A REMARK ON THE DAMPED WAVE EQUATION. Vittorino Pata. Sergey Zelik. (Communicated by Alain Miranville)

Transcription

1 COMMUNICATIONS ON Website: PURE AND APPLIED ANALYSIS Volume 5, Number 3, September 2006 pp A REMARK ON THE DAMPED WAVE EQUATION Vittorino Pata Dipartimento i Matematica F.Brioschi, Politecnico i Milano Via Bonari 9, 2033 Milano, Italy Sergey Zelik Weierstrass Institute for Applie Analysis an Stochastics Mohrenstr. 39, 07 Berlin, Germany (Communicate by Alain Miranville) Abstract. In this short note we present a irect metho to establish the optimal regularity of the attractor for the semilinear ampe wave equation with a nonlinearity of critical growth. We consier the semilinear (weakly) ampe wave equation on a boune omain Ω R 3 with smooth bounary Ω tt u + t u u + ϕ(u) = f, u(0) = u 0, t u(0) = u, () u Ω = 0. Here, f L 2 (Ω) is inepenent of time an ϕ C 2 (R), with ϕ(0) = 0, satisfies the growth an the issipation conitions ϕ (u) c ( + u ), (2) ϕ(u) lim inf u u > λ, (3) ϕ (u) l, (4) where c, l 0 an λ > 0 is the first eigenvalue of on L 2 (Ω) with Dirichlet bounary conitions. The asymptotic behavior of solutions to equation () has been the object of extensive stuies (see, e.g. [] [5], [7], [9] [4] an [7] [2]). Denoting H = H 0 (Ω) L 2 (Ω) an V = [ H 2 (Ω) H 0 (Ω) ] H 0 (Ω), problem () is known to generate a C 0 -semigroup S(t) on the phase space H, an the following result hols Mathematics Subject Classification. Primary: 35B33, 35B4; Seconary: 35L05, 35Q40. Key wors an phrases. Dampe wave equation, critical nonlinearity, global attractor. This work was partially supporte by the Italian MIUR Research Projects Aspetti Teorici e Applicativi i Equazioni a Derivate Parziali an Analisi i Equazioni a Derivate Parziali, Lineari e Non Lineari: Aspetti Metoologici, Moellistica, Applicazioni, an by the Alexaner von Humbolt Stiftung an the CRDF grant. 6

2 62 V. PATA AND S. ZELIK Theorem. The semigroup S(t) on H possesses a compact global attractor A. Besies, A is a boune subset of V. Theorem was first prove by Babin an Vishik [3]. We mention that the result is still vali if one removes conition (4), which is however very reasonable. In that case, the existence of the attractor was shown in [], whereas its V-regularity first appeare in the papers [9, 0, 20]. In particular, the argument presente in [20] allows also to treat the nonautonomous case. In all the preceing works, the V-regularity of the attractor is achieve by means of rather complicate an long proceures, requiring multiplications by fractional operators an bootstrap arguments. The aim of this note is to show how to obtain this result in a very irect way, exploiting only quite simple energy estimates. This approach can be applie to treat more complicate bounary conitions such as, for example, ynamic bounary conitions (where the use of fractional operators may be problematic), as well as to eal with stabilization problems (see the en of the paper for more etails). The key step of our proof is a suitable ecomposition of the solution u to (), which has been alreay successfully employe in the recent works [6, 8, 2]. A new proof of Theorem. We enote by, an the inner prouct an the norm on L 2 (Ω). In what follows, we will often make use without explicit mention of the Sobolev embeings an of the Young, the Höler an the Poincaré inequalities. As usual, we will perform formal estimates that can be justifie in a proper Galerkin approximation scheme. Finally, for any function z(t), we will write for short ξ z (t) = (z(t), t z(t)). We begin recalling a basic estimate. Lemma 2. For every t 0, ξ u (t) 2 H + t t u(τ) 2 τ Q( ξ u (0) H )e εt + Q( f ), for some ε > 0 an some positive increasing function Q. The proof may be foun, for instance, in [3], an it is carrie out by multiplying the equation by t u + εu, for some ε > 0 suitably small. In particular, this result yiels the existence of a boune absorbing set B 0 H for the semigroup S(t). In view of (2), for every z, ζ H 0 (Ω) we have that ϕ (ζ)z, z c ( + ζ ) 2 z z 2 z 2 + c 2 ( + ζ ) 4 z 2, for some c, c 2 0. Consequently, by Lemma 2, we can choose θ l large enough such that the inequality 2 z 2 + (θ 2l) z 2 ϕ (u(t))z, z 0 (5) hols for every z H 0 (Ω), every t 0 an every solution u(t) with ξ u (0) B 0. Then, we set ψ(r) = ϕ(r) + θr. Clearly, conition (2) still hols with ψ in place of ϕ. Besies, on account of (4), ψ (r) 0. (6)

3 DAMPED WAVE EQUATION 63 We now consier initial ata ξ u (0) B 0, an we ecompose the solution to () into the sum u = v + w, where v an w solve the equations tt v + t v v + ψ(u) ψ(w) = 0, ξ v (0) = ξ u (0), (7) v Ω = 0, an tt w + t w w + ψ(w) = θu + f, ξ w (0) = (0, 0), w Ω = 0. In the following, c 0 will stan for a generic constant epening (possibly) only on the size of B 0 (but neither on the particular ξ u (0) B 0 nor on the time t). Lemma 3. For every t 0, we have that ξ w (t) H c. Proof. The same argument of the proof of Lemma 2 applies to (8), since from Lemma 2 we know that the right-han sie belongs to L (0, ; L 2 (Ω)). Observe also that here the initial ata are null. Lemma 4. For every t s 0 an every ω > 0, t Proof. Define the functional s t w(τ) 2 τ ω(t s) + c ω. Λ = w 2 + t w Ψ(w), 2θ u, w 2 f, w, where Ψ(w) = w ψ(y)y. Note that Λ c, ue to (2) an Lemma 3. Thus, 0 multiplying (8) by t w, an applying once more Lemma 3, we obtain t Λ + 2 tw 2 = 2θ t u, w 2ω + c ω tu 2, an the claim is prove integrating in time on (s, t), exploiting the integral estimate furnishe by Lemma 2. Collecting the above results, for all initial ata ξ u (0) B 0 we have the bouns (8) ξ u (t) H + ξ w (t) H c, (9) an t [ t u(τ) 2 + t w(τ) 2] τ ω(t s) + c, ω > 0. (0) s ω In orer to conclue, we nee the following generalize version of the Gronwall lemma. Lemma 5. Let Λ : R + R + be an absolutely continuous function satisfying Λ(t) + 2εΛ(t) h(t)λ(t) + k, t where ε > 0, k 0 an t h(τ)τ ε(t s) + m, for all t s 0 an some m 0. s Then, Λ(t) Λ(0)e m e εt + kem, t 0. ε We are now in a position to prove

4 64 V. PATA AND S. ZELIK Lemma 6. For every t 0 an some ν > 0, ξ v (t) H ce νt. Proof. For ε (0, ) to be etermine later, efine the functional Λ = v 2 + t v 2 + ε v ψ(u) ψ(w), v ψ (u)v, v + 2ε t v, v. Note that, from (4) an (5), 2 ψ(u) ψ(w), v ψ (u)v, v (θ 2l) v 2 ϕ (u)v, v 2 v 2. Hence, on account of (2) an (9), Λ satisfies the inequalities 4 ξ v 2 H Λ c ξ v 2 H, () provie that ε is small enough. Multiplying (7) by t v + εv, we fin the equality t Λ + ελ + ε 2 v 2 + Γ = 2 (ψ (u) ψ (w)) t w, v ψ (u) t u, v 2, where we set Γ = ε 2 v 2 + (2 3ε) t v 2 + ε ψ (u)v, v ε 2 v 2 2ε 2 t v, v. Using (2), (6) an (9), it is apparent that Γ 0 if ε is small enough, an 2 (ψ (u) ψ (w)) t w, v ψ (u) t u, v 2 c ( t u + t w ) v 2 ε 2 v 2 + c ε( t u 2 + t w 2) Λ, by means of (). At this point, choosing ε > 0 such that the above conitions are all satisfie, we obtain the ifferential inequality t Λ + ελ c( t u 2 + t w 2) Λ. (2) In view of (0), the esire conclusion follows from Lemma 5 an (). Lemma 7. For every t 0, ξ w (t) V c. Proof. Setting q = t w, we ifferentiate (8) with respect to time, so to obtain Then, for ε > 0, we efine the functional tt q + t q q + ψ (w)q = θ t u. Λ = q 2 + t q 2 + ε q 2 + ψ (w)q, q + 2ε t q, q, which, similarly to the previous lemma, satisfies the inequalities 2 ξ q 2 H Λ c ξ q 2 H, when ε is small enough. Multiplying the above equation by t q + εq, we are le to t Λ + ελ + ε 2 q 2 + t q 2 + Γ = 2θ t u, t q + ψ (w) t w, q 2 + 2εθ t u, t w, where Γ = ε 2 q 2 + ( 3ε) t q 2 + ε ψ (w)q, q ε 2 q 2 2ε 2 t q, q.

5 DAMPED WAVE EQUATION 65 Again, Γ 0 provie that ε is small enough, whereas the right-han sie of the above ifferential equality is controlle as 2θ t u, t q + 2εθ t u, t w + ψ (w) t w, q 2 ε 2 q 2 + t q 2 + c ε tw 2 Λ + c. Hence, fixing ε small, we en up with the ifferential inequality t Λ + ελ c tw 2 Λ + c, an from Lemma 5, we get the boun t w(t) + tt w(t) c. With this information, we recover from (8) the further control w(t) c. Collecting Lemma 6 an Lemma 7, we learn that S(t)B 0 is (exponentially) attracte by a boune subset C V. In other wors, C is a compact attracting set. This, by stanar arguments of the theory of attractors (see e.g. [3, 2, 9]), yiels the existence of a compact global attractor A C for the semigroup S(t). The proof of Theorem is then complete. Further remarks. The proof of Lemma 7 repeats wor by wor the stanar argument of the issipative estimate in the phase space V for the original hyperbolic problem (). Inee, equation (8) can be rewritten in the form tt w + t w w + ϕ(w) = f + θv, (3) which coincies with (), up to the exponentially ecaying (an so nonessential) external force θv. Therefore, our technique allows to reuce the proof of the existence of an exponentially attracting set in a more regular space to verifying the issipative estimate in that space. It is worth noting that the latter problem has been always consiere much simpler (usually, it oes not require any fractional Sobolev spaces, bootstrapping, etc.). Another interesting application of our metho is the stabilization to a single equilibrium for the solutions to () as t. Inee, since the equation possesses a global Lyapunov function, the convergence to the whole set E of equilibria is immeiate, but if E is not totally isconnecte, the convergence to a single equilibrium may not take place in general. Nevertheless, when the nonlinearity ϕ is real analytic, the above convergence can be recovere by the so-calle Simon-Lojashevich technique. However, this technique naturally provies the stabilization of more regular solutions (or it requires the nonlinearity ϕ to be subcritical [5, 6]), an the convergence of weak energy solutions in the critical case is more elicate. In contrast to that, our ecomposition allows to show the convergence of the weak energy solution u by proving the convergence of the more regular solution w to problem (3). Inee, since u = w + v an v is exponentially ecaying, then the convergence of w obviously implies the convergence of u. We point out that the presence of the exponentially ecaying external force θv in (3) is nonessential for the Simon-Lojashevich technique. Finally, we mention a more recent application to the hyperbolic equation with nonlinear amping in a boune two-imensional omain tt u + σ(u) t u u + ϕ(u) = f,

6 66 V. PATA AND S. ZELIK where the amping σ is strictly positive (σ(u) > σ 0 > 0) an satisfies some natural assumptions. In this situation, the stanar bootstrapping methos seem to be inapplicable in orer to improve the regularity of the attractor (we o not know whether or not this equation is well-pose in fractional spaces). On the contrary, the above ecomposition works perfectly, an it yiels the optimal regularity of the global attractor (see [8]). REFERENCES [] J. Arrieta, A.N. Carvalho an J.K. Hale, A ampe hyperbolic equation with critical exponent, Comm. Partial Differential Equations, 7 (992), [2] A.V. Babin an M.I. Vishik, Regular attractors of semigroups an evolution equation, J. Math. Pures Appl., 62 (983), [3] A.V. Babin an M.I. Vishik, Attractors of evolution equations, North-Hollan, Amsteram, 992. [4] A. Een, A.J. Milani an B. Nicolaenko, Finite imensional exponential attractors for semilinear wave equations with amping, J. Math. Anal. Appl., 69 (992), [5] P. Fabrie, C. Galusinski, A. Miranville an S. Zelik, Uniform exponential attractors for a singularly perturbe ampe wave equation, Discrete Contin. Dynam. Systems, 0 (2004), [6] S. Gatti, M. Grasselli, A. Miranville an V. Pata, On the hyperbolic relaxation of the oneimensional Cahn-Hilliar equation, J. Math. Anal. Appl., 32 (2006), [7] J.M. Ghiaglia an R. Temam, Attractors for ampe nonlinear hyperbolic equations, J. Math. Pures Appl., 66 (987), [8] M. Grasselli, A. Miranville, V. Pata an S. Zelik, Well-poseness an long time behavior of a parabolic-hyperbolic phase-fiel system with singular potentials, Math. Nachr., to appear. [9] M. Grasselli an V. Pata, On the ampe semilinear wave equation with critical exponent, Dynamical systems an ifferential equations (Wilmington, NC, 2002). Discrete Contin. Dynam. Systems, (suppl.) (2003), [0] M. Grasselli an V. Pata, Asymptotic behavior of a parabolic-hyperbolic system, Commun. Pure Appl. Anal., 3 (2004), [] J.K. Hale, Asymptotic behavior an ynamics in infinite imensions, in Nonlinear Differential Equations (J.K. Hale an P. Martìnez-Amores Es), Pitman, Boston, 985. [2] J.K. Hale, Asymptotic behavior of issipative systems, Amer. Math. Soc., Provience, 988. [3] J.K. Hale an G. Raugel, Upper semicontinuity of the attractor for a singularly perturbe hyperbolic equation, J. Differential Equations, 73 (988), [4] A. Haraux, Two remarks on issipative hyperbolic problems, in Séminaire u Collége e France (J.-L. Lions E.) Pitman, Boston, 985. [5] A. Haraux an M. Jenoubi Convergence of boune weak solutions of the wave equation with issipation an analytic nonlinearity, Calc. Var. Partial Differential Equations, 9 (999), [6] M. Jenoubi Convergence of global an boune solutions of the wave equation with linear issipation an analytic nonlinearity, J. Differential Equations, 44 (998), [7] O.A. Layzhenskaya, Attractors of nonlinear evolution problems with issipation, J. Soviet Math., 40 (988), [8] V. Pata an S. Zelik Attractors an their regularity for 2-D wave equations with nonlinear amping, submitte. [9] R. Temam, Infinite-imensional ynamical systems in mechanics an physics, Springer, New York, 988. [20] S. Zelik, Asymptotic regularity of solutions of a nonautonomous ampe wave equation with a critical growth exponent, Commun. Pure Appl. Anal., 3 (2004), [2] S. Zelik, Asymptotic regularity of solutions of singularly perturbe ampe wave equations with supercritical nonlinearities, Discrete Contin. Dynam. Systems, (2004), Receive August 2005; revise December aress: pata@mate.polimi.it aress: zelik@wias-berlin.e