On p-Laplacian reaction–diffusion problems with dynamical boundary conditions in perforated media

Depósito de investigación de la Universidad de Sevilla https://idus.us.es/ Esta es la versión aceptada del artículo publicado en: This is a accepted manuscript of a paper published in: Mediterranean Journal of Mathematics (2023): 19/01/2023 DOI: https://doi.org/10.1007/s00009-023-02333-1 Copyright: El acceso a la versión publicada del artículo puede requerir la suscripción de la revista. Access to the published version may require subscription. “This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1007/s00009-023-02333-1” On p-Laplacian reaction-di↵usion problems with dynamical boundary conditions in perforated media Mar´ıa ANGUIANO Abstract We study the e↵ect of the p-Laplacian operator in the modelling of the heat equation through a porous medium ⇤✏⇢RN(N2). The case of p= 2 was recently published in (Anguiano, Mediterr. J. Math. 17, 18 (2020)). Using rigorous functional analysis techniques and the properties of Sobolev spaces, we managed to solve additional (nontrivial) difficulties which arise compared to the study for p= 2, and we prove a convergence theorem in appropriate functional spaces. AMS classification numbers: 35K05, 35K58. Keywords: Asymptotic analysis, p-Laplacian, heat equation. 1 Statement of the problem and the results Homogenization problems in perforated media for the p-Laplacian operator have been considered in the literature over the last decades. The homogenization of the equation div |rv✏|p2rv✏=f(1) in a periodically perforated domain is considered by Labani and Picard in [36] with Dirichlet boundary conditions. Such a problem is a generalization of the linear problem for Laplace’s equation, which corresponds to p= 2, and was studied by Cioranescu and Murat in [17]. Donato and Moscariello in [24] study the homogenization of a class of nonlinear elliptic Neumann problems in perforated domains of RN. As a consequence of [24], we are able in particular to describe the homogenization of (1) with Neumann boundary conditions. This result is a generalization of earlier related works, for instance, Cioranescu and Donato [15] and Cioranescu and Saint Jean Paulin [18]. In [41] Shaposhnikova and Podol’skii study the homogenization of (1) in an ✏-periodically perforated domain with a nonlinear boundary condition. In [21] D´ıaz et al. consider (1) with a nonlinear perturbed Robintype boundary condition in an ✏periodically perforated domain where the size of the particles is smaller than the period ✏, and the asymptotic behavior of the solution is studied as ✏!0. The closest articles to this one in the literature are [33, 34, 35], where G´omez et al. consider the case 2 <pN, and [22], where D´ıaz et al. study the case p>N. It has been discovered by physicists that, as far as the Allen-Cahn equation is concerned, for certain materials a dynamical interaction with the walls must be taken into account (see Fischer et al. [28, 29] for more details). In this sense, in the context of heat equations, dynamical boundary conditions have been rigorously derived in Gal and Shomberg [31] based on first and second thermodynamical principles and their physical interpretation was also given in Goldstein [32]. We point out that these types of boundary conditions are also used for modelling various physical situations including fluid di↵usion within a semi-permeable boundary (see Crank [20], Langer [37], and, March and Weaver [39] for more details) or several situations when the hear flow inside the domain is subject to nonlinear heating or cooling on the boundary (see Favini et al. [26, 27] for more details). In the previous literature there is no study for the homogenization of p-Laplacian parabolic models as we consider in this article. Such equations model nonlinear fluid di↵usion through a semi-permeable membrane (see 1 Duvaut and Lions [25, Ch.1]) or nonlinear heat flow with radiation on the boundary causing nonlinear cooling (see Friedman [30, Ch.7, §5]). Model problem. The heat equation that we study in this paper is the following @tv✏pv✏+↵|v✏|p2v✏=f1(v✏)in⇤ ✏⇥(0,¯ T), where v✏=v✏(x, t), x2⇤✏,t2(0,¯ T), with ¯ T>0 and ↵>0. Assume that ⇤✏⇢RN(N2) is a fixed bounded domain ⇤from which a set T✏of holes has been removed, in particular, ⇤✏is a periodically perforated domain with holes of the same size as the period (see [5] for more details on the domain). Here the di↵usion is modeled by the p-Laplacian operator pv✏:= div |rv✏|p2rv✏with p2[2,N]. On the nonlinear term f1, we assume that f12C(R), such that pq1<+1,if p=Nand 2 q1Np Np,if p2[2,N),(2) ⌘1|s|q1f1(s)s⌘2|s|q1+,for all s2R, (3) and (f1(s1)f1(s2)) (s1s2)(s1s2)2,for all s1,s 22R, (4) where ⌘i>0, i=1,2, >0, and >0. We consider the following dynamical boundary conditions on the boundary of the holes @⌫pv✏+✏@ tv✏=✏f 2(v✏) on @T✏⇥(0,¯ T), where T✏is the set of all the holes of this periodic distribution contained in ⇤✏(see [5] for more details). The “normal derivative” must be understood as @⌫pv✏=|rv✏|p2rv✏·⌫,where⌫denotes the outward normal to @T✏. This boundary equation is multiplied by ✏to compensate the growth of the surface by shrinking ✏,where the value of v✏is assumed to be the trace of the function v✏defined for x2⇤✏. On the nonlinear term f2,we assume that f22C(R), such that 2q2<+1,if p=Nand 2 q2(N1)p Np,if p2[2,N),(5) ⌘1|s|q2f2(s)s⌘2|s|q2+,for all s2R, (6) and (f2(s1)f2(s2)) (s1s2)(s1s2)2,for all s1,s 22R. (7) Moreover, we consider Dirichlet boundary condition on the boundary of ⇤ v✏=0,on @⇤⇥(0,¯ T), and the initial conditions v✏(x, 0) = v0 ✏(x),for x2⇤✏,v ✏(x, 0) = 0 ✏(x),for x2@T✏, where (v0 ✏, 0 ✏) satisfies v0 ✏2L2(⇤), 0 ✏2L2(@T✏),(8) and |v0 ✏|2 ⇤✏+✏|0 ✏|2 @T✏K, (9) where K>0 and |·| ⇤✏(respectively |·| @T✏) is the norm in L2(⇤✏)(respectivelyL2(@T✏)). Notice that on @T✏ we assume that 0 ✏(x) is equal to the trace of v0 ✏(x). 2 In summary, we study in this paper the following problem 8 > > > > < > > > > : @tv✏pv✏+↵|v✏|p2v✏=f1(v✏)in⇤ ✏⇥(0,¯ T), @⌫pv✏+✏@ tv✏=✏f 2(v✏) on @T✏⇥(0,¯ T), v✏=0,on @⇤⇥(0,¯ T), v✏(x, 0) = v0 ✏(x),for x2⇤✏, v✏(x, 0) = 0 ✏(x),for x2@T✏, (10) under the assumptions (2)-(9). In a recent article (see [5]) we addressed the problem (10) with p= 2 (for the physical motivation of this model, see, for instance, Timofte [45]). More recently, in [6] we generalize this previous study with a Laplace- Beltrami correction term and in [7] we carried out the first study on the asymptotic behavior of the solution of parabolic models in a thin porous media. We would like to highlight that analyzing a p-Laplacian problem involves additional (nontrivial) difficulties compared to the study for the Laplacian. Due to the presence of p-Laplacian operator in the domain, the variational formulation of the p-Laplacian reaction-di↵usion equation is di↵erent that in [5]. We have to work in the space Vp:= (u, (u)) : u2W1,p (⇤✏),(u) = 0 on @⇤ , where denotes the trace operator u2W1,p(⇤✏)7! u|@⇤✏2W11 p,p(@⇤✏). New W1,p-estimates are needed to deal with problem (10). To prove these estimates rigorously, we use the Galerkin approximations where we have to introduce a special basis consisting of functions in the space Vs:= (u, (u)) : u2Ws,2(⇤✏),(u) = 0 on @⇤ ,sN(p2) 2p+1, in the sense of Lions [38, p. 161]. Therefore, thanks to the assumption made on s,wehaveVs⇢Vp.Weusethe so-called energy method introduced by Tartar [44] and considered by many authors (see, for instance, Cioranescu and Donato [15]). Finally, in order to identity the limit equation, it is necessary to use monotonicity arguments. In summary, we prove that the solution of problem (10), properly extended to the whole ⇤, converges to the unique solution of a new nonlinear problem, defined all over the domain ⇤, given by a new operator and containing extra zero order terms, capturing the e↵ect of the influence of the non-homogeneous dynamical condition imposed on the boundary of T✏. Theorem 1.1 (Convergence Theorem).Assume (2)-(4), (6)–(7) and (9). On the nonlinear term f2,wesuppose that f22C1(R)and its associated exponent q2satisfies pq2<+1if p=Nand 2q2(N1)p Npif p2[2,N).(11) We suppose that (v✏, ✏)is the unique solution of (10), with (v0 ✏, 0 ✏)2Vp, and where ✏(t)=(v✏(t)) a.e. t2(0,¯ T]. Let ˆv✏be the W1,p-extension of v✏to ⇤⇥(0,¯ T). Then, we have ˆv✏(t)!v(t)in Lp(⇤),as ✏!0,8t2[0,¯ T], where “!” denotes the strong convergence and vis the unique solution of the problem given by (✓⇤+✓T)@tvdiv b(rv)+✓⇤(↵|v|p2v+f1(v)) + ✓Tf2(v)=0,in ⇤⇥(0,¯ T),(12) with Dirichlet boundary condition v=0,on @⇤⇥(0,¯ T),(13) 3 and initial condition v(x, 0) = v0(x),for x2⇤,(14) where ✓⇤=|Z⇤|/|Z|,✓T=|@T|/|Z|,Zis the representative cell in RN,Tis an open subset of Z,Z⇤=Z\¯ T and |Z|(respectively |@T|and |Z⇤|) denotes the measure of Z(respectively @Tand Z⇤). For any ⇣2RN,ifw(y)is the solution of the problem 8 < :ZZ? |ryw(y)|p2ryw(y)·r y'(y)dy =0 8'2Hper(Z⇤), w2⇣·y+Hper(Z⇤), (15) then bis defined by b(⇣)= 1 |Z|ZZ⇤ |ryw(y)|p2ryw(y)dy, (16) where Hper(Z⇤)is the space of functions from W1,p(Z⇤)which have the same trace on the opposite faces of Z. Remark 1.2. If the di↵usion is modeled by the Laplacian operator (i.e. p=2), then the limit problem (12)-(14) is the problem obtained in [5, Theorem 6.1]. Remark 1.3. It seems possible to improve the regularity assumed on the functions f1and f2when they are assumed non-increasing and H¨older continuous (for more details, see the recent book [23] where D´ıaz et al. present this improvement for the linear di↵usion case). We organize this work as follows. In the next section, we present some notations, definitions and properties of suitable spaces for the study of (10). Some preliminary results are established in Section 3, some estimates for the solution of (10) are rigorously derived in Section 4, and a convergence result is indicated in Section 5. Finally, in Section 6 we study the limit problem and a conclusion section is established in Section 7. 2 The Sobolev spaces Ws,p(⇤✏) In this section we recall the Sobolev spaces Ws,p, which will be used in this paper (see Adams and Fournier [1], Br´ezis [13, Chapter 9] and Neˇcas [40, Chapter 2] for more details about them). For any positive integer sand p1, we define the Sobolev space Ws,p(⇤✏) to be the completion of Cs(⇤✏), with respect to the norm ||u||s,p,⇤✏=0 @X 0|↵|s |D↵u|p p,⇤✏1 A 1/p . Observe that Ws,p(⇤✏) is a Banach space. By the Sobolev embedding Theorem (see [1, Chapter 4, p. 99]), we have the embedding Ws,p(⇤✏)⇢W1,r(⇤✏),(17) where s1 and 1 ps1 N1 r1 p. In particular, for p1, we define the Sobolev space W1,p(⇤✏) to be the completion of C1(⇤✏), with respect to the norm ||u||p,⇤✏:= ⇣|u|p p,⇤✏+|ru|p p,⇤✏⌘1/p , where |·| p,⇤✏is the norm in Lp(⇤✏). We set H1(⇤✏)=W1,2(⇤✏). 4 We define p?=⇢Np Npif p<N, +1if p=N. We have the continuous embedding W1,p(⇤✏)⇢⇢Lr(⇤✏)if p<N and r=p?, Lr(⇤✏)if p=Nand pr<p ?.(18) By Rellich-Kondrachov Theorem (see [13, Chapter 9, Theorem 9.16]), we have the compact embedding W1,p(⇤✏)⇢⇢Lr(⇤✏)if p<N and 1 r<p ?, Lr(⇤✏)if p=Nand pr<p ?.(19) In particular, we have the compact embedding W1,p(⇤✏)⇢L2(⇤✏),82pN. (20) One can define a family of spaces intermediate between Lpand W1,p. More precisely for p1wedefinethe fractional order Sobolev space W11 p,p(@⇤✏):=(u2Lp(@⇤✏); |u(x)u(y)| |xy|11 p+N p 2Lp(@⇤✏⇥@⇤✏)), equipped with the natural norm. We set H1/2(@⇤✏)=W1 2,2(@⇤✏). These spaces play an important role in the theory of traces. The trace operator is denoted by such that u7! u|@⇤✏. This operator belongs to L(W1,p(⇤✏),W11 p,p(@⇤✏)). We denote by |||| the norm of in this space. We will use k·k p,@⇤✏to denote the norm in W11 p,p(@⇤✏),which is given by kkp,@⇤✏=inf{kukp,⇤✏:(u)=}. We define p? b=((N1)p Npif p<N, +1if p=N. We have the continuous embedding W11 p,p(@⇤✏)⇢⇢Lr(@⇤✏)if p<N and r=p? b, Lr(@⇤✏)if p=Nand 1 r<p ? b.(21) By [40, Chaper 2, Theorem 6.2], we have the compact embedding W11 p,p(@⇤✏):=W1,p(⇤✏)⇢Lr(@⇤✏)if 1r<p ? b.(22) In particular, we have the compact embedding W11 p,p(@⇤✏)⇢L2(@⇤✏)82pN. (23) Some important notations for reading the paper: We define a few notations. For ⇤✏, we call Hp:= Lp(⇤✏)⇥Lp @⇤(@⇤✏), 8p2, where Lp @⇤(@⇤✏):={u2Lp(@⇤✏):u= 0 on @⇤},8p2. 5 On Hp, we consider the norm |(·,·)|Hpgiven by |(u, )|p Hp=|u|p p,⇤✏+✏||p p,@T✏,(u, )2Hp,8p2, where |·| p,⇤✏is the norm in Lp(⇤✏) and |·| p,@T✏is the norm in Lp(@T✏). In this paper, it is very important the following space Vp:= n(u, (u)) : u2W1,p @⇤(⇤✏)o,8p2, where W1,p @⇤(⇤✏):={u2W1,p(⇤✏):(u) = 0 on @⇤},8p2. Observe that Vpis a closed vector subspace of W1,p @⇤(⇤✏)⇥W11 p,p @⇤(@⇤✏),where W11 p,p @⇤(@⇤✏):={u2W11 p,p(@⇤✏):u= 0 on @⇤},8p2. We endow it with the norm k(·,·)kVpgiven by k(u, (u))kp Vp=kukp p,⇤✏+k(u)kp p,@T✏,(u, (u)) 2Vp. Let p0,q0 1and q0 2be the conjugate exponents of p,q1and q2, respectively. Taking into account the continuous embeddings (18) and (21), and the assumptions (2)-(5), we have the following useful continuous inclusions Vp⇢W1,p(⇤✏)⇢Lq1(⇤✏)⇢L2(⇤✏),V p⇢W11 p,p(@⇤✏)⇢Lq2(@⇤✏)⇢L2(@⇤✏),(24) and L2(⇤✏)⇢Lq0 1(⇤✏)⇢W1,p(⇤✏)0⇢V0 p,(25) where W1,p(⇤✏)0and V0 pdenote the dual of W1,p(⇤✏) and Vp, respectively. Note that W1,p(⇤✏)0is a subspace of W1,p0(⇤✏), where W1,p0(⇤✏) denotes the dual of the Sobolev space W1,p 0(⇤✏):=D(⇤✏)W1,p(⇤✏). Taking into account the compact embeddings (20) and (23), we have the compact embedding Vp⇢H282pN. (26) Finally, for s1, we consider the space Vs:= n(u, (u)) : u2Ws,2 @⇤(⇤✏)o. We note that Vsis a closed vector subspace of Ws,2 @⇤(⇤✏)⇥Ws1 2,2 @⇤(@⇤✏). Observe that by (17), we have that Ws,2(⇤✏)⇢W1,2(⇤✏) and by Rellich’s Theorem (see [40, Chapter 1, Theorem 1.4]), we obtain the compact embedding W1,2(⇤✏)⇢L2(⇤✏), and by [40, Chapter 1, Exercise 1.2], we have the compact embedding Ws1 2,2(@⇤✏):=0Ws,2(⇤✏)⇢L2(@⇤✏). Thus, we can deduce the compact embedding Vs⇢H2.(27) 6 3 Preliminary results In the latter we will need the following results: Remark 3.1 (Additional conditions on the nonlinear terms).Observe that we have the following conditions on the nonlinear terms: |f1(s)|K⇣1+|s|q11⌘,|f2(s)|K⇣1+|s|q21⌘,for all s2R,K>0.(28) Theorem 3.2 (Existence and uniqueness of solution of (10)).Assume (2)–(8), then there is a unique solution (v✏, ✏)of the problem (10) such that, for all ¯ T>0,v✏2C([0,¯ T]; L2(⇤✏)) [Lp(0,¯ T;W1,p(⇤✏)),✏2 C([0,¯ T]; L2 @⇤(@⇤✏)) [Lp(0,¯ T;W11 p,p @⇤(@⇤✏)) where (v✏(t)) = ✏(t)a.e. t2(0,¯ T]. We have that (v✏, ✏)satisfies dt(v✏(t),w)⇤✏+✏d t(✏(t),(w))@T✏+(|rv✏(t)|p2rv✏(t),rw)⇤✏+↵(|v✏(t)|p2v✏(t),w)⇤✏(29) +(f1(v✏(t)),w)⇤✏+✏(f2(✏(t)),(w))@T✏=0,8w2W1,p @⇤(⇤✏), in D0(0,¯ T),withtheinitialcondition v✏(0) = v0 ✏,and ✏(0) = 0 ✏,(30) where (·,·)⇤✏is the inner product in L2(⇤✏)or (L2(⇤✏))Nand the duality product between Lr0(⇤✏)and Lr(⇤✏) if r6=2,and(·,·)@T✏is the inner product in L2(@T✏)and the duality product between Lr0(@T✏)and Lr(@T✏)if r6=2. Moreover (v✏, ✏)satisfies the energy equality 1 2dt|(v✏(t), ✏(t))|2 H2+|rv✏(t)|p p,⇤✏+↵|v✏(t)|p p,⇤✏ +(f1(v✏(t)),v ✏(t))⇤✏+✏(f2(✏(t)), ✏(t))@T✏=0,a.e. t2(0,¯ T).(31) Proof. It is based on the theory of monotonicity of Lions [38]. We will show that operator Bp:Vp!V0 p, given by hBp((w,(w))),(u, (u))i:= (|rw|p2rw,ru)⇤✏+↵(|w|p2w, u)⇤✏,8w,u 2W1,p @⇤(⇤✏),(32) is coercive, that is hBp((w,(w)) ,(w,(w)))i k(w,(w))kVp !+1when k(w,(w))kVp!+1. We have hBp((w,(w)) ,(w,(w)))imin{1,↵}kwkp p,⇤✏(33) =1 1+kkpmin{1,↵}kwkp p,⇤✏+kkp 1+kkpmin{1,↵}kwkp p,⇤✏ 1 1+kkpmin{1,↵}k(w,(w))kp Vp,8w2W1,p @⇤(⇤✏), so Bpis coercive. By [38, Ch.2,Th.1.4], we have that (29)–(30) has a unique solution and satisfies the energy equality. Remark 3.3 (Energy inequality).By (31) and using (3) and (6), we obtain the energy inequality dt|(v✏(t), ✏(t))|2 H2+2|rv✏(t)|p p,⇤✏+2↵|v✏(t)|p p,⇤✏+2⌘1|v✏(t)|q1 q1,⇤✏(34) +2⌘1✏|✏(t)|q2 q2,@T✏2(|⇤✏|+✏|@T✏|), where |⇤✏|(respectively |@T✏|) denotes the measure of ⇤✏(respectively @T✏). 7 Remark 3.4 (Estimates for the measures of ⇤✏and @T✏).Taking into account the number of holes, we can deduce (see [6, Section 4] for more details) |@T✏|K ✏,K>0. And since |⇤✏||⇤|, we have that |⇤✏|+✏|@T✏|K, K > 0.(35) 4 Some estimates for the solution of (10) If we denote F1(s):=Zs 0 f1(r)dr and F2(s):=Zs 0 f2(r)dr, we can deduce that e⌘1|s|q1e F 1(s)e⌘2|s|q1+e 8s2R,(36) and e⌘1|s|q2e F 2(s)e⌘2|s|q2+e 8s2R.(37) with e⌘i,e >0,i=1,2. Lemma 4.1. We suppose (2)–(7) and (9). There is a constant Kindependent of ✏, such that the solution (v✏, ✏)of the problem (10) satisfies kv✏kp,⇤✏,¯ TK, sup t2[0,¯ T] kv✏(t)kp,⇤✏K, (38) for any initial condition (v0 ✏, 0 ✏)2Vp, and where || · ||p,⇤✏,¯ Tis the norm in Lp(0,¯ T;W1,p(⇤✏)). Proof. Using (35) in (34), we obtain dt|(v✏(t), ✏(t))|2 H2+2|rv✏(t)|p p,⇤✏+2↵|v✏(t)|p p,⇤✏+2⌘1|v✏(t)|q1 q1,⇤✏+2⌘1✏|✏(t)|q2 q2,@T✏K. (39) From (32)-(33), in particular, we can deduce dt|(v✏(t), ✏(t))|2 H2+2min{1,↵} 1+kkp||(v✏(t),(v✏(t)))||p VpK. (40) We integrate between 0 and tand taking into account (9), in particular, we have the first estimate in (38). In order to obtain the second estimate in (38), we have to take the inner product in the problem (10) with v0 ✏. To do this, we need that v0 ✏2Lp(0,¯ T;W1,p @⇤(⇤✏)) \Lq1(0,¯ T;Lq1(⇤✏)) with (v0 ✏)2Lq2(0,¯ T;Lq2 @⇤(@⇤✏)). As we do not have it for our solution, we have to use the Galerkin method with the properties of Bpgiven by (32). At first, we introduce a special basis consisting of functions (uj,(uj)) 2V swith sN(p2) 2p+ 1 in the sense of [38, Chapter 2, Remark 1.6, p. 161]. Therefore, thanks to the assumption made on s, taking into account (17), we have Vs⇢Vp. The scalar product in H2generates on Vs⇢H2the bilinear functional ((w, (w)),(u, (u)))H2 which can be represented in the form ((w,(w)),(u, (u)))H2=hL((w,(w))),(u, (u))iVs, where Lis a self-adjoint operator. The compact embedding (27) implies the compactness of the operator L. Hence, Lhas a complete system of eigenvectors {(uj,(uj)) : j1}. These vectors are orthonormal in H2and orthogonal in Vs. 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