissipation. We use a new method recently introduced by Martinez [13] (see also []) to study the decay rate of solutions to the wave equation u x u + g(u ) = inω R +,whereω is a bounded domain of. This method is based on a new integral inequality that generalizes a result of Haraux [6]. o we proceed with the argument combining the method in [13] with the concept of modified stable set on H 1 ( ). Here the modified stable set is the extended version of attinger s stable set.. Preliminaries and main results λ(x), σ(t), and g satisfy the following hypotheses. (i) λ(x) is a locally bounded measurable function defined on and satisfies λ(x) d ( x ), (.1) where d is a decreasing function such that lim y d(y) =. (ii) σ : R + R + is a nonincreasing function of class C 1 on R +. Consider g : R R a nondecreasing C function and suppose that there exist C i >, i = 1,,3,4, such that c 3 v m g(v) c 4 v 1/m if v 1, (.) c 1 v g(v) c v r v 1, (.3) where m 1and1 r (n +)/(n ) +. We first state two well-known lemmas, and then we state and prove two other lemmas that will be needed later. Lemma.1. Let q be a number with q<+ (n = 1,) or q n/(n )(n 3). Then there is a constant c = c(q) such that u q c u H1 ( ) for u H 1( ). (.4) Lemma. (Gagliardo-Nirenberg). Let 1 r<q + and p. Then, the inequality u p C m x u θ u 1 θ r for u ( ( ) m/ )L r (.5) holds with some constant C>and ( 1 θ = r 1 )( m p n + 1 r 1 ) 1 (.6) provided that <θ 1 (assuming that <θ<1 if m n/ is a nonnegative integer).

3 A. Benaissa and. Mokeddem 937 Lemma.3 [1]. Let E : R + R + be a nonincreasing function and assume that there are two constants p 1 and A> such that then + E (p+1)/ (t)dt AE(), <+, (.7) E(t) ce()(1 + t) /(p 1) t, if p>1, E(t) ce()e ωt (.8) t, if p = 1, where c and ω are positive constants independent of the initial energy E(). Lemma.4 [13]. Let E : R + R + be a nonincreasing function and φ : R + R + an increasing C function such that φ() =, φ(t) + as t +. (.9) Assume that there exist p 1 and A> such that then + E(t) (p+1)/ (t)φ (t)dt AE(), <+, (.1) E(t) ce() ( 1+φ(t) ) /(p 1) t, if p>1, (.11) E(t) ce()e ωφ(t) t, if p = 1, where c and ω are positive constants independent of the initial energy E(). Proof of Lemma.4. Let f : R + R + be defined by f (x):= E(φ 1 (x)), (we remark that φ 1 has a sense by the hypotheses assumed on φ). f is nonincreasing, f () = E(), and if we set x := φ(t), we obtain φ(t) φ() φ(t) f (x) (p+1)/ dx = E ( φ 1 (x) ) (p+1)/ dx = φ() AE() = Af ( φ() ), <T<+. E(t) (p+1)/ φ (t)dt (.1) etting s := φ() and letting T +,wededucethat + s Thanks to Lemma.3, we deduce the desired results. f (x) (p+1)/ dx Af(s), s<+. (.13) Before stating the global existence theorem and decay property of problem (1.1), we will introduce the notion of the modified stable set. Let K(u) = x u + u u p+1 if λ 1, I(u) = x u u p+1 if λ const, (.14)

4 938 Cauchy problem for nonlinear wave equation for u H 1 ( ). Then we define the modified stable set and by { u H 1( ) \K(u) > } {} if λ 1, { u H 1( ) \I(u) > } {} if λ const. (.15) Next, let J(u) ande(t) be thepotentialandenergyassociatedwithproblem (1.1), respectively: J(u) = 1 x u + 1 λ(x)u 1 p +1 u p+1 for u H 1( ), E(t) = 1 u + J(u). We get the local existence solution. (.16) Theorem.5. Let 1 <p (n +)/(n ) (1 <p< if n = 1,) and assume that (u,u 1 ) H 1 ( ) L ( ) and u belong to the modified stable set.thenthereexistst> such that the Cauchy problem (1.1)hasauniquesolutionu(t) on [,T) in the class satisfying u(t,x) C ( [,T);H 1( )) C 1( [,T);L ( )), (.17) and this solution can be continued in time as long as u(t). u(t), (.18) When λ const, we use the following theorem of local existence in the space H H 1, andthedecaypropertyoftheenergye(t) is necessarily required for the local solution to remain in as t ; this fact of course guarantees the global existence in H H 1 and by approximation, we obtain global existence in H 1 L 1. Theorem.6 [15]. Let (u,u 1 ) H H 1.upposethat 1 p n (1 if N 4). (.19) n 4 Then under the hypotheses (.1), (.), and (.3), problem (1.1) admits a unique local solution u(t) on some interval [,T[, T T(u,u 1 ) >,intheclassw, ([,T[;L ) W 1, ([, T[;H 1 ) L ([,T[;H ), satisfying the finite propagation property. Proof of Theorem.5 (see [15, 18]). ince the argument is standard, we only sketch the main idea of the proof. Let (u,u 1 ) H 1 L and u. Then we have a unique local solution u(t)forsomet>. Indeed, taking suitable approximate functions f j such that (see []) f j (u) = f (u) if u j, f j (u) f (u), f j (u) cj u, (.) problem (1.1) with f (u) u p 1 u replaced by f j (u) admits a unique solution u j (t) C([,T);H 1 ( )) C 1 ([,T);L ( )). Further, we can prove that u j (t),<t<t,

5 A. Benaissa and. Mokeddem 939 for sufficiently large j, and there exists a subsequence of {u j (t)} which converges to a function ũ(t) in certain senses. ũ(t) is, in fact, a weak solution in C([,T);H 1 ( )) C 1 ([,T);L ( )) (see [19, ]) and such a solution is unique by Ginibre and Velo [5] and Brenner [3]. We can also construct such a solution which meets moreover the finite propagation property, if we assume that the initial data u (x) andu 1 (x) areofcompact support: suppu suppu 1 { x, x <L }, forsomel>. (.1) Applying [9, Appendix 1] of John, then the solution is also of compact support: supp u j (t) {x, x <L+ t}.o,wehavesuppũ(t) {x, x <L+ t}. We denote the life span of the solution u(t,x) of the Cauchy problem (1.1) byt max. First we consider the case λ(x) const (λ(x) 1 without loss of generality). And construct a stable set in H 1 ( ). etting C K { } (p +1) (p 1)/, (.) (p 1) σ(τ)dτ = + if m = 1, (.3) (1 + τ) n(m 1)/ σ(τ)dτ = + if m>1. (.4) Theorem.7. Let u(t,x) be a local solution of problem (1.1)on[,T max ) with initial data u, u 1 L ( ) with sufficiently small initial energy E() so that C E() (p 1)/ < 1. (.5) Then T max =. Furthermore, the global solution of the Cauchy problem (1.1) hasthefollowing energy decay property. Under (.), (.3), and (.3), ( t ) E(t) E()exp 1 ω σ(τ)dτ t>. (.6) Under (.), (.3), and (.4), ( C ( E() ) ) /(m 1) E(t) t t>. (.7) (1 + τ) n(m 1)/ σ(τ)dτ econdly, we consider the case λ(x) const and we assume that (1) If σ(t) = ( d(t)), where d(t) = d(l + t). n +4 n p n n. (.8)

6 94 Cauchy problem for nonlinear wave equation If m = 1, we suppose that σ(τ)dτ = + (.9) with ( ( ) t (p 1)/ (4 (n )(p 1))/ d(t) exp 1 ω σ(τ)dτ) <, ( d(t) ) 1 exp ( 1 ω t σ(τ)dτ ) <. (.3) If m>1, we suppose that (1 + τ) n(m 1)/ σ(τ)dτ = (.31) with ( d(t) ) (4 (n )(p 1))/ ( t (1 + τ) n(m 1)/ σ(τ)dτ ) (p 1)/(m 1) <, ( ) 1 (.3) d(t) ( t (1 + τ) n(m 1)/ σ(τ)dτ ) 1/(m 1) <. () If d(t) = (σ(t)). If m = 1, we suppose that for some α<1, d (τ) σ α dτ = + (.33) (τ) with ( ( ) t (4 (n )(p 1))/ d d(t) exp 1 ω ) (τ) (p 1)/ σ α (τ) dτ ( ( ) 1 1 d(t) exp ω t d ) (τ) σ α (τ) dτ <. <, (.34) If m>1, we suppose that for some α<1, (1 + τ) n(m 1)/ σ ((1+α)(1+m) )/ (τ) d m+1 (τ)dτ = (.35)

7 A. Benaissa and. Mokeddem 941 with ( d(t) ) (4 (n )(p 1))/ ( t (1 + τ) n(m 1)/ σ ((1+α)(1+m) )/ (τ) d m+1 (τ)dτ ) (p 1)/(m 1) <, ( ) 1 (.36) d(t) ( t (1 + τ) n(m 1)/ σ ((1+α)(1+m) )/ (τ) d m+1 (τ)dτ ) 1/(m 1) <. We have the following theorem. Theorem.8. Let (u,u 1 ) H 1 L, u, and let the initial energy E() be sufficiently small. The following cases are considered. (i) σ(t) = ( d(t)). uppose (.), (.3), (.9), and (.3)or(.), (.3), (.31), and (.3). Then problem (1.1) admits a unique solution u(t) C([, );H 1 ) C 1 ([, );L ) and has the same decay property as Theorem.7. (ii) d(t) = (σ(t)). uppose (.), (.3), (.33), and (.34) or(.), (.3), (.35), and (.36). Then problem (1.1) admits a unique solution u(t) C([, );H 1 ) C 1 ([, );L ).Furthermore,the global solution of the Cauchy problem (1.1) has the following energy decay property: ( t d E(t) E()exp 1 ω ) (τ) σ α (τ) dτ t>if m = 1, (.37) E(t) ( C ( E() ) t (1 + τ) n(m 1)/ σ ((1+α)(1+m) )/ (τ) d m+1 (τ)dτ ) /(m 1) t> if m>1. (.38) Remark.9. In Theorem.7, the global existence and energy decay are independent, but in Theorem.8, we need the estimation of the energy decay for a local solution to prove global existence. Examples.1. (1) If σ(t) = 1/t θ,byapplyingtheorem.7 we obtain E(t) E()e 1 ωt1 θ if m = 1, E(t) C ( E() ) (1 + t) ( n(m 1) θ)/(m 1) if 1 <m<1+ θ,<θ<1, n E(t) C ( E() ) (lnt) /(m 1) if m = 1+ θ,<θ<1. n () If σ(t) = 1/t θ lnt ln t ln p t,byapplyingtheorem.7,weobtain (.39) E(t) E() ( ln p t ) ω if m = 1, θ = 1. (.4) For example, if n(m 1)/+θ = 1, that is, 1 <m<1+/n, E(t) C ( E() )( ln p t ) /(m 1). (.41)

8 94 Cauchy problem for nonlinear wave equation (3) If σ(t) = 1/t θ and d(r) = 1/r γ with θ γ by applying Theorem.8,weobtain E(t) C ( E() ) (1 + t) ( n(m 1) θ)/(m 1) E(t) C ( E() ) (lnt) /(m 1) if 1 <m<1+ θ γ + n,<θ<1, if m = 1+ θ γ + n,<θ<1. (.4) In order to show the global existence, it suffices to obtain the a priori estimates for E(t) and u(t) in the interval of existence. To prove Theorem.7 we first have the following energy identity to problem (1.1). Lemma.11 (energy identity). Let u(t,x) be a local solution to problem (1.1) on[,t max ) as in Theorem.5. Then t E(t)+ σ(s)u (s)g ( u (s) ) dsdx = E() (.43) for all t [,T max ). Next we state several facts about the modified stable set. Lemma.1. uppose that 1 <p n + n. (.44) Then (i) is a neighborhood of in H 1 ( ), (ii) for u, J(u) p 1 ( x u (p +1) + ) u. (.45) Proof of Lemma.1. (i) From Lemma.1 we have u p+1 p+1 K u p+1 H 1 ( K u p 1 H u 1 + x u ). (.46) Let { U() u H 1( R N ) u p 1 H < 1 }. (.47) 1 K Then, for any u U()\{},wededucefrom(.46)that u p+1 p+1 < u + x u, (.48) that is, K(u) >. This implies U(). (ii) By the definition of K(u)andJ(u), we have the identity (p +1)J(u) = K(u)+ (p 1) ( x u + ) u. (.49)

9 A. Benaissa and. Mokeddem 943 ince u,wehavek(u). Therefore from (.44) we get the desired in-equality (.45). Lemma.13. Let u(t) beasolutiontoproblem(1.1) on[,t max ).uppose(.44) holds. If u and u 1 L ( ) satisfy then (i) u(t) on [,T max ), (ii) u(t) I on [,T max ). C E() (p 1)/ < 1, (.5) Proof of Lemma.13. uppose that there exists a number t [,T max [suchthatu(t) on [,t [andu(t ).Thenwehave ince u(t) on [,t [, it holds that K ( u ( t )) =, u ( t ). (.51) p 1 ( x u (p +1) + ) u J(u) E(t); (.5) it follows from the nonincreasing of the energy that Hence, we obtain x u + (p +1) u p 1 E() I. (.53) u Next, from Lemma.1 and (.54)wehave (p +1) p 1 E() I on [,t ]. (.54) u p+1 p+1 K u(t) p+1 H 1 ( ) K u(t) p 1 x H 1 (R )( u n + ) u KI p 1 ( x u + ) u C E() (p 1)/( u(t) + x u(t) ) (.55) for all t [,t ], where C is the constant defined by (.). Note that from (.55) and our hypothesis η C E() (p 1)/ < 1, (.56) it follows that u(t) p+1 p+1 ( 1 η )( u(t) + x u(t) ). (.57)

10 944 Cauchy problem for nonlinear wave equation Therefore, we obtain K ( u ( t )) ( η u ( t ) + x u ( t ) ) (.58) which contradicts (.51). Thus, we conclude that u(t) on [,T max [. The assertion (ii) can be obtained by the same argument as for (.54). This completes the proof of Lemma.13. Lemma.14. Under the same assumptions as in Lemma.13, there exists a constant M depending on u H 1 and u 1 such that for all t [,T max [. u(t) H 1 + u (t) M (.59) Proof of Lemma.14. It follows from Lemma.13 that u(t) on [,T max [. o Lemma.1(ii) implies that J(u) p 1 u(t) (p +1)( + x u(t) ) on [,T max [. (.6) Hence, from Lemma.11 and (.6)weget 1 u (t) + p 1 u (p +1)( + x u(t) ) E(t) E(). (.61) o we get u(t) H 1 + u (t) M, (.6) for some M >. The above inequality and the continuation principle lead to the global existence of the solution, that is, T max =. Proof of the energy decay. From now on, we denote by c various positive constants which may be different at different occurrences. We multiply the first equation of (1.1)byE q φ u, where φ is a function satisfying all the hypotheses of Lemma.4.Weobtain = E q φ R u ( u u + u + σ(t)g(u ) u p 1 u ) dxdt n = [E q φ uu dx + E q φ ] T ( qe E q 1 φ + E q φ ) uu dxdt E q φ u R dxdt n ) p +1 u p+1 dxdt + E q φ σ(t)ug(u R )dxdt n ( ) E q φ p +1 1 u p+1 dxdt. ( u + u + u (.63)

11 A. Benaissa and. Mokeddem 945 ince ( 1 ) u p+1 dx ( ) p 1 1 η p +1 p +1 u(t) H 1 ( ) dx ( ) p 1 (p +1) 1 η p +1 p 1 E(t) = ( 1 η ) E(t), (.64) we deduce that η ] T E q+1 φ dt [E q φ uu dx + + E q φ R u dxdt n ( qe E q 1 φ + E q φ ) uu dxdt E q φ σ(t)ug(u )dxdt ] T [E q ( φ uu dx + qe E q 1 φ + E q φ ) uu dxdt + E q φ u R dxdt + c(ε) E q φ g(u ) dxdt n u 1 + ε E q φ u R dxdt + E q φ σ(t)ug(u )dxdt n u 1 (.65) for every ε>. Also, applying Hölder s and Young s inequalities, we have E q φ u >1 σ(t)ug(u )dxdt ( E q φ σ(t) c c c(ε ) Ω ) 1/(r+1) ( u r+1 dx ( E (q+1)/ φ σ 1/(r+1) (t) u >1 φ σ 1/(r+1) (t)e (q+1)/ ( E ) r/(r+1) dt u >1 g(u ) r/(r+1) dx) (r+1)/r dt σ(t)u g(u )dx) r/(r+1) dt φ ( σ 1/(r+1) (t)e (q+1)/ r/(r+1))( ( E ) r/(r+1) E r/(r+1)) dt φ ( E E)dt + ε φ σ(t)e (r+1)((q+1)/ r/(r+1)) dt c(ε )E() + ε σ()e() (rq r 1)/ φ E q+1 dt (.66)

12 946 Cauchy problem for nonlinear wave equation for every ε >. Choosing ε and ε small enough, we obtain ] T E q+1 φ dt [E q ( φ uu dx + qe E q 1 φ + E q φ ) uu dxdt + σ(t)ug(u )dxdt + c Eφ u u 1 R dxdt n ce()+c Eφ u dxdt. (.67) ince xg(x) forallx R, it follows that the energy is nonincreasing, locally absolutely continuous and E (t) = σ(t)u g(u )dx a.c. in R +. Proof of (.6). We consider the case m = 1, that is, Then we have c 3 v g(v) c4 v for all v 1. (.68) u c 13 σ(t) u ρ(t,u ) t R, x, (.69) where ρ(t,s)=σ(t)g(s)foralls R. Therefore we deduce from (.67) (applied with q = ) that E(t)φ (t)dt CE()+C φ 1 (t) σ(t) u ρ(t,u )dxdt. (.7) Define t φ(t) = σ(τ)dτ. (.71) It is clear that φ is a nondecreasing function of class C on R +. The hypothesis (.3) ensures that Then we deduce from (.7)that E(t)φ (t)dt CE()+C andthankstolemma.4we obtain φ(t) + as t +. (.7) u ρ(t,u )dxdt 3CE(), (.73) E(t) E()e (1 φ(t))/(3c). (.74) Proof of (.7). Now we assume that m>1in(.). Define φ by (.71). We apply Lemma.4 with q = (m 1)/.

13 A. Benaissa and. Mokeddem 947 We need to estimate For t, consider E q φ u dxdt. (.75) Firstwenotethatforeveryt, Ω 1 = { x, u 1 }, Ω = { x, u > 1 }. (.76) Next we deduce from (.)and(.3)thatforeveryt, (i) if x Ω 1,thenu ((1/σ(t))u ρ(t,u )) /(m+1), (ii) if x Ω,thenu (1/σ(t))u ρ(t,u ). Hence, using Hölder s inequality, we get that E q φ R u dxdt n E q φ 1 T σ(t) u ρ(t,u )dxdt + E q φ E q φ 1 T σ(t) u ρ(t,u )dxdt + E q φ E q φ 1 σ(t) u ρ(t,u )dxdt + Ω 1 Ω =. (.77) ( ) 1 /(m+1) σ(t) u ρ(t,u ) dxdt { x (L+t)} E q φ (1 + t) n(m 1)/(m+1) ( u g(u )dx) /(m+1) dt ( E ce() 1+q + c E q φ (1 + t) n(m 1)/(m+1) ) /(m+1) dt σ(t) ce() 1+q + c E q φ (1 + t) n(m 1)/(m+1) σ /(m+1) (t)( E ) /(m+1) dt. ( u g(u ) ) /(m+1) dxdt et ε>; thanks to Young s inequality and to our definitions of p and φ,weobtain E q φ u dxdt (.78) ce() 1+q + m 1 m +1 ε(m+1)/(m 1) E 1+q (φ ) (m+1)/(m 1) (1 + t) n σ /(m 1) dt m +1ε (m+1)/ E(). (.79) We choose φ such that φ /(m 1) (1 + t) n σ /(m 1) = 1, (.8)

14 948 Cauchy problem for nonlinear wave equation so Then we deduce from (.79)that and thanks to Lemma.4 (applied with c = ) we obtain t φ(t) = (1 + s) n(m 1)/ σ(s)ds. (.81) E 1+q φ dt CE(), (.8) E(t) Proof of Theorem.8. First, we see that if u,then C. (.83) φ(t) /(m 1) x u R + F ( u(t) ) dx x u n p +1 u(t) p+1 p+1 p 1 x u p +1. (.84) In the proof, we often use the following inequality: Now, we assume that I(u ) > (1/) x u.then u(t) 1 d(t) λ(x)u(t). (.85) I ( u(t) ) 1 x u(t) (.86) for some interval near t =. As long as (.86)holds,wehaveJ(t) I(u(t)). Thus ] T η E q+1 φ dt [E q ( φ uu dx + qe E q 1 φ + E q φ ) uu dxdt R n T + E q φ R u dxdt E q φ σ(t)ug(u R )dxdt n n ] T [E q ( φ uu dx + qe E q 1 φ + E q φ ) uu dxdt ( ) σ(t) + E q φ u R dxdt + c(ε) E q φ g(u ) dxdt n u 1 λ(x) + ε E q φ λ R (x)u dxdt + E q φ σ(t)ug(u )dxdt. n u 1 (.87) If σ(t) = ( d(t)), that is, σ(t) ast more rapidly than d(t), we find the same results of asymptotic behaviour as in Theorem.7.

15 A. Benaissa and. Mokeddem 949 If d(t) = (σ(t)), so, we obtain E q+1 φ dt [E q φ E q φ + c(ε) u 1 u dxdt E q φ uu dx E q φ u 1 ] T + u 1 ( qe E q 1 φ + E q φ ) uu dxdt σ(t)ug(u )dxdt ( ) σ(t) u dxdt. λ(x) (.88) We consider the case m = 1. Thus under (.)and(.3), we have E q φ for all α<1. We choose ( ) σ(t) u dxdt C λ(x) E q φ ( σ α ) (t) σ(t)u d g(u )dxdt (.89) (t) t d φ(t) (s) = σ α ds. (.9) (s) It is clear that φ is a nondecreasing function of class C on R +. Hypothesis (.33) ensures that φ(t) + as t +. (.91) By (.85), the definition of E, and the Cauchy-chwartz inequality, we have ] T [E q φ uu dx = E q ()φ () u()u ()dx E q (T)φ (T) u(t)u (T)dx R [ ] n CE q+1 φ () () d() + φ (T) CE d(t) q+1 (), ( qe E q 1 φ + E q φ ) uu dxdt q E E q φ (t) T d(t) dt + E q+1 φ (t) dt d(t) (.9) when we have (in the case m = 1) φ = d (t) σ α (t), φ (t) = d(t) d (t) σ α (t) α σ (t) d (t) σ α+1 (t). (.93) o φ (t) d (t) d α (t) d(t) d α (t) σ α (t) α d(t) σ (t) σ(t) σ α (t), (.94)

16 95 Cauchy problem for nonlinear wave equation d(t)/σ(t)isbounded,soweobtain E q+1 φ (t) dt E d(t) q+1 () [ d1 α (t)+σ 1 α (t) ] T E q+1 () ( d1 α (t)+σ 1 α (t) ) CE q+1 (). (.95) Then we deduce from (.88)that Eφ dt CE()+C σ(t)u g(u )dxdt 3CE(), (.96) andthankstolemma.4we obtain E(t) E()e (1 φ(t))/3c. (.97) Using the condition that d(t) = (σ(t)) and using Hölder s inequality, we get that E q φ u 1 σ (t) λ (x) u dxdt E q φ σ ( ) (t) 1 /(m+1) λ (x) σ(t) σ(t)u g(u ) dxdt E q φ σ ( ) (t) 1 /(m+1) { x L+t} λ (x) σ(t) σ(t)u g(u ) dxdt ( ) σ(t) ( ) /(m+1) E q φ (1 + t) d(t) n(m 1)/(m+1) u g(u )dx dt ( ) σ(t) ( E c E q φ (1 + t) d(t) n(m 1)/(m+1) ) /(m+1) dt σ(t) ( ) σ(t) c E q φ (1 + t) d(t) n(m 1)/(m+1) σ /(m+1) (t)( E ) /(m+1) dt c E q φ σα+1 (t) d (t) (1 + t)n(m 1)/(m+1) σ /(m+1) (t)( E ) /(m+1) dt. (.98) et ε>; thanks to Young s inequality and to our definitions of p and φ,weobtain ( ) σ(t) E q φ u dxdt λ(x) ce() 1+q + m 1 m +1 ε(m+1)/(m 1) E 1+q (φ ) (m+1)/(m 1) σ(1+α)(m+1)/(m 1) (t) (1+t) d n σ /(m 1) dt+ 4 (m+1)/(m 1) (t) m +1ε 1 (m+1)/ E(). (.99)

17 A. Benaissa and. Mokeddem 951 We choose φ such that t φ(t) = (1 + τ) n(m 1)/ σ ((1+α)(m+1) )/ (τ) d m+1 (τ)dτ. (.1) It is clear that φ is a nondecreasing function of class C on R +. The hypothesis (.35)ensures that φ(t) + as t +.By(.85), the definition of E, and the Cauchy-chwartz inequality we have (.9)whenwehave(thecasem>1) φ = (1 + t) n(m 1)/ σ ((1+α)(m+1) )/ (t) d m+1 (t), φ (t) = n(m 1) (1 + t) n(m 1)/ 1 σ ((1+α)(m+1) )/ (t) d m+1 (t) +(1+t) n(m 1)/ ( (1 + α)(m +1) σ (1+α)(m+1)/ (t)σ (t) d m+1 (t) +(m +1) d m (t) d ) (t)σ ((1+α)(m+1) )/ (t). (.11) Thus φ (t) C d m (t) d(t) σ m (t) σ(1 α)(m+1)/ (t) C d m (t) σ (t) σ m (t) σ ((1+α)(m+1) m)/ (t) C d ((1+α)(m+1) )/ (t) σ ((1+α)(m+1) )/ (t) (.1) d (t) d ((1+α)(m+1) m)/ (t), d(t)/σ(t)isbounded,soweobtain E q+1 φ (t) dt E d(t) q+1 () [ d(1 α)(m+1)/ (t)+σ (1 α)(m+1)/ (t) ] T E q+1 () ( d(1 α)(m+1)/()+σ (1 α)(m+1)/ () ) CE q+1 (). (.13) We deduce from this choice and thanks to Lemma.4 (applied with c = ), we obtain E(t) C( E() ) ince u and is an open set, putting E 1+q φ dt CE(), (.14). (.15) φ/(m 1) T 1 = sup { t [,+ ):u(s) for s t }, (.16) we see that T 1 > andu(t) for t<t 1.IfT 1 <T max <, wheret max is the lifespan of the solution, then u(t 1 ) ; that is, I ( u ( T 1 )) =, u ( T1 ). (.17)

18 95 Cauchy problem for nonlinear wave equation We see from Lemma. and (.85)that u(t) p+1 p+1 C u(t) (4 (n )(p 1))/ x u(t) n(p 1)/ ( d(t) ) (4 (n )(p 1))/E (p 1)/ x u(t) (.18) B(t) x u(t) for t T 1, where we have used the assumption p (n +4)/n and B(t) = C ( E() ) d (4 (n )(p 1))/ (t) ( t (1 + τ) n(m 1)/ σ ((1+α)(m+1) )/ (τ) d m+1 (τ)dτ ) (p 1)/(m 1). (.19) Next, we put T sup {t [,+ ):B(s) < 1 } for s<t, (.11) and then we see that T > andt = T 1 because B(t) < 1/ by the condition that E() is small. Then I ( u(t) ) x u(t) B(t) x u(t) 1 x u(t) (.111) for t T 1.Moreover,(.17)and(.111)imply K ( u ( )) 1 T 1 x u ( ) T 1 >, (.11) which is a contradiction, and hence, it might be T 1 = T max. Therefore, (.15)holdstrue for T T max. To prove global existence in H H 1, we need to derive the estimates for second derivatives of u(t) on the basis of the energy estimate of E(t), we utilize the differentiated equation u ttt x u + λ (x)u + σ (t)g(u )+σ(t)g(u )u + f (u)u =, (.113) where f (u) = u p 1 u.multiplying(.113)byu,wehave d dt E (t)+σ(t) g (u ) u (t) dx R n f (u) u (t) u (t) dx + σ (t) g(u ) u (t) dx, where we set (.114) E (t) = u (t) + x u(t) + λu (t). (.115) By (.)and(.3), we have g(u ) dx C u 1 u /m dx + C u 1 u r dx C(L + t) n(m 1)/m E 1/m + C E ( (n )(r 1))/ E n(r 1)/ (.116)

19 A. Benaissa and. Mokeddem 953 and by Lemma. f (u) u (t) u (t) dx C u(t) p 1 c u(t) p 1 n(p 1) n(p 1) c u(t) p 1 n(p 1) E (t). u (t) n/(n ) u (t) x u (t) u (t) (.117) ince (n +)/n p n/(n ), then Thus, we have u(t) p 1 n(p 1) d ( (n )(p 1))/ E (p 1)/. (.118) d dt E d ( (n )(p 1))/ E (p 1)/ E (t) + σ (t) ( (L + t) n(m 1)/m E 1/m + CE ( (n )(r 1))/ E n(r 1)/ ) 1/E 1/ (t). (.119) We have also, applying Young inequality, E ( (n )(r 1))/ E n(r 1)/ CE ( (n )(r 1))/( n(r 1)) (t)+e (t), (.1) hence, we deduce that o, we obtain d dt E (t) C(t) ( 1+E (t) ). (.11) E (t) C e t C(s)ds. (.1) From (.1) and the first equation of problem (1.1), we also prove easily that for all t. Indeed, we have x u(t) C (t) < (.13) x u(t) u (t) + λ u(t) + σ(t) g ( u (t) ) + f (u) (.14) and also we have f (u) C u(t) p p C u(t) (p 1) n(p 1)/ x u(t). (.15) Here, to check the last inequality of (.15), we note that if (n +4)/n p (n +)/(n ), then C u(t) (p 1) n(p 1)/ C u(t) 4 (n )(p 1) x u(t) n(p 1) 4 ( ) 4 (n )(p 1)E d(t) p 1 (t) < 1. (.16)

20 954 Cauchy problem for nonlinear wave equation Thus the solutionin the sense oftheorem.6 exists globally in time t under the assumption (u,u 1 ) H H 1. When (u,u 1 ) H 1 L we approximate (u,u 1 )by(u k,u k 1) H H 1, k = 1,,..., in the topologies of H 1 L, which satisfy suppu k suppuk 1 { x x L }. (.17) ince lim k (u k,u k 1) = (u,u 1 )inh 1 L, then for these initial data problem (1.1) has global solutions u k W, loc ([, );L ) W 1, loc ([, );H1 ) L loc ([, );H ), which satisfy (.1)and(.13). We can easily see that {u k (t)} converges uniformly on each compact interval [,T], T>. Uniqueness follow from a standard argument. The proof of Theorem.8 is now completed. References [1] A. Benaissa and. A. Messaoudi, Blow-up of solutions for the Kirchhoff equation of q-laplacian type with nonlinear dissipation,colloq. Math. 94 (), no. 1, [] A. Benaissa and L. Rahmani, Global existence and energy decay of solutions for Kirchhoff-Carrier equations with weakly nonlinear dissipation, Bull. Belg. Math. oc. imon tevin 11 (4), no. 4, [3] P. Brenner, On space-time means and strong global solutions of nonlinear hyperbolic equations, Math. Z. 1 (1989), no. 1, [4] V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms,j.differential Equations 19 (1994), no., [5] J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z. 189 (1985), no. 4, [6] A. Haraux, Two remarks on hyperbolic dissipative problems, Nonlinear Partial Differential Equations and Their Applications. Collège de France eminar, Vol. VII (Paris, ), Res. Notes in Math., vol. 1, Pitman, Massachusetts, 1985, pp [7] R. Ikehata, ome remarks on the wave equations with nonlinear damping and source terms, Nonlinear Anal. 7 (1996), no. 1, [8] R. Ikehata and T. uzuki, table and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J. 6 (1996), no. 3, [9] F. John, Nonlinear Wave Equations, Formation of ingularities, University Lecture eries, vol., American Mathematical ociety, Rhode Island, 199. [1] V. Komornik, Exact Controllability and tabilization, RAM: Research in Applied Mathematics, Masson, Paris, [11] H. A. Levine, P. Pucci, and J. errin, ome remarks on global nonexistence for nonautonomous abstract evolution equations, Harmonic Analysis and Nonlinear Differential Equations (Riverside, Calif, 1995), Contemp. Math., vol. 8, American Mathematical ociety, Rhode Island, 1997, pp [1] H. A. Levine and J. errin, Global nonexistence theorems for quasilinear evolution equations with dissipation, Arch. RationalMech. Anal. 137 (1997), no. 4, [13] P. Martinez, A new method to obtain decay rate estimates for dissipative systems, EAIM Control Optim. Calc. Var. 4 (1999), [14] M. Nakao, A difference inequality and its application to nonlinear evolution equations, J. Math. oc. Japan 3 (1978), no. 4, [15] M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, Math. Z. 14 (1993), no.,

21 A. Benaissa and. Mokeddem 955 [16], Global existence to the Cauchy problem of the semilinear wave equation with a nonlinear dissipation, Funkcial. Ekvac. 38 (1995), no. 3, [17] K. Ono, On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation, J. Math. Anal. Appl. 16 (1997), no. 1, [18] D. H. attinger, On global solution of nonlinear hyperbolic equations, Arch. RationalMech. Anal. 3 (1968), [19] W. A. trauss, On continuity of functions with values in various Banach spaces, Pacific J. Math. 19 (1966), [], Nonlinear Wave Equations, CBM Regional Conference eries in Mathematics, vol. 73, Conference Board of the Mathematical ciences, Washington, D.C., [1] G. Todorova, table and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, C. R. Acad. ci. Paris ér. I Math. 38 (1999), no., Abbès Benaissa: Département de Mathematiques, Faculté des ciences, Université Djillali Liabès, B. P. 89, idi Bel Abbès, Algeria address: benaissa abbes@yahoo.com oufiane Mokeddem: Département de Mathematiques, Faculté des ciences, Université Djillali Liabès,B.P.89,idi Bel Abbès, Algeria address: mok5 so5@yahoo.fr