Reaction–Diffusion Equation on Thin Porous 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: Bulletin of the Malaysian Mathematical Sciences Society (2021): 02/03/2021 DOI: https://doi.org/10.1007/s40840-021-01103-0 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/s40840-021-01103-0” Reaction-di↵usion equation on thin porous media Mar´ıa ANGUIANO1 Abstract We consider a reaction-di↵usion equation on a 3D thin porous media of thickness "which is perforated by periodically distributed cylinders of size ". On the boundary of the cylinders we prescribe a dynamical boundary condition of pure-reactive type. As "!0, in the 2D limit the resulting reaction-di↵usion equation has a source term coming from the dynamical-type boundary conditions imposed on boundaries of the original 3D domain. AMS classification numbers: 35K57; 35B27 Keywords: Homogenization, energy method, dynamical boundary-value problems, porous media, thin films 1 Introduction and setting of the problem We consider a parabolic problem in a thin porous media ⌦"of thickness "which is perforated by periodically distributed cylinders (obstacles) of size ". On the boundary of the cylinders, we prescribe a dynamical boundary condition of pure-reactive type. The aim of this work is to prove the convergence of the homogenization process when "goes to zero. The asymptotic behavior of the solution of linear parabolic problems with dynamical boundary conditions of pure-reactive type in periodically perforated domains was analyzed by Timofte in [14]. The author obtained a new parabolic limiting problem defined on a unified domain with extra terms coming from the influence of the dynamical boundary conditions. More recently, in [1, 2], we study the nonlinear case where the nonlinearity appears reflected in the limit equation. However, to our knowledge, there does not seem to be in the literature any study of the asymptotic behavior of the solution of parabolic models associated with dynamical boundary conditions of pure-reactive type in a thin porous media. Let us introduce the model we will be involved with in this paper. The geometrical setting. The periodic porous media is defined by a domain !and an associated microstructure, or periodic cell Y0=[1/2,1/2]2, which is made of two complementary parts: the obstacle part F0, and Y0 f=Y0\F0. More precisely, we assume that !is a smooth, bounded, connected set in R2, and that F0 is an open connected subset of Y0with a smooth boundary @F0, such that F0is strictly included in Y0. The microscale of a porous media is a small positive number ". The domain !is covered by a regular mesh of square of size ": for k02Z2, each cell Y0 k0,"="k0+"Y0is divided in an obstacle part F0 k0,"and Y0 fk0,", i.e. is similar to the unit cell Y0rescaled to size ".WedefineY=Y0⇥(0,1) ⇢R3, which is divided in an obstacle part F=F0⇥(0,1) and Yf=Y0 f⇥(0,1), and consequently Yk0,"=Y0 k0,"⇥(0,1) ⇢R3, which is also divided in an obstacle part Fk0,"and Yfk0,". We denote by ⌧(F0 k0,") the set of all translated images of F0 k0,".Theset⌧(F0 k0,") represents the obstacles in R2. 1Departamento de An´alisis Matem´atico. Facultad de Matem´aticas. Universidad de Sevilla, 41012 Sevilla (Spain) [email protected] 1 Mar´ıa Anguiano We define !"=!\Sk02K"F0 k0,"⇢R2,where K"={k02Z2:Y0 k0,"\!6=;}, and we define the thin porous media ⌦"⇢R3by ⌦"={(x1,x 2,x 3)2!"⇥R:0<x 3<"}.(1) We make the following assumption: The obstacles ⌧(F0 k0,") do not intersect the boundary @!. We define F" k0,"=F0 k0,"⇥(0,"). Denote by S"the set of the obstacles contained in ⌦".Then,S"is a finite union of obstacles, i.e. S"=[ k02K" F" k0,". We define e ⌦"=!"⇥(0,1),⌦=!⇥(0,1),⇤"=!⇥(0,").(2) We observe that e ⌦"=⌦\Sk02K"Fk0,",and we define F"=Sk02K"Fk0,"as the set of obstacles contained in e ⌦". Position of the problem. We consider the following problem for a reaction-di↵usion equation with dynamical boundary conditions of pure-reactive type on the surface of the obstacles and zero Dirichlet condition on the exterior boundary, 8 > > > > > > > < > > > > > > > : @u" @tu"+u "=f(x0,t)in⌦ "⇥(0,T), ru"·⌫"+"@u" @t="g(x0,t) on @S"⇥(0,T), u"(x, 0) = u0 "(x),for x2⌦", u"(x, 0) = 0 "(x),for x2@S", u"=0,on @⇤"⇥(0,T), (3) where u"=u"(x, t), x=(x0,x 3)2⌦",t2(0,T) and T>0. The first equation states the law of standard di↵usion in ⌦",= xdenotes the Laplacian operator with respect to the space variable and >0 is a given constant. The boundary equation (3)2is multiplied by "to compensate the growth of the surface by shrinking ", where the value of u"is assumed to be the trace of the function u"defined for x2⌦",⌫"is the outer normal to @S".Thetermru"·⌫"represents the heat flux across the boundary (see Goldstein [11] for more details). We assume that u0 "2L2(⌦), 0 "2L2(@S"),(4) are given, and that there exists a positive constant Csuch that "1|u0 "|2 ⌦"+| 0 "|2 @S"C, (5) where |·|⌦"and |·|@S"denote the norm in L2(⌦") and L2(@S"), respectively. Moreover, we assume that f2L20,T;L2(!),g2L20,T;H1 0(!),(6) are given. This choice of fand gis usual when dealing with thin domains. Since the thickness of the domain, ", is small then the exterior force can be considered independent of the vertical variable. In this paper, our main motivation is to study the asymptotic behavior of the solution u"of (3) as ", the size of the cylinders and the corresponding thickness of the thin film, tend to zero. We combine the homogenization techniques used in porous media and in thin film in order to rigorously derive the homogenized model. 2 Mar´ıa Anguiano For this purpose, we first perform a change of variables which consists in stretching in the x3-direction by a factor 1/". As in [3], we use the dilatation in the variable x3, i.e. y3=x3 ",(7) in order to have the functions defined in the open set with fixed height e ⌦"defined by (2). In this sense, we define ˜u"by ˜u"(x0,y 3,t)=u"(x0,"y 3,t), a.e. (x0,y 3,t)2e ⌦"⇥(0,T),(8) and our goal then is to describe the asymptotic behavior of this new sequence ˜u"when "tends to zero. Using the dilatation in the variable x3given by (7), we have that u"is transformed into x0˜u"+1 "2@2 y3˜u". The factor 1/"2in front of the derivate in the y3direction means a very fast di↵usion in the vertical direction. In some sense, we have substituted the thin porous media ⌦"by a non-thin domain e ⌦", but with a very strong di↵usion mechanism in the y3-direction. Because of the presence of this very strong di↵usion mechanism, it is expected that solutions of the problem which satisfies ˜u", defined in (20), become homogeneous in the y3-direction so that the limit solution will not have a dependence in this direction, and therefore, the limit problem will be two dimensional. The method we follow in this paper is the so-called energy method of Tartar [15], which has been considered by many authors (see, for instance, Cioranescu and Donato [5], Conca and Donato [9], Conca et al. [10], Timofte [14] and [1]) and the technique introduced by Vanninathan [16] for the Steklov problems which transforms surface integrals into volume integrals. We get the following limit problem to (20), as "goes to zero. Theorem 1.1 (Main Theorem).Under the assumptions (5)–(6), assume that (u0 ", 0 ")2H1(⌦")⇥H1/2(@S") and there exists a positive constant Csuch that "1/2|ru0 "|⌦"C. (9) Let (u", ")be the unique solution of the problem (3), where "is the trace of the function u"defined for x2⌦" and a.e. t2(0,T]. Then, there exists an extension e ⇧"˜u"of ˜u", where ˜u"is the dilation of u"given by (8), into all ⌦⇥(0,T), such that, as "!0, e ⇧"˜u"(t)!u(t)strongly in L2(⌦),8t2[0,T], where u2L2(0,T;H1 0(!)) is the unique solution of the following problem 8 > > > > < > > > > : ✓|Y0 f| |Y0|+|@F0| |Y0|◆@u @tdivx0(Qrx0u)+|Y0 f| |Y0|u=|Y0 f| |Y0|f+|@F0| |Y0|g, in !⇥(0,T), u(x0,0) = u0(x0),for x02!, u=0,on @! ⇥(0,T). (10) The homogenized matrix Q=((qi,j)),1i, j 2, which is symmetric and positive-definite, is given by qi,j =1 |Y0|ZY0 f (ei+ry0wi)·(ej+ry0wj)dy0,(11) where wi2Hper \R,i=1,2, is the unique solution of the cell problem 8 > > > < > > > : divy0(ei+ry0wi)=0,in Y0 f, (ei+ry0wi)·⌫0=0,on @F0, wiis Y0periodic. (12) 3 Mar´ıa Anguiano Here, eiis the ielement of the canonical basis in R2,⌫0is the outer normal to @F0,andHper is the space of functions from H1(Y0 f)which are Y0-periodic. The structure of the paper as follows. In Section 2, we introduce some notations which are used in the paper. In Section 3, we give a weak formulation of the problem, and establish the existence and uniqueness of solution. In Section 4, we establish the problem which satisfies ˜u"given by (8). Some aprioriestimates for ˜u" are rigorously obtained in Section 5. A compactness result is addressed in Section 6. Finally, the main goal of proving the asymptotic behavior of the solution ˜u"is achieved in Section 7. 2 Some notations Along this paper, the points x2R3will be decomposed as x=(x0,x 3)withx02R2,x32R. We also use the notation x0to denote a generic vector of R2. We denote by e ⌦"the characteristic function of the domain e ⌦". We denote by (·,·)⌦"(respectively, (·,·)@S") the inner product in L2(⌦")(respectively,inL2(@S")), and by |·|⌦"(respectively, |·|@S") the associated norm. We also denote (·,·)⌦"the inner product in (L2(⌦"))3. We denote by (·,·)e ⌦"(respectively, (·,·)@F") the inner product in L2(e ⌦")(respectively,inL2(@F")), and by |·|e ⌦"(respectively, |·|@F") the associated norm. We also denote (·,·)e ⌦"the inner product in (L2(e ⌦"))N,with N=2,3. We denote by (·,·)⌦the inner product in L2(⌦), and by |·|⌦the associated norm. We also denote (·,·)⌦the inner product in (L2(⌦))3.Byk·k⌦we denote the norm in H1(⌦). We denote by (·,·)!"the inner product in L2(!") and by |·|!"the associated norm. We also denote (·,·)!"the inner product in (L2(!"))2.By||·||!"we denote the norm in H1(!"), which is associated to the inner product ((·,·))!":= (rx0·,rx0·)!"+(·,·)!". By k·ke ⌦"we denote the norm in H1(e ⌦"), which is associated to the inner product ((·,·))e ⌦":= (rx0·,rx0·)e ⌦"+(@y3·,@ y3·)e ⌦"+(·,·)e ⌦". By ||·||e ⌦",T we denote the norm in L2(0,T;H1(e ⌦")). By |·|e ⌦",T (respectively |·|@F",T ), we denote the norms in L2(0,T;L2(e ⌦")) and L2(0,T;(L2(e ⌦"))3)(respectively L2(0,T;L2(@F"))). By |·|⌦,T , we denote the norms in L2(0,T;L2(⌦)) and L2(0,T;(L2(⌦))3). We denote by 0the trace operator u7! u|@⌦". The trace operator belongs to L(H1(⌦"),H1/2(@⌦")), and we will use k0kto denote the norm of 0in this space. Analogously, we denote by 0the trace operator ˜u7! ˜u|@ e ⌦". We will use k·k@ e ⌦"to denote the norm in H1/2(@e ⌦"),which is given by k˜ k@ e ⌦"=inf{k˜vke ⌦":0(˜v)=˜ }. We recall that with this norm, H1/2(@e ⌦") is a Hilbert space. We denote by Hr @⇤"(⌦") and Hr @⇤"(@⌦"), for r0, the standard Sobolev spaces which are closed subspaces of Hr(⌦") and Hr(@⌦"), respectively, and the subscript @⇤"means that, respectively, traces or functions in @⌦", vanish on this part of the boundary of ⌦", i.e. Hr @⇤"(⌦")={v2Hr(⌦"):0(v) = 0 on @⇤"}, and Hr @⇤"(@⌦")={v2Hr(@⌦"):v= 0 on @⇤"}. Let us notice that, in fact, we can consider an element of H1/2(@S") as an element of H1/2 @⇤"(@⌦"), and we can consider the given 0 "as an element of L2 @⇤"(@⌦"). 4 Mar´ıa Anguiano We denote by Hr @⌦(e ⌦") and Hr @⌦(@e ⌦"), for r0, the standard Sobolev spaces which are closed subspaces of Hr(e ⌦") and Hr(@e ⌦"), respectively, and the subscript @⌦means that, respectively, traces or functions in @e ⌦", vanish on this part of the boundary of e ⌦", i.e. Hr @⌦(e ⌦")={˜v2Hr(e ⌦"):0(˜v) = 0 on @⌦}, and Hr @⌦(@e ⌦")={˜v2Hr(@e ⌦"):˜v= 0 on @⌦}. Let us consider the space H:= L2(e ⌦")⇥L2 @⌦(@e ⌦"), with the natural inner product ((˜v, ˜ ),(˜w, ˜'))H=(˜v, ˜w)e ⌦"+"(˜ ,˜')@F",which in particular induces the norm |(·,·)|Hgiven by |(˜v, ˜ )|2 H=|˜v|2 e ⌦"+"|˜ |2 @F",(˜v, ˜ )2H. Let us also consider the space V1:= n(˜v,0(˜v)) : ˜v2H1 @⌦(e ⌦")o. We note that V1is a closed vector subspace of H1 @⌦(e ⌦")⇥H1/2 @⌦(@e ⌦"),and therefore, with the norm k(·,·)kV1 given by k(˜v,0(˜v))k2 V1=k˜vk2 e ⌦"+k0(˜v)k2 @F",(˜v,0(˜v)) 2V1, V1is a Hilbert space. For a vectorial function v=(v0,v 3) and a scalar function w, we introduce the operators: r"and div",by r"w=(rx0w, 1 "@y3w)t,div"v=div x0v0+1 "@y3v3.(13) We denote by |O| the Lebesgue measure of |O| (3-dimensional if Ois a 3-dimensional open set, 2-dimensional of Ois a curve). Finally, we denote by Ca generic positive constant, independent of ", which can change from line to line. 3 Existence and uniqueness of solution of the problem (3) We state in this section a result on the existence and uniqueness of solution of problem (3). Definition 3.1. A weak solution of (3) is a pair of functions (u", "), satisfying u"2C([0,T]; L2(⌦")), "2C([0,T]; L2 @⇤"(@⌦")),for all T>0,(14) u"2L2(0,T;H1 @⇤"(⌦")),for all T>0,(15) "2L2(0,T;H1/2 @⇤"(@⌦")),for all T>0,(16) 0(u"(t)) = "(t),a.e. t2(0,T],(17) 8 > > > < > > > : d dt(u"(t),v)⌦"+"d dt( "(t), 0(v))@S"+(ru"(t),rv)⌦"+(u"(t),v)⌦" =(f(x0,t),v)⌦"+"(g(x0,t), 0(v))@S" in D0(0,T),forallv2H1 @⇤"(⌦"), (18) u"(0) = u0 ",and "(0) = 0 ".(19) 5 Mar´ıa Anguiano Thanks to [1, Theorem 3.3], we have the following result. Theorem 3.2. Under the assumptions (4)–(6), there exists a unique solution (u", ")of the problem (3). Moreover, this solution satisfies the energy equality 1 2 d dt |u"(t)|2 ⌦"+"| "(t))|2 @S"+|ru"(t)|2 ⌦"+|u"(t)|2 ⌦" =(f(x0,t),u "(t))⌦"+"(g(x0,t), "(t))@S",a.e. t2(0,T). 4 Dilatation in the variable x3: definition of ˜u" In this section, we use the dilatation in the variable x3, i.e. we use (7) in order to have the functions defined in the open set with fixed height e ⌦"defined by (2). In this sense, we define ˜u"2L2(0,T;H1 @⌦(e ⌦")), for all T>0, by (8), and using the transformation (7), the system (3) can be rewritten as 8 > > > > > > > < > > > > > > > : @˜u" @tdiv"(r"˜u")+˜u"=f(x0,t)in e ⌦"⇥(0,T), r"˜u"·˜⌫"+"@˜u" @t="g(x0,t) on @F"⇥(0,T), ˜u"(x0,y 3,0) = ˜u0 "(x0,y 3),for (x0,y 3)2e ⌦", ˜u"(x0,y 3,0) = ˜ 0 "(x0,y 3),for (x0,y 3)2@F", ˜u"=0,on @⌦⇥(0,T), (20) where div"and r"are given by (13), and ˜⌫"is the outer normal to @F". Taking in (3) as test function ˜v(x0,x 3/")with˜v2H1 @⌦(e ⌦"), applying the change of variables (7) and taking into account that dx ="dx0dy3and d(x)="d(x0)dy3, the variational formulation of problem (20) is then the following one 8 > > > > > < > > > > > : d dt ✓Ze ⌦" ˜u"(t)˜vdx0dy3◆+"d dt ✓Z@F" 0(˜u"(t))0(˜v)d(x0)dy3◆+Ze ⌦"r"˜u"(t)·r"˜vdx0dy3 +Ze ⌦" ˜u"(t)˜vdx0dy3=Ze ⌦" f(x0,t)˜vdx0dy3+"Z@F" g(x0,t)0(˜v)d(x0)dy3 in D0(0,T), for all ˜v2H1 @⌦(e ⌦"). (21) Moreover, ˜u"satisfies the energy equality 1 2 d dt |(˜u"(t), 0(˜u"(t)))|2 H+|r"˜u"(t)|2 e ⌦"+|˜u"(t)|2 e ⌦"(22) =(f(x0,t),˜u"(t))e ⌦"+"(g(x0,t), 0(˜u"(t)))@F",a.e. t2(0,T). Considering in (5) the change of variables given in (7), and taking into account that dx ="dx0dy3and d(x)= "d(x0)dy3, we can deduce that |˜u0 "|2 e ⌦"+"|˜ 0 "|2 @F"C. (23) 5A priori estimates for ˜u" Let us obtain some aprioriestimates for ˜u". Lemma 5.1. Under the assumptions (4)–(6), there exists a positive constant Cindependent of ", such that the solution ˜u"of the problem (20) satisfies |r"˜u"|e ⌦",T C, |˜u"|e ⌦",T C, (24) 6 Mar´ıa Anguiano |˜u"(t)|e ⌦"C, p"|0(˜u"(t))|@F"C, (25) for all t2(0,T). Proof. By (22), we have d dt |(˜u"(t), 0(˜u"(t)))|2 H+2|r"˜u"(t)|2 e ⌦"+2|˜u"(t)|2 e ⌦"|(˜u"(t), 0(˜u"(t)))|2 H+|f(t)|2 !"+"|g(t)|2 @F0 ". Integrating between 0 and t, we obtain |(˜u"(t), 0(˜u"(t)))|2 H+ 2min{1,}Zt 0⇣|r"˜u"(s)|2 e ⌦"+|˜u"(s)|2 e ⌦"⌘ds (26) |(˜u0 ", 0(˜u0 "))|2 H+Zt 0|(˜u"(s), 0(˜u"(s)))|2 Hds +ZT 0⇣|f(s)|2 !"+"|g(s)|2 @F0 "⌘ds. By [1, Lemma 4.1] with p= 2, we can deduce "|g(s)|2 @F0 "C|g(s)|2 !"+"2|rg(s)|2 !"C||g(s)||2 !", which together with (6) gives ZT 0⇣|f(s)|2 !"+"|g(s)|2 @F0 "⌘ds C. (27) Taking into account (23) and (27) in (26) and applying Gronwall Lemma, in particular we obtain (25). Finally, taking into account (23), (25) and (27) in (26), we get (24). Now, if we want to take the inner product in (20) with ˜u0 ", we need that u0 "2L2(0,T;H1 @⌦(e ⌦")). However, we do not have it for our weak solution. Therefore, we use the Galerkin method in order to prove, rigorously, new aprioriestimates for ˜u". On the space V1we define a continuous symmetric linear operator A1:V1!V0 1, given by hA1((˜v,0(˜v))),(˜w, 0(˜w))i=(r"˜v,r"˜w)e ⌦"+(˜v, ˜w)e ⌦", (28) for all ˜v, ˜w2H1 @⌦(e ⌦"). We observe that A1is coercive. In fact, taking into account that "<1, we have hA1((˜v,0(˜v)) ,(˜v,0(˜v)))i=|rx0˜v|2 e ⌦"+1 "2|@y3˜v|2 e ⌦"+|˜v|2 e ⌦"(29) >|rx0˜v|2 e ⌦"+|@y3˜v|2 e ⌦"+|˜v|2 e ⌦" min {1,}k˜vk2 e ⌦" =1 1+k0k2min {1,}k˜vk2 e ⌦" +k0k2 1+k0k2min {1,}k˜vk2 e ⌦" 1 1+k0k2min {1,}k(˜v,0(˜v))k2 V1, for all ˜v2H1 @⌦(e ⌦"). Let us observe that the space H1 @⌦(e ⌦")⇥H1/2 @⌦(@e ⌦") is compactly imbedded in H, and therefore, for the symmetric and coercive linear continuous operator A1:V1!V0 1,whereA1is given by (28), there exists a non-decreasing sequence 0 < 12... of eigenvalues associated to the operator A1with limj!1 j=1, 7 Mar´ıa Anguiano and there exists a Hilbert basis of H,{(wj, 0(wj)) : j1}⇢D(A1), with span{(wj, 0(wj)) : j1}densely embedded in V1, such that A1((wj, 0(wj))) = j(wj, 0(wj)) 8j1. Taking into account the above facts, we denote by (˜u",m(t), 0(˜u",m(t))) = (˜u",m(t;0,˜u0 ",˜ 0 "), 0(˜u",m(t;0,˜u0 ",˜ 0 "))) the Galerkin approximation of the solution (˜u"(t;0,˜u0 ",˜ 0 "), 0(˜u"(t;0,˜u0 ",˜ 0 "))) to (20) for each integer m1, which is given by (˜u",m(t), 0(˜u",m(t))) = m X j=1 "mj(t)(wj, 0(wj)),(30) and is the solution of d dt((˜u",m(t), 0(˜u",m(t))),(wj, 0(wj)))H +hA1((˜u",m(t), 0(˜u",m(t)))),(wj, 0(wj))i =(f(t),w j)e ⌦"+"(g(t), 0(wj))@F",j=1, . . . , m, (31) with initial data (˜u",m(0), 0(˜u",m(0))) = (˜u0 ",m, 0(˜u0 ",m)),(32) where "mj(t)=(˜u",m(t),w j)e ⌦"+(0(˜u",m(t)), 0(wj))@F", and (˜u0 ",m, 0(˜u0 ",m)) 2span{(wj, 0(wj)) : j=1,...,m}converge (when m!1)to(˜u0 ",˜ 0 ") in a suitable sense which will be specified below. Lemma 5.2. Suppose the assumptions (4)–(6). Then, for any initial condition (u0 ", 0 ")2H1(⌦")⇥H1/2(@S") of the problem (3) such that satisfies (9), there exists a positive constant Cindependent of ", such that the solution ˜u"of the problem (20) satisfies sup t2[0,T ]k˜u"(t)ke ⌦"C, (33) |˜u0 "|e ⌦",T C, p"|0(˜u0 ")|@F",T C. (34) Proof. First, considering in (9) the change of variables given in (7), and taking into account that dx ="dx0dy3 and @y3="@x3, we can deduce that (˜u0 ",˜ 0 ")2V1such that |r"˜u0 "|e ⌦"C. (35) For all m1, there exists (˜u0 ",m, 0(˜u0 ",m)) 2span{(wj, 0(wj)) : 1 jm}, such that the sequence {(˜u0 ",m, 0(˜u0 ",m))}converges to (˜u0 ",˜ 0 ")inV1. Then, taking into account (23) and (35), we know that there exists a positive constant Csuch that |r"˜u0 ",m|2 e ⌦"+|˜u0 ",m|2 e ⌦"C. (36) For each integer m1, we consider the sequence {(u",m(t), 0(u",m(t)))}defined by (30)-(32) with these initial data. Multiplying by the derivative 0 "mj in (31), and summing from j=1tom, we obtain |(˜u0 ",m(t), 0(˜u0 ",m(t)))|2 H +1 2 d dt(hA1((˜u",m(t), 0(˜u",m(t)))),(˜u",m(t), 0(˜u",m(t)))i) =(f(t),˜u0 ",m(t))e ⌦"+"(g(t), 0(˜u0 ",m(t)))@F". 8 Mar´ıa Anguiano Using (58) in (59), we have d dt ✓Z⌦ e ⌦"e ⇧"˜u"vP" i"dx0dy3◆+"d dt ✓Z@F" 0(˜u")v0( i")d(x0)dy3◆ +Z⌦ ˜ ⇠"·rx0vP" i"dx0dy3Z⌦ ^ rx0 i"·rx0ve ⇧"˜u"dx0dy3+Z⌦ e ⌦"e ⇧"˜u"vP" i"dx0dy3 =Z⌦ e ⌦"f(x0,t)vP" i"dx0dy3+"Z@F" g(x0,t)v0( i")d(x0)dy3, in D0(0,T). We consider '2C1 c([0,T]) such that '(T) = 0 and '(0) 6= 0. Multiplying by 'and integrating between 0 and T,wehave '(0) ✓Z⌦ e ⌦"e ⇧"˜u"(0)vP" i"dx0dy3◆ZT 0 d dt'(t)✓Z⌦ e ⌦"e ⇧"˜u"(t)vP" i"dx0dy3◆dt "'(0) ✓Z@F" 0(˜u"(0))v0( i")d(x0)dy3◆"ZT 0 d dt'(t)✓Z@F" 0(˜u"(t))v0( i")d(x0)dy3◆dt +ZT 0 '(t)Z⌦ ˜ ⇠"·rx0vP" i"dx0dy3dt ZT 0 '(t)Z⌦ ^ rx0 i"·rx0ve ⇧"˜u"dx0dy3dt (60) +ZT 0 '(t)Z⌦ e ⌦"e ⇧"˜u"(t)vP" i"dx0dy3dt =ZT 0 '(t)Z⌦ e ⌦"f(x0,t)vP" i"dx0dy3dt +"ZT 0 '(t)Z@F" g(x0,t)v0( i")d(x0)dy3dt. Now, we have to pass to the limit, as "!0. We will focus on the terms which involve the gradient. Taking into account (55), we reason as in steps 1 and 2 for the others terms. Firstly, using (46) and (55), we have ZT 0 '(t)Z⌦ ˜ ⇠"·rx0vP" i"dx0dy3dt !ZT 0 '(t)Z⌦ ⇠·rx0vx 0 idx0dy3dt, and by (45), (56) and Lebesgue’s Dominated Convergence Theorem, we obtain ZT 0 '(t)Z⌦ ^ rx0 i"·rx0ve ⇧"˜u"dx0dy3dt !1 |Y0|ZT 0 '(t)Z! ZY0 f (ei+ry0wi)dy0!·rx0vudx 0dt. 15 Mar´ıa Anguiano Therefore, when we pass to the limit in (60), we obtain '(0) ✓|Y0 f| |Y0|+|@F0| |Y0|◆✓Z! u(0)vx0 idx0◆✓|Y0 f| |Y0|+|@F0| |Y0|◆ZT 0 d dt'(t)Z! u(t)vx0 idx0dt +ZT 0 '(t)Z⌦ ⇠·rx0vx 0 idx0dy3dt 1 |Y0|ZT 0 '(t)Z! ZY0 f (ei+ry0wi)dy0!·rx0vu(t)dx0dt +|Y0 f| |Y0|ZT 0 '(t)Z! u(t)vx0 idx0dt =|Y0 f| |Y0|ZT 0 '(t)Z! f(t)vx0 idx0dt +|@F0| |Y0|ZT 0 '(t)Z! g(t)vx0 idx0dt. Using Green’s formula and equation (53), we have ZT 0 '(t)Z⌦ ⇠·rx0x0 ivdx0dy3dt +1 |Y0|ZT 0 '(t)Z! ZY0 f (ei+ry0wi)dy0!·rx0uvdx 0dt =0. The above equality holds true for any v2D(⌦) independent of y3, and '2C1 c([0,T]). This implies that ✓Z1 0 ⇠dy3◆·rx0x0 i+1 |Y0| ZY0 f (ei+ry0wi)dy0!·rx0u=0,in !⇥(0,T). We conclude that divx0✓Z1 0 ⇠dy3◆=div x0(Qrx0u),(61) where Q=((qij)), 1 i, j 2, is given by qij =1 |Y0|ZY0 f (ei+ry0wi)·ejdy0. Observe that if we multiply system (12) by the test function wj, integrating by parts over Y0 f, we obtain ZY0 f (ei+ry0wi)·ry0wjdy0=0, then we conclude that qij is given by (11). Step 5. Finally, thanks to (53) and (61), we observe that usatisfies the first equation in (10). A weak solution of (10) is any function u, satisfying u2C([0,T]; L2(!)),for all T>0, u2L2(0,T;H1 0(!)),for all T>0, ✓|Y0 f| |Y0|+|@F0| |Y0|◆d dt(u(t),v)+(Qrx0u(t),rx0v)+|Y0 f| |Y0|(u(t),v)=|Y0 f| |Y0|(f(t),v)+|@F0| |Y0|(g(t),v),in D0(0,T), for all v2H1 0(!), and u(0) = u0. Since the homogenized matrix Qis positive-definite (see [5, Theorem 4.7]), applying a slight modification of [12, Chapter 2,Theorem 1.4], we obtain that the problem (10) has a unique solution, and therefore Theorem 1.1 is proved. 16 Mar´ıa Anguiano References [1] M. Anguiano, Existence, uniqueness and homogenization of nonlinear parabolic problems with dynamical boundary conditions in perforated media, Mediterr. J. Math., (2020) 17:18. [2] M. 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