Asymptotic Behavior for Semi-Linear Wave Equation with Weak Damping

Transcription

1 Int. Journal of Math. Analysis, Vol. 7, 2013, no. 15, HIKARI Ltd, Asymptotic Behavior for Semi-Linear Wave Equation with Weak Damping Ducival Carvalho Pereira State University of Pará Belém - PA - Brasil ducival@oi.com.br Carlos Alberto Raposo Federal University of São João del-rei São João del - Rei - MG - Brasil raposo@ufsj.edu.br Abstract In this work we study the asymptotic behavior as t of the solutions for the initial boundary value problem associated to the semilinear wave equation with weak damping. Keywords: Semi-linear wave, weak damping, exponential decay, Lemma of Nakao 1 Introduction We shall consider the initial boundary value problem associated to the damped semi-linear wave equation, u Δu + F (u)+αu =0, (1) in the cylinder Q =Ω (0,T), 0 T <, where Ω is a smooth bounded domain in R n, α is a real positive constant, F is a continuous real function with sf (s) G(s) 0, for all s R n and G is a primitive of F. The asymptotic behavior as t of the classical solution of equation with F (u) =u 3 was studied by Sattinger [7], for small initial data, and Nakao [5] without any restrictions concerning the smallness of the initial data. The exponential decay in the energy norm of weak solutions was proved by Strauss [8] and Nakao [6].

2 714 D. C. Pereira and C. A. Raposo When F (u) = u ρ 1 u, ρ 1, the existence of global weak solutions was studied by Lions [3] and the asymptotic behavior was proved by Avrin [2]. Existence of global weak solutions when α = 0 was proved by Strauss [9]. He also proved that the local energy decay for zero as t. In this work we prove that the global weak solutions of equation (1) decay exponentially. For our goal we use the following result Lemma 1.1 Let E(t) be a nonnegative function on [0, ) satisfying sup E(s) C 0 ( E(t) E(t +1)) s [t,t+1] where C 0 is a positive constant. Then we have Proof. 1.1 See page 748 of [6]. E(t) Ce wt with w = 2 Asymptotic behavior 1 C We use the standard Lebesgue space and Sobolev space with their usual properties as in [1] and in this sense and denote the norm in L 2 and H 1 0 respectively. Lemma 2.1 For F Lipschitz, derivable except on a finite number of points with sf (s) G(s) 0, for all s R, the solution of the initial boundary value problem associated to the equation (1) with initial data u 0 H0 1(Ω), u 1 L 2 (Ω) and G(u 0 ) L 1 (Ω) satisfies u (t) 2 + u(t) 2 Ce wt, for all t 1, where C and w are positive constants. Proof. 2.1 If F is Lipschitz and derivable except on a finite number of points, then there exists a unique solution u of (1), see [9] in the class satisfying u L (0,T; H0 1 (Ω)), (2) u L (0,T; L 2 (Ω)), (3) u Δu + F (u)+αu =0 in L 2 (0,T; L 2 (Ω)) = L 2 (Q). (4) Taking the inner product in L 2 (Ω) of (4) with u we obtain d dt E(t)+2α u (t) 2 =0, (5)

3 Asymptotic behavior for semi-linear wave equation 715 where E(t) = u (t) 2 + u(t) 2 +2 G(u(t)) dx with G(s) = Integrating (5) from t to t +1, we get s Ω 0 F (ξ) dξ. t+1 t u (s) 2 ds = 1 2α [E(t) E(t + 1)] D2 (t). (6) The mean value theorem for integrals applied in (6) implies that there exists [ t, t + 1 ] and t 2 [t + 34 ] 4,t+1 such that u (t i ) 2D(t), t i =1, 2. The inner product in L 2 (Ω) of (4) with u implies d dt (u (t),u(t)) u (t) 2 + u(t) 2 +(F (u(t)),u(t)) + α(u (t),u(t)) = 0. Integrating to t 2 and applying Cauchy-Schwarz s inequality and Poincare s inequality, we get [ u(s) 2 +(F (u(s)),u(s))]ds C 0 [ u ( ) u( ) + u (t 2 ) u(t 2 ) ] + u (s) 2 ds + α u (s) u(s) ds where C 0 denote Poincare s constant in Ω. Using sf (s) G(s) 0, u (t i ) 2D(t), t i =1, 2, and (6) we have [ u(s) 2 +(F (u(s)),u(s))]ds 8C 0 D(t) E(t) + (2 + α 2 C0 2 )D(t)2 H 2 (t), from where follows t 2 E(s)ds D 2 (t)+h 2 (t), and by mean value theorem for integrals, there exists t [,t 2 ] such that E(t ) 2D 2 (t)+h 2 (t). (7) Integrating (5) from t to t, using (6) and (7) follows that E(t) E(t )+2α + t+1 t u (s) 2 ds C 1 D 2 (t) = C 1 [E(t) E(t + 1)], 2α

4 716 D. C. Pereira and C. A. Raposo where C 1 = 4 + (32 + α 2 )C0 2 +2α. Then ess sup s [t,t+1] and finally by Lemma 1.1 follows that E(s) =E(t) C 1 [E(t) E(t + 1)], 2α E(t) Ce wt, for all t 1 with C and w positive constants, and by defintion of E(t) we have u (t) 2 + u(t) 2 Ce wt. Now we are in the position to present our principal result. Theorem 2.2 If F is continuous with sf (s) G(s) 0, for all s R then the solution of the initial boundary value problem associated to the equation (1) with initial data u 0 H 1 0(Ω), u 1 L 2 (Ω) and G(u 0 ) L 1 (Ω) satisfies u (t) 2 + u(t) 2 Ce wt, for all t 1, where C and w are positive constants. Proof. 2.3 For F continuous there exists a sequence (F k ) k N with each F k Lipschitiz and derivable except on a finite number of points satisfying sf k (s) G k (s) 0, for all s R where G k (s) = s F 0 k(ξ) dξ, such that F k F uniformly on the bounded sets of R. For each k N let u k be the solution of (1) when we replace F by F k.foru solution of (1), u k satisfies (see [9]) u k u weakly-star in L (0,T; H0(Ω)), 1 (8) u k u weakly-star in L (0,T; L 2 (Ω)), (9) F k (u k ) F (u) weakly in L 1 (Ω), (10) u k Δu k + F k (u k )+αu k =0 in L 2 (Q) (11) From Lemma 2.1 we get u k (t) 2 + u k (t) 2 Ce wt, for all t 1. Letbe [1,T], then u k( ) 2 + u k ( ) 2 Ce w, (12) u k ( ) λ weakly in H0 1 (Ω) as k, (13) u k () η weakly in L 2 (Ω) as k. (14) (15) Using (8), (9) and Arzelá-Ascoli s theorem we have λ = u( ). Now, we are going to prove that η = u ( ). For this we consider θ [,T[ defined for δ>0 by θ(t) = 1 if t =, 1 δ (t δ) if t + δ, 0 if t + δ.

5 Asymptotic behavior for semi-linear wave equation 717 It follows from (11) that (u k,v)+(u k,v)+(f k (u k ),v)+α(u k,v)=0for all v L 2 (Ω), inl 2 (0,T). Now multiplying by θ and integrating from to T we get for all v L 2 (Ω) (u k( ),v)+ 1 δ + (u k(t),v)dt + (F k (u k (t)),v)θ(t)dt + α a(u k (t),v)θ(t)dt (u k (t),v)θ(t)dt =0, and taking the limit as k using (10) and (14) we obtain + (η,v)+ 1 δ (u (t),v)dt + (F (u(t)),v)θ(t)dt + α a(u(t),v)θ(t)dt (u (t),v)θ(t)dt =0. Now taking again the limit as δ we get (η,v)+(u ( ),v)=0and then η = u ( ). Therefore, the lim inf as k in (12) implies u ( ) 2 + u( ) 2 Ce w. (16) If we consider θ C 0 [0,T] defined for δ>0 by 1 if 0 t T δ, 1 θ(t) = (t T + δ) if t δ t T, δ 0 if t = T, and repeating the same process we prove that (16) holds for t = T. Finally, u (t) 2 + u(t) 2 Ce w. for all t 1 and the proof of theorem is complete. References [1] A. Adams, Sobolev Spaces, Academic Press, New York, [2] J. Avrin, Energy decay of underdamped nonlinear wave equations, Houston Journal of Mathematics. 13(4) (1987) [3] J. L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod, Paris, 1969.

6 718 D. C. Pereira and C. A. Raposo [4] M. Nakao, Convergence of solutions of the wave equation with a nonlinear dissipative term to the steady state, Mem. Fac. Science, Kyushu Univ. 30 (1976) [5] M. Nakao, Decay of classical solutions of a semilinear wave equation, Math. Rep. XI-7 (1977) [6] M. Nakao, A difference inequalit and its application to nonolinear evolution equation, J. Math. Soc. Japan. 30(4) (1978) [7] D. H. Sattinger, Sytability of nonlinear hyperbolic equations, Arch. Rat. Mech. Anal. 28 (1968) [8] W. A. Strauss, The energy method in nonlinear partial differential equations, Notas de Matematica, N. 47, IMPA, Rio de Janeiro, Brasil, [9] W. A. Strauss, On wealk solutions of semilinear hyperbolic equations, An, Acad. Bras. Ciencias. 42(4) (1970) Received: October, 2012