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Abstract and Applied Analysis On the strongly damped wave equation and the heat equation with mixed boundary conditions
On the strongly damped wave equation and the heat equation with mixed boundary conditions
Neves, Aloisio F.Quanto ti piace questo libro?
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Volume:
5
Anno:
2000
Lingua:
english
Rivista:
Abstract and Applied Analysis
DOI:
10.1155/s1085337500000348
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PDF, 1,86 MB
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ON THE STRONGLY DAMPED WAVE EQUATION AND THE HEAT EQUATION WITH MIXED BOUNDARY CONDITIONS ALOISIO F. NEVES Received 10 October 2000 We study two onedimensional equations: the strongly damped wave equation and the heat equation, both with mixed boundary conditions. We prove the existence of global strong solutions and the existence of compact global attractors for these equations in two different spaces. 1. Introduction In this paper, we study existence of strong solutions and existence of global compact attractors for the following onedimensional problems. The strongly damped wave equation, utt − uxx − utxx = g(t), u(t, 0) = 0, 0<x < , 0<t <T; ux (t, ) + utx (t, ) = ρ ut (t, ) , (1.1) and the heat equation zt − zxx + G(z) = h(t), z(t, 0) = 0, zx (t, ) = ρ z(t, ) . (1.2) Here and T are positive constants, ρ : R → R is a nonincreasing and bounded function, g, h ∈ L1 (0, T ; L2 (0, )), and G is an operator from a subspace of H 1 into L2 . In the case where ρ is not continuous, we will understand ρ(x0 ), at a point of discontinuity x0 , as being the whole interval [ρ(x0 +0), ρ(x0 −0)]. In this case ρ will be a multivalued function, and the “equal signs” in the last equations of (1.1) and (1.2) will be changed to “belong signs.” So, the boundary conditions at x = will be written, respectively, as ux (t, ) + utx (t, ) ∈ ρ ut (t, ) , zx (t, ) ∈ ρ z(t, ) , Copyright © 2000 Hindawi Publishing Corporation Abstract and Applied Analysis 5:3 (2000) 175–189 2000 Mathematics Subject Classification: 47H06, 35B65, 35B41 URL: http://aaa.hindawi.com/volume5/S1085337500000348.html (1.3) 176 Strongly damped wave and heat equations or equivalently, ut (t, ), ux (t, ) + utx (t, ) ∈ , z(t, ), zx (t, ) ∈ , (1.4) where is the graph of the multivalued function ρ. The existence of global solutions for these two problems can be obtained using the theory of monotone operators. The problem (1.2) gives rise to a maximal monotone operator A that is of subdifferential type, A =; ∂ϕ, where ϕ is a lower semicontinuous and convex functional. This problem was studied in [1] under some conditions on G, in particular the existence of strong solutions was proved. Our goal is to obtain existence of global compact attractor. To reach this goal, first of all, we will obtain a relation between the solutions of the two problems. With this relation we can use one problem to get the properties of the other, in particular this relation will be used to prove the existence of strong solutions for the problem (1.1). Once we have existence of solutions, we will start working in order to get the existence of the attractors. For our purpose, we will study the problem (1.2) in two different spaces L2 and H 1 and using the relation between the solutions we will prove the existence of attractors for the problems. More specifically, setting ut = v, where u(t) is solution operator given by (1.1), we will study the evolution of three operators, z(t) given by (1.2), in the spaces L2 and H 1 , u(t) + v(t) in the space H 1 and v(t) in the space L2 . In order to obtain the results we will use the following procedures: to prove the bounded dissipativeness of the problem (1.1) we will construct an appropriate equivalent norm in the space. The bounded dissipativeness of (1.2) in H 1 will be obtained using the uniform Gronwall lemma with some appropriate estimates. The proof of the compactness of the operators will be done using arguments of AubinLion’s type. Asymptotic behavior of parabolic equations with monotone principal part was recently studied by Carvalho and Gentile in [3], the main difference with our case, problem (1.2), is that our functional ϕ is not equivalent to the norm of the space. 2. Abstract formulation and existence of solutions As usual in wave equations context, setting v = ut , (1.1) can be seen as a system: ut = v, vt = (u + v)xx + g, u(t, 0) = 0, 0<x < , 0<t <T; ux (t, ) + vx (t, ) = ρ v(t, ) . (2.1) Therefore, our problem (1.1) can be viewed as an evolution equation ẇ + Aw = f (t) (2.2) in the Hilbert space Ᏼ = H1,0 × L2 (0, ), H1,0 = u ∈ H 1 (0, ) : u(0) = 0 , (2.3) with the inner product u 1 , v1 , u 2 , v 2 Ᏼ = 0 u1 u2 + v1 v2 dx, (2.4) Aloisio F. Neves where f (t) = is given by 177 0 , g(t) A : Ᏸ(A) ⊂ Ᏼ :−→ Ᏼ, A(u, v) = − v, −(u + v) (2.5) (2.6) on the domain Ᏸ(A) = (u, v) ∈ H1,0 × H1,0 : (u + v) ∈ H 2 (0, ) and (u + v) ( ) ∈ ρ v( ) . (2.7) Throughout the paper we denote by ·, · and  ·  the usual inner product and norm of L2 , respectively. We use the terminology of Brézis [2] and Hale [4] Lemma 2.1. The operator A is maximal monotone. Proof. If w1 = (u1 , v1 ) and w2 = (u2 , v2 ) are in Ᏸ(A), we have by integrating by parts that w1 − w2 , Aw1 − Aw2 (2.8) 2 = − v 1 ( ) − v 2 ( ) u1 + v 1 ( ) − u 2 + v 2 ( ) + v1 − v2 dx. 0 Since ρ is nonincreasing and (ui + vi ) ( ) ∈ ρ(vi ( )), i = 1, 2, we have w1 − w2 , Aw1 − Aw2 ≥ 0, (2.9) therefore, A is a monotone operator. We prove that A is maximal by showing that R(I + A) = Ᏼ. In fact, if (f, g) ∈ Ᏼ we consider z as being the unique solution of the ODE problem: z − 2z = f + 2g := h ∈ L2 (0, ), z(0) = 0, z (0) = a ∈ R, (2.10) where a is chosen conveniently. Since z ∈ H 2 (0, ) ∩ H1,0 and f ∈ H1,0 , setting 1 u = (z + f ), 2 1 v = (z − f ), 2 (2.11) we have that u, v ∈ H1,0 , u + v = z ∈ H 2 (0, ), and u − v = f, v − (u + v) = g. (2.12) Therefore, it remains to be proved that (u + v) ( ) ∈ ρ(v( )) or equivalently z ( ) ∈ ρ̃(z( )), where 1 ρ̃(x) = ρ x −f ( ) . (2.13) 2 178 Strongly damped wave and heat equations We obtain that condition by choosing the constant a appropriately. Setting 0 1 , M = 1 0 2 we have from the variation constant formula 1 z( ) M 0 − = ae e( 1 2 0 z( ) Since √ −s)M (2.14) 0 ds. h(s) (2.15) 2 sinh √ 0 2 , eM = (2.16) 1 cosh √ 2 we have that the righthand side of (2.15) is a straight line in plane, parametrized by a, with positive slope. Therefore, there will be a unique a that gives the intersection with the nonincreasing graph of ρ̃. The lemma is proved. Solutions of abstract evolution equations will be considered in the sense of Brézis [2], that is we have the following definition. Definition 2.2. Let f be in L1 (0, T ; Ᏼ). A continuous function w : [0, T ] → Ᏼ is a solution (or strong solution) of ẇ(t) + Aw(t) = f (t) (2.17) if w satisfies (i) w(t) ∈ Ᏸ(A), ∀t ∈ (0, T ), (ii) w(t) is absolutely continuous (AC) on every compact set K ⊂ (0, T ) (therefore ẇ(t) exists a.e. in (0, T )), (iii) ẇ(t) + A(w(t)) = f (t), a.e. in (0, T ). Moreover, w ∈ C([0, T ]; Ᏼ) is a weak solution of (2.17) if there exist sequences (fn ) ∈ L1 (0, T ; Ᏼ) and (wn ) ∈ C([0, T ]; Ᏼ) such that wn are strong solutions of (2.18) w˙n (t) + A wn (t) = fn (t), fn → f in L1 (0, T ; Ᏼ), and wn → w uniformly in [0, T ]. We have from Theorem 3.4 of [2] the existence of weak solution for the problem (2.1). In order to prove that this weak solution is in fact strong, we will look for a relation between the solutions of (2.1) and the solutions of (1.2). The problem (1.2) was studied in [1], where G is an operator G : H1,0 −→ L2 (0, ) (2.19) Aloisio F. Neves 179 not necessarily local and h ∈ L2 (0, T ; L2 (0, )). The problem can be written as the abstract evolution problem in L2 (0, ) ż + Ꮽz = F (t, z), (2.20) where F (t, z) = −G(z) + h(t) and Ꮽ : Ᏸ(Ꮽ) ⊂ L2 (0, ) → L2 (0, ) is the operator given by Ꮽz = −z (2.21) on the domain Ᏸ(Ꮽ) = z ∈ H1,0 ∩ H 2 (0, ) : z ( ) ∈ ρ z( ) . (2.22) From Lemmas 2.1 and 2.2 of [1] we have that the operator Ꮽ is strongly monotone, that is, 2 Ꮽz1 − Ꮽz2 , z1 − z2 ≥ z1 − z2 (2.23) and of subdifferential type, Ꮽ = ∂ϕ, where ϕ : L2 (0, ) → R ∪ {+∞} is a proper, convex, and lower semicontinuous function defined by 1 z (x)2 dx if z ∈ H1,0 , p z( ) + (2.24) ϕ(z) = 2 0 +∞ otherwise, where p is given by z p(z) = −ρ(s) ds. (2.25) 0 We should observe that ϕ may assume negative values, but the following estimate is true: 2 z ≤ k1 ϕ(z) + k2 , ∀z ∈ H1,0 , (2.26) where k1 , k2 are constants; in particular ϕ is bounded below. Indeed, since ρ(s) is bounded (by a constant k), we have for z ∈ H1,0 z (x) dx, p z( ) ≥ −kz( ) = −k z (x) dx ≥ −k 0 and then 0 1 z (x)2 − k 2 dx ≤ 4 0 (2.27) 0 1 z (x)2 − k z (x) dx ≤ ϕ(z) 2 (2.28) implies the estimate (2.26). When G is Lipschitz continuous and h ∈ L2 (0, T ; L2 (0, )), it was proved, [1, Theorems 3.2 and 4.1], that the solutions of (1.2) are strong, in particular z(t) ∈ Ᏸ(Ꮽ), for every t ∈ (0, T ). Moreover, from Theorem 3.6 of [2] the solution z satisfies √ dz t (t) ∈ L2 0, T ; L2 (0, ) dt (2.29) 180 Strongly damped wave and heat equations and when z(0) ∈ Ᏸ(ϕ) = H1,0 , dz (t) ∈ L2 0, T ; L2 (0, ) . dt (2.30) Consider the following relations between the problems (2.1) and (1.2): z(t, x) = u(t, x) + v(t, x) − u(t, )ξ(x), (2.31) G(z) = z( )ξ, (2.32) h(t, x) = g(t, x) + v(t, x) + u(t, )ξ (x), (2.33) where ξ : [0, ] → R is a smooth function satisfying ξ(0) = 0, ξ( ) = 1, and ξ ( ) = 0. The operator G, given in (2.32), can be considered as an operator from H 1 (0, ) with values in L2 (0, ), and also with values in H1,0 . In both of these cases G is Lipschitz continuous and satisfies G(z) ≤ c z , (2.34) since z (x) dx ≤ z L1 . z( ) = (2.35) 0 It is easy to see that if (u, v) is a solution of (2.1) then z, given by (2.31), is a solution of (1.2) with h given by (2.33) and with initial condition z(0) = u(0) + Sv(0). Conversely, if z is a solution of (1.2), we consider the problem in H1,0 given by du (t) + u(t) − J (t)u(t) = 0, dt u(0) = 0, (2.36) where J (t)u(t) = G(u(t)) + z(t). Since J (t) : H1,0 → H1,0 , for t > 0, is globally Lipschitz, this problem has existence and uniqueness of solutions, see [2, Theorem 1.4]. If u(t) is this unique solution, then considering v(t) given by the relation (2.31) and g by the relation (2.33) we have that (u, v) satisfies the problem (2.1) with u(0) = 0 and v(0) = z(0). Under these conditions we can prove the following result. Theorem 2.3. If g ∈ L2 (0, T ; L2 (0, )), then for every w0 = (u0 , v0 ) ∈ Ᏼ there exists a unique strong solution w = (u, v) ∈ C([0, T ]; Ᏼ) of (2.1) such that w(0) = w0 . Moreover, the solution w = (u, v) satisfies √ d t (u + v)(t) ∈ L2 0, T ; L2 (0, ) dt (2.37) d (u + v)(t) ∈ L2 0, T ; L2 (0, ) . dt (2.38) and, for v(0) ∈ H1,0 , Aloisio F. Neves 181 Proof. Since z, given by (2.31), is a strong solution of (1.2), in particular z(t) ∈ Ᏸ(Ꮽ), Ᏸ(Ꮽ) given in (2.22), for every t ∈ (0, T ). It is easy to see that (u, v) is a strong solution of (2.1). From (2.31) d dz (u + v) = + v(t, )ξ, dt dt v(t, ) = z(t, ), (2.39) and z(t, ) ∈ L2 (0, T ), according to the trace theorem for Lipschitz domain, [7, page 15], therefore (2.37) and (2.38) follow, respectively, from (2.29) and (2.30). The proof is complete. Although we are interested in studying the influence of the nonlinear boundary condition in the problems, we should observe that we have existence of strong solution in more general situations. In fact, we can consider utt − uxx − utxx + q t, x, u, ut = 0, 0 < x < , 0 < t < T ; (2.40) u(t, 0) = 0, ux (t, ) + utx (t, ) = ρ ut (t, ) , where (q1 ) the application (t, x) → q(t, x, w) belongs to L2 (0, T ; L2 (0, )), for every fixed w ∈ Ᏼ; (q2 ) there exists k > 0, such that q t, x, w1 −q t, x, w2 2 ≤ k w1 −w2 Ᏼ , ∀t ∈ [0, T ], ∀w1 , w2 ∈ Ᏼ. (2.41) L (0, ) This problem can be viewed as an abstract evolution equation in the Hilbert space Ᏼ ẇ + Aw + B(t, w) = 0, (2.42) where B : [0, T ] × Ᏼ → Ᏼ is given by B = (0, q). From the assumptions (q1 ) and (q2 ), we have that B satisfies (B1 ) for every w ∈ Ᏼ, the application t → B(t, w) belongs to L2 (0, T ; Ᏼ); (B2 ) there exists k > 0, such that B t, w1 − B t, w2 ≤ k w1 − w2 , ∀t ∈ [0, T ], ∀w1 , w2 ∈ Ᏼ. (2.43) (2.44) Under the above assumptions we have the following result. Theorem 2.4. For every w0 ∈ Ᏼ there exists a unique strong solution w ∈ C([0, T ]; Ᏼ) of (2.42) satisfying w(0) = w0 . Proof. We use the method of Brézis [2]. Since, for every w ∈ C([0, T ]; Ᏼ), B(t, w(t)) ∈ L2 (0, T ; Ᏼ), we can consider the sequence wn in C([0, T ]; Ᏼ), defined by w0 (t) = w0 and wn+1 is the weak solution of (2.45) ẇn+1 (t) + A wn+1 (t) = −B t, wn (t) , wn+1 (0) = w0 182 Strongly damped wave and heat equations which exists by Theorem 2.3. Using the first inequality of Lemma 3.1 of [2], we obtain wn+1 (t) − wn (t) ≤ t 0 B σ, wn (σ ) − B σ, wn−1 (σ ) dσ t ≤k 0 wn (σ ) − wn−1 (σ ) dσ, (2.46) therefore n wn+1 (t) − wn (t) ≤ (kt) w1 − w0 ∞ . (2.47) L n! Thus, the sequence wn converges uniformly to w in [0, T ], so w is a weak solution of ẇ(t) + A w(t) = −B t, w(t) , w(0) = w0 . (2.48) Now, since B(t, w(t)) = (0, q(t, ·, w(t))) and it is easy to see that q(t, ·, w(t)) ∈ L2 (0, T ; L2 (0, )), we have from Theorem 2.3 that w is a strong solution of (2.42). The proof is complete. It is not difficult to see that the strong solutions, given by this theorem, depend continuously on the initial data. More specifically, we have that there exists a positive constant c such that (2.49) w(t) − w̃(t)L∞ ([0,T ];Ᏼ) ≤ cw0 − w̃0 Ᏼ , where w(t) and w̃(t) are solutions of (2.42) with initial conditions w0 and w̃0 , respectively. 3. Existence of attractors in L2 We start by constructing an equivalent norm in the space Ᏼ. Lemma 3.1. If W (w) is given by W (w) = W (u, v) = 0 1 2 1 2 u + v + 2βuv dx, 2 2 (3.1) where 0<β < 2 2 2 +1 , (3.2) then W 1/2 is an equivalent norm in Ᏼ. Moreover, there exists a positive constant λ such that β 0 2 − 1 f 2 − 2βg 2 − 2βfg dx ≤ −λ f 2 + g2 ∀f, g ∈ L2 (0, ). (3.3) Aloisio F. Neves √ Proof. Using Poincaré (u ≤ ( / 2)u ) and Schwarz inequalities, we have β 2 β 2 − √ u + v2 ≤ 2βuv dx ≤ √ u + v2 . 2 2 0 183 (3.4) Using (3.2) we can see that 1 β √ < . 2 2 (3.5) √ Therefore, if η = 1/2 − β / 2, we have 2 2 η u + v2 ≤ W (u, v) ≤ u + v2 , (3.6) then W 1/2 is an equivalent norm in Ᏼ. The second part of the lemma follows by noticing that 2 β − 1 f 2 − 2βg 2 − 2βfg dx ≤ β 2 − 1 f 2 − 2βg2 + 2βf g (3.7) 0 and, for β satisfying (3.2), the righthand side of this inequality is a negative definite form. Theorem 3.2. If g, h ∈ L∞ (R+ ; L2 (0, )), then the problems (1.2) and (2.1) are bounded dissipative. More precisely, if (u, v) and z are the solutions of (1.2) and (2.1), with initial conditions (u0 , v0 ) and z0 , respectively, then there exist positive constants c1 , c2 , and µ such that u(t), v(t) ≤ c1 u0 , v0 e−µt + c2 , (3.8) Ᏼ −µt Ᏼ + c2 . (3.9) z(t) ≤ c1 z0 e Moreover, for z0 ∈ H1,0 and r positive, there exist positive constants a, b, with b = b(r) depending on r, such that t+r 2 ϕ z(s) ds ≤ a z0 e−µt + b, t ≥ 0. (3.10) t Proof. From the relation between the two problems the estimate (3.9) follows from (3.8). To prove (3.8) it is enough to consider initial data in the domain Ᏸ(A). Using (2.1) and Poincaré inequality (v2 ≤ ( 2 /2)v 2 ), we obtain after an integration by parts that for almost every t d W u(t), v(t) dt 2 2 2 ≤ β − 1 v − 2β u − 2βu v dx Ẇ (t) = 0 (3.11) + 2βu( ) + v( ) (u + v) ( ) + [2βu + v]g(t) dx. 0 (3.12) 184 Strongly damped wave and heat equations The first integral, line (3.11), can be estimated using Lemma 3.1 2 2 2 2 2 β − 1 v − 2β u − 2βu v dx ≤ −λ u + v . (3.13) 0 To estimate the terms in line (3.12), we observe that (u + v) ( ) satisfies the boundary condition, so it is bounded by some constant M, then using (2.35) we can show that there exists a positive constant c, such that, for every δ > 0 2 2 1 2βu( ) + v( ) (u + v) ( ) ≤ c δ u + v + M 2 . (3.14) δ Using Poincaré inequality we also obtain 2 2 1 (2βu + v)g(t) dx ≤ c δ u + v + g2 . δ 0 (3.15) Choosing δ sufficiently small, we obtain positive constants µi = 1, 2, and K, such that 2 2 Ẇ (t) ≤ −µ1 u + v + K ≤ −µ2 W (t) + K. (3.16) Solving this differential inequality, we obtain W (t) ≤ e−µ2 t W (0) + K µ2 (3.17) that implies (3.8). In order to prove inequality (3.10) we have that Ꮽ is the subdifferential of the functional ϕ and ϕ(0) = 0, therefore ϕ(z) ≤ Ꮽz, z. So, multiplying (1.2) by z we obtain 1 d 2 z + ϕ(z) ≤ −G(z), z + h, z. (3.18) 2 dt The operator G satisfies (2.34), then, using (2.26), we obtain for every δ > 0 a constant M depending on δ such that G z(t) , z(t) ≤ δϕ z(t) + M z(t)2 + 1 (3.19) and, since h(t), z(t) ≤ c z(t)2 + 1 , (3.20) we obtain by grouping the equivalent terms and choosing a convenient small value for δ that d z(t)2 + ϕ z(t) ≤ a1 + a2 z(t)2 , (3.21) dt for some positive constants a1 , a2 . Integrating this inequality from t to t +r we obtain t+r t+r ϕ z(s) ds ≤ z(t)2 + a1 r + a2 z(s)2 ds. (3.22) t This inequality and (3.9) imply (3.10). t Aloisio F. Neves 185 Theorem 3.3. If h ∈ L∞ (R+ ; L2 (0, )), then the solution operator Th (t) : L2 (0, ) → L2 (0, ), associated to the solution of (1.2), is a compact operator for each t > 0. Proof. Multiplying (1.2) by φ ∈ H1,0 , we obtain zt , φ = zx (t, )φ( ) − zx , φx − G(z), φ + h, φ, (3.23) therefore (3.10) and (3.23) imply that zt ∈ L2 (0, T ; H1,0 ) and T 0 2 zt H1,0 dt ≤ C z(0), T . (3.24) To prove the compactness it is enough to consider initial data in a dense subset of L2 (0, ). Let B be the bounded set B = B(r)∩H1,0 , where B(r) the ball of L2 (0, ) with center at zero and radius r, and Th (t)(z0 ) the solution of (1.2) with initial condition z0 . From (3.10) and (3.24), B̄ = Th (·) z0 ; z0 ∈ B (3.25) is a bounded set in the Banach space dv ∈ L2 0, T ; H1,0 . W = v ∈ L2 0, T ; H1,0 ; vt = dt (3.26) Therefore, from Theorem 5.1 of [6], B̄ is a precompact set in L2 (0, T ; L2 (0, )). Then, if (zn ) is a sequence in B, taking subsequences if necessary, we can suppose that (Th (·)(zn )) converges to some function z(·) ∈ L2 (0, T ; L2 (0, )), and also, for almost every τ ∈ (0, T ), Th (τ ) zn −→ z(τ ) as n −→ ∞. (3.27) Consider now the evolution operator S(·)(z, h) given by S(t)(z, h) = Th (t)z, ht , (3.28) where ht is the translation of h, ht (τ ) = h(t +τ ). From [8], S(t) : t ≥ 0 is a dynamical system. Therefore, for t > 0, there exists τ ∈ (0, t) such that (3.27) is true, then Th (t)zn , ht = S(t) zn , h = S(t − τ )S(τ ) zn , h = S(t − τ ) Th (τ )zn , hτ −→ S(t − τ ) z(τ ), hτ (3.29) = Thτ (t − τ )z(τ ), ht implies the compactness of Th (t). Denoting by vu0 (t) the dynamical system given by the problem (2.1), when the initial condition u(0) = u0 ∈ H1,0 is fixed. Using Theorems 3.2 and 3.3 and the relation (2.31), we can state the next result that is a consequence of Theorem 2.2 of Ladyzhenskaya [5]. Theorem 3.4. Under the above conditions the two dynamical systems z(t) and vu0 (t) have compact global attractors in L2 (0, ). 186 Strongly damped wave and heat equations 4. Existence of attractors in H1,0 We start doing some estimates of the solution z(t) of (1.2) when the initial condition z(0) ∈ H1,0 . Using Theorem 3.6 of [2], we have that t → ϕ(z(t)) is absolutely continuous and d ϕ z(t) = Ꮽz(t), zt (t) , a.e., (4.1) dt then d ϕ z(t) = −Ꮽz(t)2 + Ꮽz(t), −G(z) + h, (4.2) dt and integrating on t we obtain t Ꮽz(s)2 ds + ϕ z(t) 0 t (4.3) Ꮽz(s) h(s) − G z(s) ds ≤ ϕ z(0) + 0 t t t 1 G z(s) 2 ds + Ꮽz(s)2 ds + h(s)2 ds. ≤ ϕ z(0) + 2 0 0 0 Using (2.26) and (2.34), we obtain, for t ∈ [0, T ], that t 1 t Ꮽz(s)2 ds + ϕ z(t) ≤ ϕ z(0) + c1 + c2 ϕ z(s) ds 2 0 0 (4.4) for some constants c1 , c2 . Thus, from Gronwall inequality, there exists a constant C(ϕ(z(0)), T ) depending on ϕ(z(0)) and T such that ϕ z(t) ≤ C ϕ z(0) , T , (4.5) t Ꮽz(s)2 ds ≤ C ϕ z(0) , T , (4.6) 0 in particular, we have z ∈ L∞ (0, T ; H1,0 ) ∩ L2 (0, T ; H 2 (0, )). Moreover, if z1 (t) and z2 (t) are solutions with initial condition on H1,0 we have, using (2.23), z1 (t) − z2 (t) 2 ≤ Ꮽz1 (t) − Ꮽz2 (t), z1 (t) − z2 (t) 2 1 d z1 (t) − z2 (t) − G z1 (t) − G z2 (t) , z1 (t) − z2 (t) . =− 2 dt (4.7) Since G is Lipschitz, we obtain after an integration on t 2 2 1 1 t z (s) − z (s) 1 ds + z1 (t) − z2 (t) 2 2 0 2 t 2 1 z1 (s) − z2 (s)2 ds, ≤ z1 (0) − z2 (0) + c 2 0 (4.8) Aloisio F. Neves 187 therefore, from Gronwall inequality, there exists a constant C depending on T , such that z1 (t) − z2 (t) ≤ C z1 (0) − z2 (0), (4.9) t z1 (s) − z2 (s) 2 ds ≤ C z1 (0) − z2 (0)2 (4.10) 0 for t ∈ [0, T ]. Now we study the evolution of the problem (1.2) in H1,0 . Our first result is concerned with continuity with respect to time and initial data. Lemma 4.1. The solution operator, z(t) = T (t)z0 , of the problem (1.2) is continuous in the variables t and z0 in the H1,0 norm. More precisely, the operator R+ × H1,0 −→ H1,0 , t, z0 −→ T (t)z0 , (4.11) is continuous separately in each variable. Proof. Fix z0 ∈ H1,0 and let (tn ) be a sequence in R+ converging to t, we know that the solution (z(tn )) converges to z(t) in L2 (0, ) and, using Lemma 3.6 of [2], (ϕ(z(tn ))) converges to ϕ(z(t)). Then, from (2.26), (z(tn ))  is bounded, therefore, there exists a subsequence of (z(tn )), that we keep denoting by (z(tn )), that converges weakly to z(t) in H1,0 . First of all, we claim that the weak convergence implies the convergence of (z(tn , )). In fact considering a smooth function φ such that φ(0) = 0 and φ( ) = 0, we obtain by integrating by parts z tn , x φ(x) dx = z tn , φ( ) − z tn , x φ (x) dx, 0 0 (4.12) z (t, x)φ(x) dx = z(t, )φ( ) − 0 z(t, x)φ (x) dx. 0 Thus, passing to the limit, z(tn , ) → z(t, ), what proves our claim. Next, since p is continuous and 2 z t n = 2 ϕ z t n − p z tn , , (4.13) H 1,0 we have z(tn )H1,0 → z(t)H1,0 that implies the strong convergence of (z(tn )) to z(t) and the continuity of the operator in the variable t. Now we prove the continuity of the operator in the second variable. In fact, what we have is a stronger result: Theorem 4.2. If (z0n ) is a bounded sequence in H1,0 and converges to z0 in the L2 (0, )norm, then the corresponding solutions of (1.2) zn (t) = T (t)z0n converges to z(t) = T (t)z0 in H1,0 , for fixed t > 0, as n → ∞. In particular, for t > 0, the operator T (t) : H1,0 → H1,0 is compact. 188 Strongly damped wave and heat equations Proof. We have (ϕ(z0n )) bounded, then from (4.5) and (2.26) both sequences (ϕ(zn (t))) and ((zn (t)) ) are uniformly bounded for t ∈ [0, T ]. The convergence z0n → z0 in L2 (0, ) and (4.10) imply the convergence (4.14) zn −→ z in L2 0, T ; H1,0 , therefore zn (τ ) → z(τ ) in H1,0 for almost every τ ∈ [0, T ]. For t ∈ [0, T ], ϕ zn (t) − ϕ z(t) ≤ ϕ zn (t) − ϕ zn (τ ) + ϕ zn (τ ) − ϕ z(τ ) + ϕ z(τ ) − ϕ z(t) . (4.15) The first term in the righthand side satisfies ϕ zn (t) − ϕ zn (τ ) = t τ d ϕ zn (s) ds ds (4.16) and from (4.2) 2 d ϕ z(t) ≤ G z(t) + h(t)2 , dt (4.17) therefore the sequences (d/dt (ϕ(zn (t)))) are uniformly bounded in L2 (0, ) for every t ∈ [0, T ]. Then (4.15) implies ϕ(zn (t)) → ϕ(z(t)) for every t ∈ [0, T ], as n → ∞. Therefore, the same argument we have just used in the first part of the theorem implies that zn (t) → z(t) in H1,0 norm, as n → ∞. Theorem 4.3. If h ∈ L∞ (0, ∞; L2 (0, )), then there exists a bounded set in H1,0 that attracts all the solutions of the problem (1.2) with initial condition in a subset of H1,0 that it is bounded in L2 (0, ). In particular, the problem (1.2) is bounded dissipative in H1,0 . Proof. If z(t) is a solution of the problem (1.2) with initial condition in H1,0 we have, using (4.17), that ϕ(z(t)) satisfies the differential inequality d ϕ z(t) ≤ a1 ϕ z(t) + a2 + h(t)2 , dt t > 0, (4.18) where a1 , a2 are constants. For solution with initial conditions in H1,0 and bounded in L2 (0, ), (3.10) implies t+r that t ϕ(z(s))ds is less than a fixed constant for t sufficiently large, then we can use the uniform Gronwall lemma, see [9, page 89], to obtain the result of the theorem. As a consequence of the two previous theorems and the relation (2.31) we have the following theorem. Theorem 4.4. Under the above conditions, the dynamical system z(t) given by (1.2) has a compact global attractor in H1,0 . Moreover, for v(0) ∈ H1,0 , u(t)+v(t) given by (2.1) has also a compact global attractor in H1,0 . Aloisio F. Neves 189 References [1] [2] [3] [4] [5] [6] [7] [8] [9] V. Bonfim and A. F. Neves, A onedimensional heat equation with mixed boundary conditions, J. Differential Equations 139 (1997), no. 2, 319–338. MR 98i:35098. Zbl 887.35078. H. Brézis, Opérateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces de Hilbert, no. 5, Notas de Matematica (50), NorthHolland Publishing, Amsterdam, 1973 (French), NorthHolland Mathematics Studies. MR 50#1060. Zbl 252.47055. A. N. Carvalho and C. B. Gentile, Asymptotic Behaviour of Nonlinerar Parabolic Equations with Monotone Principal Part, vol. 1, Cadernos de Matemática, no. 1, São Carlos, Brazil, 2000. J. K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Rhode Island, 1988. MR 89g:58059. Zbl 642.58013. O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. MR 92k:58040. Zbl 755.47049. J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, 1969 (French). MR 41#4326. Zbl 189.40603. J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson et Cie, Éditeurs, Paris, 1967 (French). MR 37#3168. G. R. Sell, Nonautonomous differential equations and topological dynamics. I. The basic theory, Trans. Amer. Math. Soc. 127 (1967), 241–262. MR 35#3187a. Zbl 189.39602. R. Temam, Infinitedimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, vol. 68, SpringerVerlag, New York, 1988. MR 89m:58056. Zbl 662.35001. Aloisio F. 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