PREPUBLICACIONES DEL DEPARTAMENTO DE MATEMÁTICA APLICADA UNIVERSIDAD COMPLUTENSE DE MADRID MA-UCM

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1 PREPUBLICACIONES DEL DEPARTAMENTO DE MATEMÁTICA APLICADA UNIVERSIDAD COMPLUTENSE DE MADRID MA-UCM Asymptotic behaviour of a parabolic problem with terms concentrated in the boundary A. Jiménez-Casas and A. Rodriguez-Bernal Mayo matemática_aplicada@mat.ucm.es

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3 Asymptotic behaviour of a parabolic problem with terms concentrated in the boundary Ángela Jiménez-Casas 1 Aníbal Rodríguez-Bernal 2 1 Grupo de Dinámica No Lineal. Universidad Pontificia Comillas de Madrid. C/Alberto Agulilera 23, 2815 Madrid. 2 Departamento de Matemática Aplicada. Universidad Complutense de Madrid, 284 Madrid and Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM. Abstract We analyze the asymptotic behavior of the attractors of a parabolic problem when some reaction and potential terms are concentrated in a neighborhood of a portion Γ of the boundary and this neighborhood shrinks to Γ as a parameter ε goes to zero. We prove that this family of attractors is upper continuous at ε =. 1 Introduction Let Ω be an open bounded smooth set in IR N with a C 2 boundary Ω. Let Γ Ω be a smooth subset of the boundary, isolated from the rest of the boundary, that is, dist(γ, Ω \ Γ) >. Define the strip of width ε and base Γ as ω ε = {x σ n(x), x Γ, σ [, ε)} for sufficiently small ε, say ε ε, where n(x) denotes the outward normal vector. We note that for small ε, the set ω ε is a neighborhood of Γ in Ω, that collapses to the boundary when the parameter ε goes to zero. We are interested in the behavior, for small ε, of the solutions of the nonlinear parabolic problem u ε t + A εu ε + µu ε = F ε (x, u ε ) in (, T) Ω u (P ε ) ε n = on Γ (1.1) u ε () = u H 1 (Ω) Partially suppported by Projects MTM , CCG7-UCM/ESP-2393 UCM-CAM, Grupo de Investigación CADEDIF, PHB26-3PC Spain and 775-UCIII, M.E.C. SPAIN.

4 Γ ω ε Ω Figure 1: The set ω ε where A ε u ε = u ε + 1 ε X ω ε V ε u ε, µ > is such that the elliptic problem associated is positive and F ε (x, u ε ) = f (x, u ε ) + 1 ε X ω ε g (x, u ε ), where we denote by X ωε the characteristic function of the set ω ε. Thus the potential functions and the effective reaction are concentrated in ω ε. Note that we assume that both f and g are defined on Ω IR. We are interested in the behavior, for small ε, of the attractor of this parabolic problem. Below we will assume several hypotheses that imply for some function V defined on Γ and 1 ε X ω ε V ε X Γ V F ε (x, u) F (x, u) = f (x, u) + X Γ g (x, u) in some sense (see [4]). So, we consider the limit parabolic problem given by u t u + µu = f (x, u) u (P ) n + V u = g (x, u) u() = u H 1 (Ω). in (, T) Ω on Γ (1.2) Our goal is to prove that the family of sets A ε global attractor of (P ε ) is upper semicontinuous at ε = in H 1 (Ω), that is: dist H 1 (Ω)(A ε, A ), if ε with dist H 1 (Ω)(A ε, A ) := sup u ε A ε 2 inf { u ε u H u 1 (Ω)} A

5 where A is the global attractor associated to the limit problem (1.2). In order to prove this result about the global attractor for the parabolic problem, we use some previous results about the concentrating integral and the elliptic problem associated to the parabolic problem, see [4, 5]. 2 Upper Semicontinuity of Attractors We consider the family of parabolic problems (1.1), for ε (, ε ] u ε t + A ε u ε + µu ε = F ε (x, u ε ) u (P ε ) ε n = u ε () = u H 1 (Ω) in (, T) Ω on Γ where A ε u ε = u ε + 1 ε X ω ε V ε u ε, and F ε (x, u ε ) = f (x, u ε ) + 1 ε X ω ε g (x, u ε ). Throughout this section we will assume that 1 V ε ρ K 1 ε ω ε with ρ > N 1, and K 1 a positive constant independent of ε, and that there exists a function V L ρ (Γ) such that for any smooth function ϕ, we have 1 lim V ε ϕ = V ϕ. ε ε ω ε Γ Next, we consider the parabolic problem given by u t u + µu = f (x, u) u (P ) n + V u = g (x, u) u() = u H 1 (Ω). in (, T) Ω on Γ We assume also the the following conditions for the nonlinearity functions f, g : Growth conditions (G) : f, g : Ω IR IR, ε [, ε ] are locally Lipschitz uniformly in x Ω and x Ω respectively and satisfy the following growth conditions: If N > 2 we assume that j(x, u) j(x, v) c u v ( u σ 1 + v σ 1 + 1) with j = f or j = g and with exponents σ f and σ g respectively, such that σ f N + 2 N 2 and σ g N N 2. If N = 2 we assume that for every η > there exists c η > such that j(x, u) j(x, v) c η u v (e η u 2 + e η v 2 ). 3

6 These conditions imply the local existence and uniqueness of solutions of (1.1) and (1.2), see J.Arrieta, A. Rodriguez-Bernal et al.[2]. We assume also the following conditions which ensure that local solutions of the nonlinear parabolic problems (1.1) and (1.2) are globally defined and we have well defined semigroups in H 1 (Ω), T ε (t)u = u ε (t, x; u ), ε ε, see [3]. Note that the nonlinear semigroups are given by the variation of constants formula T ε (t, u ) = e A εt u + with A ε = A ε + µi and ε [, ε ), see [7]. e A ε(t s) F ε (, T ε (s, u ))ds Sign conditions (S) Assume in addition that there exist C L p (Ω), D L p (Ω), p > N 2 and E Lq (Ω), F L q (Ω), q > N 1 such that and sf (x, s) C(x)s 2 + D(x) s, x Ω, s IR, ε ε sg (x, s) E(x)s 2 + F(x) s, x Ω, s IR, ε ε. Moreover we assume there exist positive constants K i, i = 2, 3 independent of ε such that 1 E ρ K 2, with ρ > N 1, ε ω ε and 1 F r 2(N 1) K 3, with r > max{1, }. ε ω ε N Dissipative condition (D) Finally we assume the first eigenvalue, λ 1, of the following problem is positive { ϕ Cϕ + µϕ = (P 1 λ1 ϕ in Ω ) ϕ + V n ϕ = Eϕ on Γ. (2.1) With these assumptions our goal is to prove the upper semicontinuity of the family of global attractors. In order to prove this, we use the previous result for the elliptic problem (see J.Arrieta, A. Jimenez-Casas, A.Rodriguez-Bernal [4]), and the following lemmas. Lemma 2.1 Under the above hypotheses on V ε, f, g, for sufficiently small ε, problems (1.1) and (1.2) have global attractors A ε. Moreover, there exists R > independent of ε, such that sup ε ε sup u A ε u L (Ω) R. 4

7 Proof We denote by λ ε 1 the first eigenvalue of the following elliptic problem { φ ε + 1 ε X ω ε (V ε E)φ ε + (µ C)φ ε = λ ε 1 φ on Ω φ ε n = in Γ. By the spectral convergence obtained in [4], we have λ ε 1 λ 1 with λ 1 the first eigenvalue of the elliptic limit problem (2.1). Hence for small enough ε we have λ ε 1 > for every ε ε. We split the proof in several steps. Step 1: Since, λ ε 1 >, for ε ε, following the arguments in [3], see also [9], we prove that lim sup t u ε (t, x; u ) Φ ε (x) uniformly in x Ω and for u B in a bounded set B in H 1 (Ω), where Φ ε (x) is the unique solution of { Φ ε + 1 ε X ω ε (V ε E)Φ ε + (µ C)Φ ε = D + 1 ε X ω ε F on Ω Φ ε n = in Γ. Step 2: From the convergence results for elliptic problems in [4], we prove that Φ ε (x) Φ (x), as ε, in C β ( Ω), for some β >, where Φ (x) the unique solution of the following problem { Φ (C µ)φ = D in Ω Φ n + V Φ = EΦ + F in Γ. Thus, from the smoothing effect of the equations and the results in [6] we get that problems (1.1) and (1.2) have global compact attractors A ε in H 1 (Ω), for ε ε. Also, there exists R independent of ε > and ε enough small, such that sup ε ε sup u Aε u L (Ω) R. With this and the variation of constants formula we get Lemma 2.2 Under the above hypotheses on V ε, f, g we have that, there exists R > such that sup sup u H 1 (Ω) L (Ω) R. ε ε u A ε In particular, A attracts ε (,ε )A ε in H 1 (Ω). The following Lemma 2.3 proved in [8] shows several technical results on the behavior of concentrating function as ε. This will allow us to prove the convergence of the nonlinearities F ε given by the Lemma 2.4. Note that below we make use of the intermediate sobolev spaces H s (Ω) and their dual spaces which we denote H s (Ω) for 1 2 s 1, that is H s (Ω) (H s (Ω)). 5

8 Lemma 2.3 Under the above hypotheses, if v H 1 (Ω) L (Ω) R, then we have that: i) For any 1 < s 1 there exists M(R) a positive constant independent of ε such that for 2 any smooth function ϕ up to the boundary of Ω, 1 ε ω ε g (v)ϕ M(R) ϕ H s (Ω). ii) There exists M(ε, R) if ε such that for any smooth function ϕ up to the boundary of Ω, 1 g (v)ϕ g (v)ϕ M(ε, R) ϕ H ε 1 (Ω). ωε Γ Proof The proof of this Lemma can be found in [8]. With this, we get the following result that states the convergence of the nonlinear terms of the problems. Lemma 2.4 Under the above hypotheses we have that for any 1 2 < s < 1: i) There exists C > independent of ε > such that sup v A ε { F ε (v) H s (Ω), F (v) H s (Ω)} C. ii) There exists M(ε) with M(ε) if ε, such that sup v A ε F ε (v) F (v) H s (Ω) M(ε). Proof Part i) follows from Lemmas 2.2 and 2.3 i). For part ii), note that from Lemmas 2.1 and 2.4, we obtain that: sup v A ε F ε (v) F (v) H 1 (Ω) M(ε) as ε. Now, fix 1 2 < s < 1. Then for any s such that 1 2 < s < s < 1, by interpolation F ε (v) F (v) H s (Ω) F ε (v) F (v) θ H s (Ω) F ε(v) F (v) 1 θ H 1 (Ω) for some < θ < 1. Again by Lemmas 2.2 and 2.3 i), the first term in the right hand side above is bounded while the second goes to zero, both uniformly for v A ε, and we conclude. On the other hand, from the spectral convergence of the linear operators we get, see [1] for a similar result. Lemma 2.5 Under the above hypotheses, let 1 < s < 1. Then, there exist α (1+s, 1) 2 2 and a function C (ε) with C (ε) if ε, such that for all h H s (Ω) we have that e A εt h e A t h H 1 (Ω) C (ε)t α h H s (Ω), t >. 6

9 With all the above we can then obtain the convergence of the nonlinear semigroups. Lemma 2.6 Under the above hypothesis let 1 2 < s < 1 and some fixed τ >. Then, there exists a function C(ε) with C(ε) if ε, such that for u ε A ε, ε (, ε ), for some α ( 1+s 2, 1). T ε (t, u ε ) T (t, u ε ) H 1 (Ω) M(τ)C(ε)t α for t (, τ] Proof We consider the nonlinear semigroup given by the variation of constant formula: T ε (t, u ε ) = e A εt u ε + where A ε = A ε + µi, ε [, ε ] and T (t, u ε ) = e A t u ε + e A ε(t s) F ε (x, T ε (s, u ε ))ds (2.2) e A (t s) F (x, T (s, u ε ))ds. (2.3) From (2.2) and (2.3), together with the previous results, we will get below that T ε (t, u ε ) T (t, u ε ) H 1 (Ω) M C(ε)t α + + M (t s) α T ε (s, u ε ) T (s, u ε ) H 1 (Ω)ds (2.4) for some M depending on τ. Hence, applying the singular Gronwall Lemma, Lemma in [7], to (2.4), we get the result. We now split the proof of (2.4) in several steps. In effect, from (2.2) and (2.3) we have that: T ε (t, u ε ) T (t, u ε ) H 1 (Ω) e A ε t u ε e A t u ε H 1 (Ω)+ + e A ε (t s) F ε (x, T ε (s, u ε )) e A (t s) F ε (x, T ε (s, u ε )) H 1 (Ω)ds+ + e A (t s) [F ε (x, T ε (s, u ε )) F (x, T ε (s, u ε ))] H 1 (Ω)ds+ + e A (t s) [F (x, T ε (s, u ε )) F (x, T (s, u ε ))] H 1 (Ω)ds = I 1 + I 2 + I 3 + I 4. Step 1.- From Lemma 2.5 together with Lemma 2.2, we obtain: I 1 = e A ε t u ε e A t u ε H 1 (Ω) C (ε)t α u ε H s (Ω) C (ε)t α K with < C (ε) if ε and K a positive constant independent of ε. Step 2.- Again Lemma 2.5 gives I 2 = e A ε(t s) F ε (x, T ε (s, u ε )) e A (t s) F ε (x, T ε (s, u ε )) H 1 (Ω)ds 7

10 C (ε) (t s) α F ε (x, T ε (s, u ε )) H s (Ω) ds. Now, from Lemma 2.4 and using thet the attractor A ε is invariant for the semigroup T ε, we obtain a positive constant K 1 independent of ε such that F ε (, T ε (s, u ε )) H s (Ω) K 1. From this I 2 C (ε) K 1 1 α t1 α C (ε)k 2 t α since t τ. Step 3.- I 3 = e A (t s) (F ε (x, T ε (s, u ε )) F (x, T ε (s, u ε )) H 1 (Ω)ds K 2 (t s) α F ε ((x, T ε (s, u ε ) F (x, T ε (s, u ε )) H s (Ω) ds. Using again Lemma 2.4 and the invariance of the attractor, we obtain that F ε (, T ε (s, u ε )) F (, T ε (s, u ε )) H s (Ω) M(ε), with M(ε) if ε and I 3 M(ε)K 3 t α, since t τ, with K 3 a positive constant independent of ε and depending on τ. Step 4.- I 4 = e A (t s) (F (x, T ε (s, u ε )) F (x, T (s, u ε ))) H 1 (Ω)ds K 2 (t s) α F (x, T ε (s, u ε )) F (x, T (s, u ε )) H s (Ω) ds. Now, from the bounds in Lemma 2.2 and the regularity of the nonlinear terms f and g we get that F (u) F (v) H s (Ω) L u v H 1 (Ω) with L = L(R) if the norm of both u and v in H 1 (Ω) L (Ω) is bounded by R. Therefore, we get I 4 K 2 L (t s) α T ε (s, u ε ) T (s, u ε ) H 1 (Ω)ds. Puting all the estimates above together, we get (2.4) and the proof is complete. Theorem 2.7 Under the above hypothesis about V ε, f, g the family of global attractors of (P ε ), A ε, is upper semicontinuous at ε = in H 1 (Ω), that is: dist H 1 (Ω)(A ε, A ), if ε where dist H 1 (Ω)(A ε, A ) := sup u ε A ε inf { u ε u H u A 1 (Ω)} Proof In effect, from Lemma 2.2, A attracts <ε ε A ε, since the latter is a bounded set in H 1 (Ω). Hence, given δ >, there exists τ = τ(δ) such that dist H 1(T (τ)u ε, A ) δ 2 for every u ε A ε with ε (, ε ). Next, using that A ε is invariant, given v ε A ε, there exists u ε such that T ε (τ)u ε = v ε. Therefore, dist H 1(v ε, A ) v ε T (τ)u ε H 1 (Ω) + dist H 1(T (τ)u ε, A ). 8

11 Then from Lemma 2.6 it is clear that if ε is small enough we get and we conclude. v ε T (τ)u ε H 1 (Ω) = T ε (τ)u ε T (τ)u ε H 1 (Ω) δ 2, References [1] J.M. Arrieta, Spectral Behavior and Uppersemicontinuity of Attractors International Conference on Differential Equations (EQUADIFF 99) Berlin. World Scientific, (2). [2] J.M. Arrieta, A. N. Carvalho, A. Rodríguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities, J. Differential Equations 156, (1999). [3] J.M. Arrieta, A. N. Carvalho, A. Rodríguez-Bernal, Attractors of Parabolic Problems with Critical Nonlinearities. Uniform Bounds. Comm.P.D.E. s 25, vol 1-2, 1-37 (2). [4] J.M. Arrieta, A. Jiménez-Casas, A. Rodríguez-Bernal Nonhomogeneous flux condition as limit of concentrated reactions, Revista Iberoamericana de Matematicas vol 24, no 1, (28). [5] J.M.Arrieta, A. Rodríguez Bernal, J. Rossi, The best Sobolev trace constant as limit of the usual Sobolev constant for small strips near the boundary. Proceeding of the Royal Society of Edinburgh. Section A. Mathematics, vol 138A, (26). [6] J.K. Hale, Asymptotic Behavior of Dissipative System, American Mathematical Society (1988). [7] D. Henry, Geometry Theory of Semilinear Parabolic Equations, Springer-Verlag, (1981). [8] A. Jiménez-Casas, A. Rodríguez-Bernal Upper Semicontinuity of Attractors for Parabolic Problem with terms concentrating on the boundary, in preparation. [9] A. Rodríguez-Bernal, A. Vidal-López Extremal equilibria for nonlinear parabolic equations in bounded domains and applications, Journal of Differential Equations 244, (28). 9

12 PREPUBLICACIONES DEL DEPARTAMENTO DE MATEMÁTICA APLICADA UNIVERSIDAD COMPLUTENSE DE MADRID MA-UCM SHAPE OPTIMIZATION OF GEOTEXTILE TUBES FOR SANDY BEACH PROTECTION, D. Isebe, P. Azerad, F. Bouchette, B. Ivorra and B. Mohammadi 2. ON THE FINITE TIME EXTINTION PRHENOMENON FOR SOME NONLINEAR FRACTIONAL EVOLUTION EQUATIONS, J.I. Diaz, T. Pierantozzi and L. Vázquez 3. SEMICLASSICAL MEASURES AND THE SCHRÖDINGER FLOW ON COMPACT MANIFOLDS, F.Macià 4. INVERSE PROBLEMS IN HEAT EXCHANGE PROCESSES, A. Fraguela, J.A. Infante, A.M.Ramos and J.M.Rey 5. EXTREMAL EQUILIBRIA FOR DISSIPATIVE PARABOLIC EQUATIONS IN LOCALLY UNIFORM SPACES, J. Cholewa and A. Rodríguez-Bernal 6. OPTIMIZING INITIAL GUESSES TO IMPROVE GLOBAL MINIMIZATION, B. Ivorra, B. Mohammadi and A. M. Ramos. 7. STABILITY OF THE BLOW-UP TIME AND THE BLOW-UP SET UNDER PERTURBATIONS, J. M. Arrieta, R. Ferreira, A. de Pablo, and J. D. Rossi 8. PERTURBATION OF THE EXPONENTIAL TYPE OF LINEAR NONAUTONOMOUS PARABOLIC EQUATIONS AND APPLICATIONS TO NONLINEAR EQUATIONS, A.Rodriguez-Bernal. 9. PERMANENCE AND ASYMPTOTICALLY STABLE COMPLETE TRAJECTORIES FOR NON-AUTONOMOUS LOTKA-VOLTERRA MODELS WITH DIFFUSION, J. Langa, J. Robinson, A. Rodríguez-Bernal and A. Suárez 1. EQUILIBRIA AND GLOBAL DYNAMICS OF A PROBLEM WITH BIFURCATION FROM INFINITY, J.M. Arrieta, A. Rodríguez-Bernal, and R. Pardo 11. VALIDATION OF A VARIANCE-EXPECTED COMPLIANCEMODEL FOR TRUSS DESIGN, M. Carrasco, B. Ivorra and A.M. Ramos 12. CONSTRUCTION OF THE MAXIMAL SOLUTION OF BACKUS PROBLEM, G. Díaz, J.I. Díaz and J.Otero

13 PREPUBLICACIONES DEL DEPARTAMENTO DE MATEMÁTICA APLICADA UNIVERSIDAD COMPLUTENSE DE MADRID MA-UCM DESIGN OF CODE DIVISION MULTIPLE ACCESS FILTERS USING GLOBAL OPTIMIZATION TECHNIQUES, B. Ivorra, B. Mohammadi, and A. M.Ramos 2. DYNAMICS IN DUMBBELL DOMAINS II. THE LIMITING PROBLEM, J.M. Arrieta, A. N. Carvalho and G. Lozada-Cruz 3. DYNAMICS IN DUMBBELL DOMAINS III. CONTINUITY OF ATTRACTORS, J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz 4. GEOMETRIC VERSUS SPECTRAL CONVERGENCE FOR THE NEUMANN LAPLACIAN UNDER EXTERIOR PERTURBATIONS OF THE DOMAIN, J. M. Arrieta and D. Krejcirik 5. ON THE MODELLING AND SIMULATION OF HIGH PRESSURE PROCESSES AND INACTIVATION OF ENZYMES IN FOOD ENGINEERING, J.A. Infante, B. Ivorra, Á.M. Ramos and J.M. Rey 6. CHARACTERIZATION OF THE MARTIAN SURFACE LAYER, G.Martínez, F. Valero, L. Vazquez. 7. CHARACTERIZATION OF THE MARTIAN CONVECTIVE BOUNDARY LAYER, G.Martínez, F. Valero, L. Vazquez. 8. INFINITE RESONANT SOLUTIONS AND TURNING POINTS IN A PROBLEM WITH UNBOUNDED BIFURCATION, J.M. Arrieta, R.Pardo, A.Rodríguez Bernal 9. CASCADES OF HOPF BIFURCATIONS FROM BOUNDARY DELAY, J.M. Arrieta, N. Cónsul, S. Oliva 1. QUENCHING PHENOMENA FOR A NON-LOCAL DIFFUSION EQUATION WITH A SINGULAR ABSORPTION, R. Ferreira 11. WELL-POSEDNESS OF THE EINSTEIN EULER SYSTEM IN ASYMPTOTICALLY FLAT SPACETIMES: THE CONSTRAINT EQUATIONS, U. Brauer and L. Karp 12. WELL-POSEDNESS OF THE EINSTEIN EULER SYSTEM IN ASYMPTOTICALLY FLAT SPACETIMES: THE EVOLUTION EQUATIONS, U. Brauer and L. Karp 13. EXTREMAL EQUILIBRIA FOR MONOTONE SEMIGROUPS IN ORDERED SPACES WITH APPLICATIONS TO EVOLUTIONARY EQUATIONS, J. W. Cholewa and A. Rodriguez-Bernal 14. ASYMPTOTIC BEHAVIOUR OF A PARABOLIC PROBLEM WITH TERMS CONCENTRATED IN THE BOUNDARY, A. Jiménez-Casas and A. Rodriguez-Bernal