Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions

Transcription

1 Local existence and exponential growth for a seilinear daped wave equation with dynaic boundary conditions Stéphane Gerbi, Belkace Said-Houari To cite this version: Stéphane Gerbi, Belkace Said-Houari. Local existence and exponential growth for a seilinear daped wave equation with dynaic boundary conditions. Advances in Differential Equations, Khayya Publishing, 28, 3-2, pp <hal > HAL Id: hal Subitted on 6 Oct 28 HAL is a ulti-disciplinary open access archive for the deposit and disseination of scientific research docuents, whether they are published or not. The docuents ay coe fro teaching and research institutions in France or abroad, or fro public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de docuents scientifiques de niveau recherche, publiés ou non, éanant des établisseents d enseigneent et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Local existence and exponential growth for a seilinear daped wave equation with dynaic boundary conditions Stéphane Gerbi and Belkace Said-Houari Abstract In this paper we consider a ulti-diensional daped seiliear wave equation with dynaic boundary conditions, related to the Kelvin-Voigt daping. We firstly prove the local existence by using the Faedo-Galerkin approxiations cobined with a contraction apping theore. Secondly, the exponential growth of the energy and the L p nor of the solution is presented. AMS Subject classification : 35L45, 35L7, 35B4. Keywords : Daped wave equations, Kelvin-Voigt daping, dynaic boundary conditions, local existence, Faedo-Galerkin approxiation, exponential growth. Introduction In this paper we consider the following seilinear daped wave equation with dynaic boundary conditions: u tt u α u t = u p 2 u, x, t > ux, t =, x Γ, t > [ u u tt x, t = ν x, t + α u ] t ν x, t + r u t 2 u t x, t x, t > ux, = u x, u t x, = u x x. where u = ux, t, t, x, denotes the Laplacian operator with respect to the x variable, is a regular and bounded doain of R N, N, Laboratoire de Mathéatiques, Université de Savoie, Le Bourget du Lac, France, e-ail:stephane.gerbi@univ-savoie.fr, corrresponding author. Laboratoire de Mathéatiques Appliquées, Université Badji Mokhtar, B.P. 2 Annaba 23, Algérie, e-ail:saidhouarib@yahoo.fr

3 = Γ, esγ >, Γ = and denotes the unit outer ν noral derivative, 2, a, α and r are positive constants, p > 2 and u, u are given functions. Fro the atheatical point of view, these probles do not neglect acceleration ters on the boundary. Such type of boundary conditions are usually called dynaic boundary conditions. They are not only iportant fro the theoretical point of view but also arise in several physical applications. In one space diension, the proble can odelize the dynaic evolution of a viscoelastic rod that is fixed at one end and has a tip ass attached to its free end. The dynaic boundary conditions represents the Newton s law for the attached ass, see [5, 2, ] for ore details. In the two diension space, as showed in [26] and in the references therein, these boundary conditions arise when we consider the transverse otion of a flexible ebrane whose boundary ay be affected by the vibrations only in a region. Also soe dynaic boundary conditions as in proble appear when we assue that is an exterior doain of R 3 in which hoogeneous fluid is at rest except for sound waves. Each point of the boundary is subjected to sall noral displaceents into the obstacle see [3] for ore details. This type of dynaic boundary conditions are known as acoustic boundary conditions. In the one diensional case and for r =, that is in the absence of boundary daping, this proble has been considered by Grobbelaar-Van Dalsen [6]. By using the theory of B-evolutions and the theory of fractional powers developped in [27, 28], the author showed that the partial differential equations in the proble gives rise to an analytic seigroup in an appropriate functional space. As a consequence, the existence and the uniqueness of solutions was obtained. In the case where r and = 2, Pellicer and Solà-Morales [25] considered the one diensional proble as an alternative odel for the classical spring-ass daper syste, and by using the doinant eigenvalues ethod, they proved that for sall values of the paraeter a the partial differential equations in the proble has the classical second order differential equation u t + d u t + k ut =, as a liit where the paraeter, d and k are deterined fro the values of the spring-ass daper syste. Thus, the asyptotic stability of the odel has been deterined as a consequence of this liit. But they did not obtain any rate of convergence. We recall that the presence of the strong daping ter u t in the proble akes the proble different fro that considered in [5] and widely studied in the litterature [32, 29, 3, 4, 3] for instance. For this reason less results were known for the wave equation with a strong daping and any probles reained unsolved, specially the blow-up of solutions in the presence of a strong daping and nonlinear daping at the sae tie. Here we will give a partial answer to 2

4 this question. That is to say, we will prove that the solution is unbounded and grows up exponentially when tie goes to infinity. Recently, Gazzola and Squassina [4] studied the global solution and the finite tie blow-up for a daped seilinear wave equations with Dirichlet boundary conditions by a careful study of the stationnary solutions and their stability using the Nehari anifold and a ountain pass energy level of the initial condition. The ain difficulty of the proble considered is related to the non ordinary boundary conditions defined on. Very little attention has been paid to this type of boundary conditions. We ention only a few particular results in the one diensional space and for a linear daping i.e. = 2 [8, 25, 2]. A related proble to is the following: u tt u + gu t = f in, T u ν + Kuu tt + hu t =, on, T ux, = u x in u t x, = u x in where the boundary ter hu t = u t ρ u t arises when one studies flows of gaz in a channel with porous walls. The ter u tt on the boundary appears fro the internal forces, and the nonlinearity Kuu tt on the boundary represents the internal forces when the density of the ediu depends on the displaceent. This proble has been studied in [2, 3]. By using the Fadeo-Galerkin approxiations and a copactness arguent they proved the global existence and the exponential decay of the solution of the proble. We recall soe results related to the interaction of an elastic ediu with rigid ass. By using the classical seigroup theory, Littan and Markus [2] established a uniqueness result for a particular Euler-Bernoulli bea rigid body structure. They also proved the asyptotic stability of the structure by using the feedback boundary daping. In [22] the authors considered the Euler-Bernoulli bea equation which describes the dynaics of claped elastic bea in which one segent of the bea is ade with viscoelastic aterial and the other of elastic aterial. By cobining the frequency doain ethod with the ultiplier technique, they proved the exponential decay for the transversal otion but not for the longitudinal otion of the odel, when the Kelvin-Voigt daping is distributed only on a subinterval of the doain. In relation with this point, see also the work by Chen et al. [9] concerning the Euler-Bernoulli bea equation with the global or local Kelvin-Voigt daping. Also odels of vibrating strings with local viscoelasticity and Boltzann daping, instead of the Kelvin-Voigt one, were considered in [23] and an exponential energy decay rate was established. Recently, Grobbelaar-Van Dalsen [7] considered an extensible thero-elastic bea which is hanged at one end with rigid body attached to its free end, i.e. one diensional hybrid theroelastic structure, and showed that the ethod 3

5 used in [24] is still valid to establish an unifor stabilization of the syste. Concerning the controllability of the hybrid syste we refer to the work by Castro and Zuazua[6], in which they considered flexible beas connected by point ass and the odel takes account of the rotational inertia. In this paper we consider the proble where we have set for the sake of siplity a =. Section 2 is devoted to the local existence and uniqueness of the solution of the proble. We will use a technique close to the one used by Georgiev and Todorova in [5] and Vitillaro in [33, 34]: a Faedo-Galerkin approxiation coupled to a fix point theore. In section 3, we shall prove that the energy is unbounded when the initial data are large enough. In fact, it will be proved that the L p -nor of the solutions grows as an exponential function. An essential ingredient of the proof is a lower bound in the L p nor and the H seinor of the solution when the initial data are large enough, obtained by Vitillaro in [32]. The other ingredient is the use of an auxillary function L which is a sall perturbation of the energy in order to obtain a linear differential inequality, that we integrate to finally prove that the energy is exponentially growing. To this end, we use Young s inequality with suitable coefficient, interpolation and Poincaré s inequalities. Let us recall that the blow-up result in the case of a nonlinear daping 2 is still an open proble. 2 Local existence In this section we will prove the local existence and the uniqueness of the solution of the proble. We will adapt the ideas used by Georgiev and Todorova in [5], which consists in constructing approxiations by the Faedo-Galerkin procedure in order to use the contraction apping theore. This ethod allows us to consider less restrictions on the initial data. Consequently, the sae result can be established by using the Faedo-Galerkin approxiation ethod coupled with the potential well ethod [7]. 2. Setup and notations We present here soe aterial that we shall use in order to prove the local existence of the solution of proble. We denote H = { u H / u Γ = }. By.,. we denote the scalar product in L 2 i.e. u, vt = ux, tvx, tdx. Also we ean by. q the L q nor for q, and by. q,γ the L q nor. 4

6 Let T > be a real nuber and X a Banach space endowed with nor. X. L p, T; X, p < denotes the space of functions f which are L p over, T with values in X, which are easurable and f X L p, T. This space is a Banach space endowed with the nor T /p f L p,t;x = f p X dt. L, T; X denotes the space of functions f : ], T[ X which are easurable and f X L, T. This space is a Banach space endowed with the nor: f L,T;X = ess sup f X. <t<t We recall that if X and Y are two Banach spaces such that X Y continuous ebedding, then L p, T; X L p, T; Y, p. We will also use the ebedding see [, Therore 5.8]. 2N H L q, 2 q q where q = N 2, if N 3 +, if N =, 2. Let us denote V = H L. In this work, we cannot use directly the existence result of Georgiev and Todorova [5] nor the results of Vitillaro [33, 34] because of the presence of the strong linear daping u t and the dynaic boundary conditions on. Therefore, we have the next local existence theore. q Theore 2. Let 2 p q and ax 2, q. q + p Then given u H and u L 2, there exists T > and a unique solution u of the proble on, T such that u C [, T], H C [, T], L 2, u t L, 2 T; H L, T We will prove this theore by using the Fadeo-Galerkin approxiations and the well-known contraction apping theore. In order to define the function for which a fixed point exists, we will consider first a related proble. For u C [, T], H Γ C [, T], L 2 given, let us consider the following 5

7 proble: v tt v α v t = u p 2 u, x, t > vx, t =, x Γ, t > [ v v tt x, t = ν x, t + α v ] t ν x, t + r v t 2 v t x, t x, t > vx, = u x, v t x, = u x x. We have now to state the following existence result: q Lea 2. Let 2 p q and ax 2, q. Then given q + p u H, u L 2 there exists T > and a unique solution v of the proble 2 on, T such that v C [, T], H C [, T], L 2, v t L, 2 T; H L, T and satisfies the energy identity: [ ] v v t v t 2 t 2, s for s t T. + α = t s t s t v t τ 2 2dτ + r v t τ, dτ s uτ p 2 uτv t τdτdx In order to prove lea 2., we first study for any T > and f H, T; L 2 the following proble: v tt v α v t = fx, t, x, t > vx, t =, x Γ, t > [ v v tt x, t = ν x, t + α v ] t ν x, t + r v 3 t 2 v t x, t x, t > vx, = u x, v t x, = u x x. At this point, as done by Doronin et al. [3], we have to precise exactly what type of solutions of the proble 3 we expected. Definition 2. A function vx, t such that v L, T; H Γ, v t L 2, T; H Γ L, T, v t L, T; H Γ L, T; L 2, v tt L, T; L 2 L, T; L 2, vx, = u x, v t x, = u x, 6 2

8 is a generalized solution to the proble 3 if for any function ω H Γ L and ϕ C, T with ϕt =, we have the following identity: T f, wt ϕt dt = + T T [ ] v tt, wt + v, wt + α v t, wt ϕt dt [ ] ϕt v tt t + r v t t 2 v t t w dσ dt. Lea 2.2 Let 2 p q and 2 q. Let u H 2 V, u H 2 and f H, T; L 2, then for any T >, there exists a unique generalized solution in the sense of definition 2., vt, x of proble Proof of the lea 2.2 To prove the above lea, we will use the Faedo-Galerkin ethod, which consists in constructing approxiations of the solution, then we obtain a priori estiates necessary to guarantee the convergence of these approxiations. It appears soe difficulties in order to derive a second order estiate of v tt. To get rid of the, and inspired by the ideas of Doronin and Larkin in [2] and Cavalcanti et al. [8], we introduce the following change of variables: ṽt, x = vt, x φt, x with φt, x = u x + t u x. Consequently, we have the following proble with the unknown ṽt, x and null initial conditions: ṽ tt ṽ α ṽ t = fx, t + φ + α φ t, x, t > ṽx, t =, x Γ, t > [ ṽ + φ ṽ tt x, t = x, t + α ṽ t + φ t ] x, t ν ν r ṽ t + φ t 2 ṽ t + φ t x, t x, t > ṽx, =, ṽ t x, = x. 4 Reark 2. It is quite clear that if ṽ is a solution of proble 4 on [, T], then v is a solution of proble 3 on [, T]. Moreover writing the proble in ter of ṽ shows exactly the regularity needed on the initial conditions u and u to ensure the existence. Now we construct approxiations of the solution ṽ by the Faedo-Galerkin ethod as follows. For every n, let W n = span{ω,..., ω n }, where {ω j x} j n is a basis in the space V. By using the Grah-Schidt orthogonalization process we can take 7

9 ω = ω,...,ω n to be orthonoral in L 2 L 2. We define the approxiations: n ṽ n t = g jn tw j 5 j= where ṽ n t are solutions to the finite diensional Cauchy proble written in noral for since ω is an orthonoral basis: ṽ ttn tw j dx + ṽ n + φ w j + α ṽ n + φ w t j dx + ṽttn t + r ṽ n + φ t 2 ṽ n + φ t wj dσ = fw j dx. 6 g jn = g jn =, j =,...,n According to the Caratheodory theore, see [], the proble 6 has solution g jn t j=,n H 3, t n defined on [, t n. We need now to show: firstly that forall n N,, t n = T, secondly that these approxiations converge to a solution of the proble 4. To do this we need the two following a priori estiates: first-order a priori estiates to prove the first point. But we will show that the presence of the nonlinear ter u t 2 u t forces us to derive a second order a priori estiate to pass to the liit in the nonlinear ter. Indeed the key tool in our proof is the Aubin-Lions lea which uses the copactness of the ebedding H 2 L First order a priori estiates Multiplying equation 6 by g jn t, integrating over, t and using integration by parts we get: for every n, 2 + α + r t [ ] ṽn t ṽ tn t ṽ tn 2 2, + t t t φ t ṽ tn dx + α ṽ tn s 2 2 ds ṽ n + φ t 2 ṽ n + φ t ṽ tn dσds t = ft, xṽ tn s dxds φ ṽ n dx Unfortunately, the presence of the nonlinear boundary conditions excludes us to use the spatial basis of eigenfunctions of in H as done in [4] 7 8

10 By using Young s inequality, there exists δ >, in fact sall enough such that t α φ t ṽ tn dx δ t ṽ tn 2 dx + t 4δ φ t 2 dx 8 and t t φ ṽ n dx δ ṽ n 2 dx + t φ 2 dx. 9 4δ By Young s and Poincaré s inequalities, we can find C >, such that t ft, xṽ tn sdxds C t f 2 + ṽ tn s 2 dxds. The last ter in the left hand side of equation 7 can be written as follows: = t t ṽ n + φ t 2 ṽ n + φ t ṽ tndσds t ṽ n + φ t dσds ṽ n + φ t 2 ṽ n + φ t φ t dσds, Then Young s inequality gives us, for δ 2 > : t ṽ n + φ t 2 ṽ n + φ t φ t dσds t δ 2 ṽ n + φ t t dσds + δ / 2 φ t dσds. Consequently, using the inequalities 8, 9, and in the equation 7, choosing δ and δ 2 sall enough, we ay conclude that: [ ] ṽn t ṽ tn t ṽ tn t 2 2,Γ 2 t t +α ṽ tn s 2 2 ds + r ṽ n + φ t dσds C T, 2 where C T is a positive constant independent of n. Therefore, the last estiate 2 gives us, n N, t n = T, and: ṽ n n N is bounded in L, T; H Γ, 3 9

11 ṽ tn n N is bounded in L, T; L 2 L 2, T; H Γ Now, by using the following algebraic inequality: L, T; L 2. 4 we can find c, c 2 >, such that: t A + B λ 2 λ A λ + B λ, A, B, λ, 5 ṽ n + φ t dσds c t ṽ tn dσds c 2 t φ t dσds. 6 Then, by the ebedding H 2 L 2 q, we conclude that u L. Therefore, fro the inequalities 2 and 6, there exists C T > such that: t ṽ tn dσds C T. Consequently, Second order a priori estiate ṽ tn is bounded in L, T. 7 In order to obtain a second a priori estiate, we will first estiate ṽ ttn 2 2 and ṽ ttn 2 2,. For this purpose, considering w j = ṽ ttn and t = in the equation 6, we get ṽ ttn ṽ ttn 2 2, + φ ṽ ttn dx = +α φ t ṽ ttn dx + r f, xṽ ttn dxds. Since the following equalities hold: φ = u, φ t = u, φ t ṽ ttn = φ t 2 φ t ṽ ttndσds 8 v ttn φ t ṽ ttn + φ t ν dσ, as f H, T; L 2 and u, u H 2, by using Young s inequality and the ebedding H 2 L, we conclude that there exists C > independent of n such that: ṽ ttn ṽ ttn 2 2, C. 9

12 Differentiating equation 6 with respect to t, ultiplying the result by g jnt and suing over j we get: d [ ṽtn t dt + ṽ ttnt ṽ ] ttnt 2 2, + φ t ṽ ttn dx +α ṽ ttn s r ṽ n + φ t 2 ṽ n + φ tt ṽ ttn dσ 2 = f t t, xṽ ttn sdxds. Since φ tt =, the last ter in the left hand side of the equation 2 can be written as follows: ṽ n + φ t 2 ṽ n + φ ttṽ ttndσ = ṽ n + φ t 2 ṽ ttn + φ tt 2 dσ, But we have, ṽ n + φ t 2 ṽ ttn + φ tt 2 dσ = ṽ tn t + φ t 2 2 ṽ tn t + φ t dσ. t Now, integrating equation 2 over, t, using the inequality 9 and Young s and Poincaré s inequalities as in, there exists C T > such that: ] [ ṽ tn t 22 + ṽ ttn t 22 + ṽ ttn t 22,Γ +α 2 4r + 2 Consequently,we deduce the following results: t ṽ ttn s ṽ tn t + φ t 2 2 ṽ tn t + φ t dσ t C T. ṽ ttn t n N is bounded in L, T; L 2, ṽ ttn t n N is bounded in L, T; L 2, ṽ tn t n N is bounded in L, T; H Γ. 2 Fro 3, 4, 7 and 2, we have ṽ n n N is bounded in L, T; H Γ. Then ṽ n n N is bounded in L 2, T; H Γ. Since ṽ tn n N is bounded in L, T; L 2, ṽ tn n N is bounded in L 2, T; L 2. Consequently ṽ n n N is bounded in H, T; H. Since the ebedding H, T; H L 2, T; L 2 is copact, by using Aubin-Lions theore, we can extract a subsequence ṽ µ µ N of ṽ n n N such that ṽ µ ṽ strongly in L 2, T; L 2.

13 Therefore, Following [9, Lee 3.], we get: ṽ µ ṽ strongly and a.e on, T. ṽ µ p 2 ṽ µ ṽ p 2 ṽ strongly and a.e on, T. On the other hand, we already have proved in the preceding section that: ṽ tn n N is bounded in L, T; L 2 Fro 3 and 2, since ṽ n t H 2 Γ C ṽ nt 2 and ṽ tn t H 2 Γ C ṽ tnt 2 we deduce that: ṽ n n N is bounded in L, 2 T; H 2 Γ ṽ tn n N is bounded in L, 2 T; H 2 Γ ṽ ttn n N is bounded in L 2, T; L 2. Since the ebedding H 2 L 2 is copact, again by using Aubin-Lions theore, we conclude that we can extract a subsequence also denoted ṽ µ µ N of ṽ n n N such that: Therefore fro 7, we obtain that: ṽ tµ ṽ t strongly in L 2, T; L ṽ tµ 2 ṽ tµ χ weakly in L, T, It suffices to prove now that χ = ṽ t 2 ṽ t. Clearly, fro 22 we get: ṽ tµ 2 ṽ tµ ṽ t 2 ṽ t strongly and a.e on, T. Again, by using the Lions s lea, [9, Lee.3], we obtain χ = ṽ t 2 ṽ t. The proof now can be copleted arguing as in [9, Théorèe 3.] Uniqueness Let v, w two solutions of the proble 3 which share the sae initial data. Let us denote z = v w. It is staightforward to see that z satisfies: t +2r t z z t z t 2 2, + 2α z 2 2ds [ ] v t s 2 v t s w t s 2 w t s v t s w t sdsdσ =. 23 2

14 By using the algebraic inequality: [ ] 2, c >, a, b R, a 2 a b 2 b a b c a b 24 equation 23 yields to: t z t z z t 2 2, + 2α t +c z t 2 2ds v t s w t s dsdσ. Then, this last inequality yields to z =. This finishes the proof of the lea Proof of lea 2. We first approxiate u C [, T], H C [, T], L 2 endowed with the standard nor u = ax u tt 2 + ut H, by a sequence u k k N t [,T] C [, T] by a standard convolution arguents see [4]. It is clear that f u k = u k p 2 u k H, T; L 2. This type of approxiation has been already used by Vitillaro in [33, 34]. Next, we approxiate the initial data u L 2 by a sequence u k k N C. Finally, since the space H2 V H is dense in H for the H nor, we approxiate u H by a sequence u k k N H 2 V H. We consider now the set of the following probles: vtt k v k α vt k = u k p 2 u k, x, t > v k x, t =, x Γ, t > [ ] v k vtt k x, t = ν x, t + α vk t ν x, t + 25 r vk t 2 vt k x, t x, t > v k x, = u k, vk t x, = uk x. Since every hypothesis of lea 2.2 are verified, we can find a sequence of unique solution v k k N of the proble 25. Our goal now is to show that v k, vt k k N is a Cauchy sequence in the space { Y T = v, v t /v C [, T], H C [, T],L 2, v t L 2, T; H } L, T endowed with the nor [ ] v, v t 2 Y T = ax v t t T v v t 2 + L,T Γ 3 t v t s 2 2 ds.

15 For this purpose, we set U = u k u k, V = v k v k. It is straightforward to see that V satisfies: V tt V α V t = u k p 2 u k u k p 2 u k x, t > V x, t = x Γ, t > [ V V tt x, t = ν x, t + α V ] t ν x, t r vt k 2 vt k x, t vk t 2 vt k x, t x, t > V x, = u k uk, V tx, = u k uk x. We ultiply the above differential equations by V t, we integrate over, t and we use integration by parts to obtain: 2 t Vt V V t 2 2, + α t + r = 2 + t vt k 2 vt k vk t 2 vt k V t 2 2 ds v k t vk t Vt V V t 2 2, u k p 2 u k u k p 2 u k v k t vk t By using the algebraic inequality 24, we get: 2 t Vt V V t 2 2, + α dσds dxdτ, t, T. V t 2 2 ds + c V t, Vt V V t 2 2,Γ 2 t + u k p 2 u k u k p 2 u k vt k vk t dxdτ, In order to find a ajoration of the ter: t, T. t u k p 2 u k u k p 2 u k v k t vk t dxdτ, t, T in the previous inequality, we use the result of Georgiev and Todorova [5] specifically their equations 2.5 and 2.6 in proposition 2.. The hypothesis on p 4

16 ensures us to use exactly the sae arguent. Thus by applying Young s inequality and Gronwall inequality, there exists C depending only on and p such that: V YT C V t V V t 2 2, + CT U YT. Let us now reark that fro the notations used above, we have: V = u k u k, V t = u k u k and U = u k u k. Thus, since u k k N is a converging sequence in H Γ, u k k N is a converging sequence in L 2 and u k k N is a converging sequence in C [, T],H Γ C [, T],L 2 so in Y T also, we conclude that v k, v k t k N is a Cauchy sequence in Y T. Thus v k, v k t converges to a liit v, v t Y T. Now by the sae procedure used by Georgiev and Todorova in [5], we prove that this liit is a weak solution of the proble 2. This copletes the proof of the lea Proof of theore 2. In order to prove theore 2., we use the contraction apping theore. For T >, let us define the convex closed subset of Y T : Let us denote: X T = {v, v t Y T such that v = u, v t = u }. B R X T = {v X T ; v YT R}. Then, lea 2. iplies that for any u X T, we ay define v = Φ u the unique solution of 2 corresponding to u. Our goal now is to show that for a suitable T >, Φ is a contractive ap satisfying Φ B R X T B R X T. Let u B R X T and v = Φ u. Then for all t [, T] we have: t v t v v t 2 2, + 2 = u u u 2 2, + 2 t v t, ds + 2α t v t 2 2 ds u τ p 2 u τ v t τ dxdτ. 26 Using Hölder inequality, we can control the last ter in the right hand side of the inequality 26 as follows: t u τ p 2 u τ v t τ dxdτ t u τ p 2N/N 2 v t τ 2N/ 3N Np+2p dτ 5

17 Since p 2N 2N, we have N 2 3N Np + 2p 2N N 2 Thus, by Young s and Sobolev s inequalities, we get δ >, Cδ >, such that: t t t, T, u τ p 2 u τ v t τ dxdτ CδtR 2p + δ v t τ 2 2dτ. Inserting the last estiate in the inequality 26 and choosing δ sall enough in order to counter-balance the last ter of the left hand side of the inequality 26 we get: v 2 Y T 2 R2 + CTR 2p. Thus, for T sufficiently sall, we have v YT R. This shows that v B R X T. Next, we have to verify that Φ is a contraction. To this end, we set U = u ū and V = v v, where v = Φu and v = Φū are the solutions of proble 2 corresponding respectively to u and v. Consequently we have: V tt V α V t = u p 2 u ū p 2 ū x, t > V x, t = x Γ, t > [ V V tt x, t = ν x, t + α V ] t ν x, t r v t 2 v t x, t v t 2 v t x, t x, t > V x, =, V t x, = x.. 27 By ultiplying the differential equation 27 by V t and integrating over, t, we get: 2 t Vt V V t 2 2, + α t r = t V t 2 2 ds+ vt 2 v t v t 2 v t vt v t dσds u p 2 u ū p 2 ū v t v t dxdτ, t, T. Again, by using the algebraic inequality 24, we have: t Vt V V t 2 2, + α t V t 2 2 ds + c V t, u p 2 u ū p 2 ū v t v t dxdτ, t, T. 28 6

18 To estiate the ter in the right hand side of the inequality 28, let us denote: It := t u p 2 u ū p 2 ū v t v t dxdτ. Using the algebraic inequality: u p 2 u ū p 2 ū cp u ū u p 2 + ū p 2, which holds for any u, ū R, where c p is a positive constant depending only on p, we find: T It c p u ū u p 2 + ū p 2 V t dxdτ. Following the sae arguent as Vitillaro in [34, eq 77], choosing p < r < q such that: q q p + < r r p + <, let s > such that: + + r s =. Using Hölder s inequality we obtain: T /s It c p u ū r V t. u p 2 + ū p 2 s. 29 Therefore, the algebraic inequality 5 gives us: /s u p 2 + ū p 2 s /s 2 s u p 2s p 2s p 2s + ū p 2s. But since we get A + B β A β + B β, A, B and < β < u p 2 + ū p 2 s /s 2 s u p 2 p 2s + ū p 2 p 2s. 3 Consequently, inserting the inequality 29 in 3 and using Poincaré s inequality, we obtain: T It c 2 R p 2 u ū r V t 2 ds. 7

19 Applying Hölder s inequality in tie, we finally get: It c 2 R p 2 T /2 u ū L,T;L r c 2 2 Rp 2 T /2 T V t 2 2 /2 T u ū 2 L,T;L r + V t 2 2. Lastly, by choosing T sall enough in order to have: α c 2 2 Rp 2 T /2 >, we conclude by inserting the estiate 3 in the estiate 28 that: t Vt V V t 2 2,Γ 2 + α V t 2 2ds + c V t, c 2 2 Rp 2 T /2 u ū 2 L,T;L r. Since r < q, using the ebedding 3 32 in the estiate 32, we finally have: L, T; H Γ L, T; L r By choosing T sall enough in order to have V 2 Y T c 3 R p 2 T /2 U 2 Y T. 33 c 3 R p 2 T /2 <. the estiate 33 shows that Φ is a contraction. Consequently the contraction apping theore guarantees the existence of a unique v satisfying v = Φv. The proof of theore 2. is now copleted. Reark 2.2 To prove the existence and uniqueness of the solution to the ore general proble: u tt u α u t = fu, x, t > ux, t =, x Γ, t > [ u u tt x, t = ν x, t + α u ] t ν x, t + gu t x, t > ux, = u x, u t x, = u x x. we can use the sae ethod, provided that the functions f and g satisfy respectively the conditions H 3 H 7 and H 8 H 9 of the paper of Calvacanti et al. [8]. 8

20 3 Exponential growth In this section we consider the proble and we will prove that when the initial data are large enough in the energy point of view, the energy grows exponentially and thus so the L p nor. In order to state and prove the result, we introduce the following notations. Let B be the best constant of the ebedding H Lp defined by: B = inf { u 2 : u H, u p = }. We also define the energy functional: Eut = Et = 2 u 2 2 p u p p + 2 u t u t 2 2,. 34 Finally we define the following constant which will play an iportant role in the proof of our result: α = B p/p 2, and d = 2 p α2. 35 In order to obtain the exponential growth of the energy, we will use the following lea see Vitillaro [32], for the proof: Lea 3. Let u be a classical solution of. Assue that E < d and u 2 > α. Then there exists a constant α 2 > α such that u., t 2 α 2, t, 36 and u p Bα 2, t. 37 Let us now state our new result. Theore 3. Assue that < p where 2 < p q. Suppose that E < d and u 2 > α. Then the solution of proble growths exponentially in the L p nor. Proof: By setting Ht = d Et 38 we get fro the definition of the energy 34: [ < H Ht d 2 u t u t 2 2, + 2 u 2 2 ] p u p p, 39 9

21 using the fundaental estiate 36 and the equality 35, we get: d 2 u 2 2 < d 2 α2 = p α2 <, t. Hence we finally obtain the following inequality: < H Ht p u p p, t. For ε sall to be chosen later, we then define the auxillary function: Lt = Ht + ε u t udx + ε u t udσ + εα 2 u Let us reark that L is a sall perturbation of the energy. By taking the tie derivative of 4, we obtain: = α u t r u t, + ε u t εα u t udx dlt dt +ε u tt udx + ε Using proble, the equation 4 takes the for: dlt dt u tt udσ + ε u t 2 2,. 4 = α u t r u t, + ε u t 2 2 ε u 2 2 +ε u p p + ε u t 2 2, εr u t u t ux, tdσ. 42 To estiate the last ter in the right hand side of the previous equality, let δ > be chosen later. Young s inequality leads to: u t u t ux, tdσ δ u, + δ / u t,. This yields by substitution in 42: dlt dt α u t r u t, + ε u t 2 2 ε u 2 2 +ε u p p + ε u t 2 2, εr εr δ / u t,γ. δ u, 43 Let us recall the inequality concerning the continuity of the trace operator here and in the sequel, C denotes generic positive constant which ay change fro line to line: u,γ C u H s, 2

22 which holds for: and < s <, s N 2 N > and the interpolation and Poincaré s inequalities see [2] Thus, we have the following inequality: u H s C u s 2 u s 2 C u s p u s 2 u,γ C u s p u s 2. If s < 2/, using again Young s inequality, we get: [ sµ u, C u p p p + u 2 2 sθ 2 ] 44 for /µ + /θ =. Here we choose θ = 2/s, to get µ = 2/ 2 s. Therefore the previous inequality becoes: [ ] s2 u, C u p 2 sp p + u Now, choosing s such that: we get: < s 2 s 2 s p 2 p p 2,. 46 Once the inequality 46 is satisfied, we use the classical algebraic inequality: z ν z + + z + ω, z, < ν, ω, ω to obtain the following estiate: s2 u p 2 sp p D u pp + H D u pp + H t, t 47 2

23 where we have set D = + /H. Inserting the estiate 47 into 44 we obtain the following iportant inequality: ] u, C [ u pp + u 22 + H t. In order to control the ter u 2 2 in equation 43, we preferely use as Ht >, the following estiate: ] u, C [ u pp + u H t. which gives finally: u, C [ 2d ] u pp p u t 22 u t 22,Γ. 48 Consequently inserting the inequality 48 in the inequality 43 we have: dl t dt α u t ε r δ / r u t,γ +ε + r C δ u t 2 2 ε u ε + 2 r C δ u pp p + ε + r C δ u t 2 2,Γ Fro the inequality 39 we have: u 2 2 2Ht 2d + u t u t 2 2, 2 p u p p. Thus inserting it in 49, we get the following inequality: dl t dt α u t ε r δ / r u t,γ +ε 2 + r C δ u t ε + r C δ u t 2 2,Γ 5 +ε 2ε p + 2 r C δ u p p p +2 ε Ht d + r C δ 22

24 Finally, using the definition of α 2 and d see equation 35 and the lea 3., we obtain: dl t α u t 2 2 dt + ε r δ / r u t,γ +ε 2 + r C δ u t ε + r C δ u t 2 2,Γ 5 +ε 2 [ p 2d Bα 2 p + 2 ] r C δ + 4d Bα 2 p p u p p }{{} :=c +ε 2Ht + r C δ d. Setting c = 2 p 2d Bα 2 p, we have c > since α 2 > B p/p 2. We choose now δ sall enough such that: [ c + 2 ] r C δ + 4d Bα 2 p p >. Once δ is fixed, we choose ε sall enough such that: r Therefore, the inequality 5 becoes: dl t dt εr δ / > and L >. ] εη [H t + u t 22 + u t 22,Γ + u pp + d for soe η > 52 Next, it is clear that, by Young s inequality and Poincaré s inequality, we get [ ] L t γ H t + u t u t 2 2, + u 2 2 for soe γ >. 53 Since Ht >, we have: t >, 2 u 2 2 p u p p + d. Thus, the inequality 53 becoes: ] L t ζ [Ht + u t 22 + u t 22,Γ + u pp + d, for soe ζ >. 54 Fro the two inequalities 52 and 54, we finally obtain the differential inequality: dl t µl t, for soe µ >. 55 dt 23

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