Neus Consul y. stable equilibria. Such type of solutions are denoted patterns. In our setting, IR 2

Transcription

1 Stable boundary layers in a diusion problem with nonlinear reaction at the boundary Jose M. Arrieta Neus Consul y Anbal Rodrguez-Bernal 1 Introduction In this paper we consider the evolution problem with a nonlinear reaction acting on the boundary 8 u t "u = in t > >< + g(x u) = on t > >: u( ) = () H 1 (): and study the existence and behavior as the parameter " goes to zero of non constant stable equilibria. Such type of solutions are denoted patterns. In our setting, IR is a bounded domain with a C 1 and connected boundary denotes the normal derivative. The nonlinearity isgiven by g(x u) =u(1 u)(c(x) u) x u IR where c C 1 () satises <c(x) < 1 for all x, and the transversality (z) 6= for all z with c(z) =1=, where is the unit tangent vector at the boundary. Letus note that this condition implies that there exist only a nite and even number of points x i, i =1 ::: N satisfying c(x i )=1=. partially supported by BFM-798 y partially supported by CICYT PB98-93-C-1 1

2 In [4] we showed the existence of patterns, u ",for" small enough which are close to a limit prole u. These equilibria satisfy 8 < : "u = + g(x u) = on (1.3) and they present layers at the boundary located at the points x i,thatwe will denote by transition points. These patterns are obtained as global minimizers of the appropriate energy functional related to problem (1.1). Moreover the trace of u at the boundary is a characteristic function with values where c(x) > 1= and 1 where c(x) < 1=. Observe that there are some results on the literature about the existence or non existence of patterns for some other similar problems. If the nonlinearity at the boundary does not have a spatial dependence, that is, g(x u) =g(u) andisaballinir N, N, it is known that there are no patterns for any ">, that is, only constant solutionscan be stable equilibria, see [5]. Also, it is known that for problem (1.1) with " =1,the formation of patterns can be achieved with a typical bistable cubic nonlinearity with no spatial dependence, g(u) =u u 3, when the domain looses its convexity. This is the case of the so called dumbbell domains which consist of two disjoint xed domains joined by a thin channel, see [6, 7]. Patterns also appear when the boundary of the domain is disconnected, like a domain with holes, see [5]. Therefore here we show that spacedependent boundary ux provides a mechanism for pattern formation even on convex domain with connected boundary. These results have been obtained very much in the spirit of the famous results presented in [9, 1] which assert the non existence of patterns in convex domains for some reaction diusion equations with Neumann boundary conditions. In this direction, the one dimensional problem given by 8 >< >: u t = " u xx + f(x u) <x<1 t > u x ( t)=u x (1 t)= t> u(x ) = (x) x 1 : (1.4) with " IR + and f(x u) = u(1 u)(u a(x)) was considered in [1]. The function a() C 1 ([ 1]) satises <a(x) < 1, for all x [ 1] and a transversality condition similar to (1.). For " small, a complete classication of all stable equilibrium solutions of problem (1.4) was given in that paper. In [11] some similar results are obtained allowing the function a() being discontinuous. Recently in [8], similar results have been proven for higher dimensional problems of the form 8 >< >: u t = "r(d(x)ru)+(1 u = u(x ) = (x) : (t x) IR t> (1.5)

3 where IR N, N. Here the function a() C () and satises 1 <a(x) < 1 and vanishes on some hypersurfaces. There are some obvious dierences between problems (1.4) and (1.1) due to the dimension of and the presence of a nonlinear Neumann boundary condition. In certain sense, problem (1.3) is a dimensional problem. Hence, most of the basic tools for the one dimensional problem, e.g. sub and super solutions, the underlying second order boundary value problem, etc., are no longer available. In this paper we study the convergence of the patterns u " to u in dierent norms, including C 1() and in loc L1 ( nfx 1 ::: x N g). That is, we show that the patterns converge, as "!, strongly in the sup norm outside small neighborhoods of the transition points. Notice that the limit prole has jumps at these points and therefore it is not possible to obtain an L 1 convergence in the whole domain. To obtain this result we need to perform a careful analysis of the behavior of the patterns near the boundary. This is done in Section 3. Once the patterns have been obtained it is important to study their stability regions as the parameter goes to zero. We will show the existence of "-independent nested regions, R n, around the limiting prole u with the property that if an initial condition is given in R n and if " is small enough the solution of (1.1) with this initial condition will remain in R n1 for all time. The fact that these regions are independent of" means that the prole of the patterns is persistent as" approaches zero. This is done in Section 4. We also include in Section a brief summary of the results obtained previously in [4], which are needed for the rest of the paper. There still remains many interesting questions for this problem, like for instance, the asymptotic stability and uniqueness of the patterns, the monotonicity of the layers at the boundary and, in general, the shape of the layers. Existence of non-constant stable equilibria In this section we recall and summarize earlier results announced by the authors in [4]. These results state the existence of nontrivial stable equilibria for (1.1) and also some preliminary description of their shape as "!. by Associated to problem (1.1) we have the energy functional J " : H 1 ()! IR dened where G(x u) = u J " (u) = " jruj + G(x u) dx (.1) g(x s)ds. This energy acts as a Lyapunov function for the solutions of (1.1), since it strictly decreases in time except at equilibria. 3

4 Note that natural candidates for stable equilibrium solutions are absolute minimizers of the energy functional (.1), whose existence is guaranteed by the direct method of calculus of variations. Moreover, by maximum principles, these equilibria take values between and 1. To prove that global minimizers are non constant we actually show that, as "!, they are close to some non constant prole, u (x), which is obtained as follows. Observe rst that for xed x, the function s IR! G(x s) has an absolute minimum at s = if c(x) > 1= and at s =1ifc(x) < 1=. This implies that for any function u() dened onwehave G(x u(x))dx G(x (x))dx = m, where (x) denotes characteristic function on the boundary given by ( 1 x with c(x) < 1= (x) = x with c(x) > 1= : Since this function minimizes the boundary term in the energy (.1), it is natural to guess that a global minimizer of the full energy would be obtained by considering the harmonic extension to of, u which satises ( u = in u (x) =(x) on : However, 6 H 1= () and therefore u 6 H 1 (), so it is not actually a minimizer of the energy. Moreover, it can be shown that u 1, u H s () for any s<1, u C 1 () and in fact u C ( n x 1 ::: x N ). To prove that global minimizers approach somehow the prole u, as "!, we proceed as follows. By mollication on the boundary we construct functions " C 1 () with " 1suchthat " = in ", where " =n[ N B(x i=1 i "). Using Fourier transform we show that the harmonic extension to of ", that we denote by v ", satises krv " k L () Cj log(")j and therefore J " (v " ) m + C "j log(")j for some constant C independent of". In particular this implies that global minimizers lie in the set K " = fu H 1 () J " (u) < m + C "j log(")jg which is open, nonempty and positively invariant for (1.1). Moreover absolute minimizers satisfy that jru " j Cj log(")j (.) for some constant C independent of". From here and the transversality condition on the coecient c(x) we are able to show that ku " k L () + ku " u k L () C("j log(")j) 1 4 : (.3) This implies the convergence in L p () and L p () for any 1 p<1. 4

5 3 L 1 -convergence of minimizers outside transition points In the previous section we have obtained the convergence of minimizers in L p () and in L p () for any 1 p<1. Since the limit function u is discontinuous at the transition points, it is not possible to obtain convergence in L 1 () or L 1 (). Nevertheless, we will show in this section that if we avoid the transition points, the minimizers will converge in the norm of the supremum. To obtain this result we will need to perform a careful analysis of the behavior of the function u " near the boundary and outside the transition points fx 1 ::: x N g. As a matter of fact we can show the following result Proposition 3.1 The convergence ofu " to u holds in C 1 loc () and in L1 loc ( nfx 1 x N g). Before proving the result, let us state and prove the following technical lemma that will be needed below. Lemma 3. If C 1 ( ) and u is a harmonic function which is regular on supp() we have jr(u)j = u jrj : Proof of Lemma 3.. Integrating by parts one obtains jr(u)j = u @n u u @n u u 1 ru r u : Again, integrating by parts in the last integral we have u = = r r(u u ru r u jrj + Putting these two inequalities together we obtain u : Proof of Proposition 3.1. We start by perfoming an analysis in the interior of. For any > small enough, denote by U = fx :d(x ) g. We are going to show that there exists a constant C independent of" and such that j" log(")j1=4 ku " u k L 1 (U ) C (3.1) 5

6 Let C 1 () a function with the property that =1inU, =inn U = = fx :d(x ) =g and j (x)j 1, jr (x)j C= and j (x)j C=. Denote by z " = u " u and by w = z ". Notice that w depends on both and ". The function w satises the equation: ( w =rz" r + z " in w = on : (3.) Now we estimate both terms in the right hand side of (3.). Applying Lemma 3. with u = z " and = we obtain that U jrz " j jr(z " )j = z " jr j" j log(")j1= C for any > where we have used (.3). Appying this estimate to U = weobtain 1 j" krz " r k L () Ckrz " k log(")j1= 1 j" log(")j1= L (U = ) C C : (=) 4 Similarly, from (.3). 1 j" kz " k L () Ckz " k log(")j1= L () C : 4 4 With these estimates, applying elliptic regularity theory to (3.) we get j" log(")j1=4 kwk H () C : Using the embedding H (),! L 1 () we obtain j" log(")j1=4 ku " u k L 1 (U ) kwk L 1 () C where the constant C is independent of" and. This shows (3.1). To show thec 1 loc()-convergence we proceed as follows. Applying again elliptic regularity theory to equation (3.) we obtain that for any k 1 ku " u k H k+1 (U ) C()ku " u k H k (U = ): (3.3) Given now any xed open set U with U, and any r 1we can choose a sequence of < < 1 < < r with the property that U i+1 = U i, i = ::: r 1, and U U r. 6

7 Using (3.3) we get that ku " u k H i (U i ) C( i )ku " u k H i1 (U i = ) C( i )ku " u k H i1 (U i1 ) which, by repeated iterations, implies that ku " u k H r (U ) C( r )ku " u k L (U ) C(U)j" log(")j 1=4! as "! : Since this is obtained for arbitrary large r we obtain the convergence of the minimizers in C 1 (U). Since U is also arbitrary we get the C 1 loc ()-convergence. We remark that the constant C(U) depends only on the distance of U to the boundary of. We analyse now the convergence near but outside the transition points. We parameterize with a map :[ S]!, using for instance the arclength. Observe thatby the regularity of@ the map (s ) [ S] ( )! (s) ~n, where ~n denotes the outward normal direction, is a smooth parameterization of the set n U as long as is small enough. For any two points < and for small enough we denote by R = fx = () ~n : g K = fx = (s) ~n s [ ] g = S s R s = fx = (s) s [ ]g 1 = fx = (s) ~n s [ ]g: Notice that with this = [1 [R [R. Denote by s i [ S] the values such that(s i )= x i for i =1 ::: N. Without loss of generality wemay assume that s 1 =. We also denote by x N +1 = x 1 = () = (S), see Figure 1. Let us choose in (3.1) (") =" for some <1=8. From (3.1) we still have ku " u k L 1 (U(") )! as "! : (3.4) We are going to show thatforany < with [ ] (s i s i+1 ), for some i = 1 ::: N +1,wehave ku " u k L 1 (K (") )! as "! : (3.5) Since the function u is either 1 or in and, therefore, it is constant on s i s i+1, i =1 ::: N +1, wewillhave that in, u is either constantly or constantly 1. Let us assume that u in. The other case is done analogously. Hence, by the continuity ofu outside the transition points and since (")!, we willhave that ku k L 1 (K (") )! as"!. Hence, to show (3.5) it will be enough to show ku " k L 1 (K (") )! as "! : (3.6) 7

8 s i+1 σ R σ,δ K σ,σ,δ σ,σ Γ Γ 1 σ,σ,δ σ R σ,δ U δ s i Ω Figure 1: Let us show now that for any and 1 with [ 1 ] (s i s i+1 )wehave that min ku " k L 1 (R (") )! as "! : (3.7) [ 1 ] Assume this is not true, then there will exist a positivenumber >andasequenceof "!, that we will still denote by ",such that for all [ 1 ]wehave ku " k L 1 (R (") ) On the other hand, since we also have ku k L 1 (K!, from (3.4) we obtain that (")) ku " k L 1 ( 1 1!. In particular for " small enough we will have that 1 (")) ku " k L 1 ( 1 =. 1 (")) But then, using the parameterization of K 1 (") given by(s ) [ 1 ]( ("))! x = (s) ~n we have that jru " j dx K 1 (") jru " j dx C 1 (") "(x(s )) j But notice that for s [ 1 ] xed, the function! " () u " (x(s )) has ((")) = and there exists a point for which " (). This implies that (") "() j d = "(x(s )) j d (") s [ 1 ]: 8

9 Taking into account thatwehave chosen (") =" for some <1=8, we obtain that which contradicts the fact that shows (3.7). jru " j dx C (=) (") C " jru " j dx Cj log(")j, asitwas shown in (.). This We are now in position to prove (3.6). For given [ ] (s i s i+1 ), choose (s i ) and ( s i+1 ). Applying (3.7) to [ ]andto[ ]we obtain that min ku " k L 1 (R (") )! as "! [ ] min [ ] ku " k L 1 (R (") )! as "! : In particular we canchoose (") [ ]and (") [ ] such that ku " k L 1 (R(") (") )! as "! ku " k L 1 (R (") (") )! as "! : Moreover, from (3.5) and the fact that ku k L 1 ( 1 ("))!, as "! wehave that ku " k L 1 ( 1 (") (") (") )! as "! : These facts mean that if we consider the set K (") (") ("), which contains K ("),in the three parts of the boundary given by 1 (") (") ("), R (") (") and R (") (") the function u " goes to in L 1. If it does not go to zero in L 1 in the whole of K (") (") (") we will reach toacontradiction with the fact that u " is a minimizer. To see this, consider 1 minfg(x 1) : x g. Then if we choose " small enough such that and if we dene the function ku " k L 1 (R(") (") ) = ku " k L 1 (R (") (") ) = ku " k L 1 ( 1 ) = (") (") (") ( u" (x) outside K w " (x) = (") (") (") minfu " (x) g K (") (") (") then, w " H 1 (). If there exists a point ink (") (") (") for which u " >,thenwehave that jrw " j dx < jru " j dx and G(x w " (x))dx G(x u " (x))dx: 9

10 This implies that J " (w " ) <J(u " ), which contradicts the fact that u " is a minimizer. This shows that neccesarily This proves the proposition. ku " k L 1 (K(") (") (") )! as "! : 4 Stability of proles The patterns u " wehave found for " small enough are near the prole given by the harmonic extension, u, of the characteristic function dened on the boundary. By its denition a pattern is a stable equilibrium of the system and this means that, for " xed, if we start near u " then the ow will remain in a neighborhood of the pattern. It is now important to study the stability regions of this equilibria as "!. As a matter of fact in this section we will construct a sequence of nested regions R n, that is R n R n1, around the limiting prole u with the property that, for < " < "(n), if we pick up an initial condition in R n the solution of (1.1) does not leave R n1 for all positive time. Moreover this regions are independent of" and their intersection contains only the limiting prole u, see Theorem 4.1. This does not imply that the only equilibrium inside R n is u ". There may coexistinr n with other equilibria and even some of them may be unstable. But it really ensures that if we start near the limiting prole u then, for " small enough, we will remain close to it for all positive time. In this sense is that we refer to stability of proles. The analysis performed so far has been done for the prototype nonlinearity g(x u) = u(u 1)(u c(x)), but all the results apply for the most general nonlinearity g a b (x u) = (u a)(u b)(u ~c(x)) where a<~c(x) <bfor all x. The role of the constants solutions u =,u = 1 is played now by u = a and u = b. The transitions will lie now in the points x i that satisfy ~c(x i )=(a + b)=. That is, the points where (a+b)= a g a b (x i u)du + b (a+b)= g a b (x i u)du =: Again we will need to (x i) 6= at the transition points. We goback again to our nonlinearity g(x u) =u(1 u)(c(x) u) in (1.1), which satises that <c(x) < 1 on. Hence there exists > such that the function c(x) satises <c(x) < 1 for all x : We consider now the functions g (x u) =(u + )(1 u)(c(x)+ u) g + (x u) =(u )(1 + u)(c(x) u) 1

11 for any strictly positive. Observe that with the notation above, g = g 1 with ~c(x) =c(x)+ and g + = g 1+ with ~c(x) =c(x). Let us remark that the functions g (x ), g(x ) and g + (x ) are ordered, that is, g + (x ) <g(x ) <g (x ). Since the function c is C (x i) 6=,for small enough the functions g (x ) and g + (x ) satisfy the same hypothesis of the function g(x ). That is, both are cubic functions with the middle zero moving with x under the given c(x), there exist exactly N transition points for each of them, that we denote by x i and x + i, i =1 ::: N, respectively and the transversality condition is satised for the transition points of each of them. Let us note that the transition points satisfy now c(x i )=1= for g (x ) and c(x + i )=1=+ for g + (x ), i =1 ::: N. Let us observe that, we have x i < x i < x + (x i) <. Now, we take the problem (1.1) for g (x v), that is, 8 >< >: v t "v = in t + g (x v) = on t > v( ) = () H (x i) > or x + i < x i < x i if (4.1) and for g + (x v), that is, 8 >< >: v t "v = in t + g+ (x v) = on t > v( ) = + () H 1 (): (4.) We consider as well the energy functionals J " J+ : " H 1 ()! IR, dened as in the rst section: J (v) = " " jrvj + G (x v) dx J + " (v) = " jrvj + where G (x v) = v g (x s)ds and G+ (x v) = v G + (x v) dx g + (x s)ds. For a given ">, let u " and u + " be absolute minimizers of J " and J ", + respectively. Finally, let us denote by and + the characteristic functions corresponding to the extreme zeros of g (x u) andg + (x u), dened in the same way as the function with respect to g(x u), in Section 1. Let u and u+ be the harmonic extensions to of the functions and + both dened on, respectively. By the construction we obviously 11

12 have that <<+,which implies by the maximum principle that u <u <u +. Similarly, ifwehave < we also obtain the orderings < <<+ <+, and again by the maximum principles we get the same ordering relations for the harmonic extensions, that is u <u <u <u + <u+, see Figure. χ + δ χ δ χ 1 x+ i x + i+1 x i x i+1 x i x i+1 c(x) Figure : The main result in this section is the next theorem. Theorem 4.1 Given any > small, there exists "() > such that for any! " [u u+ ] the solution of (1.1) with initial condition! ", that we will denote by u " (t! " ), satises u " (t! " ) [u u+ ], for all <"<"(), t. Remark 4. Observe that the ordered intervals [u u+ ] arenested and that T >[u u+ ] = fu g.inparticular Theorem 4.1 implies a stability of the prole given by u, in the sense that if we start with a prole near u, then for " small enough, we will remain close to it. Before proving this result we construct, for a given ">, a region which is positively invariant for the ow dened by (1.1), that is, a region with the property that if an initial condition is taken in this region the solution of (1.1) does not leave the region for any t>. Lemma 4.3 Given any > small enough, the functions u + ", u " are a super and subsolution of the problem (1.3), respectively. Moreover, there exists"() > such that u " <u " <u + " for all <"<"(). We remind that u + is a supersolution of (1.3) if it satises 8 >< >: "u + + g(x u+ ) on : 1

13 If u satises the reverse inequalities, u is a subsolution of (1.3). Proof. Since g + (x u) <g(x u) <g (x u) for all x andu, " u+ " are solutions of (4.1) and (4.), respectively, we conclude that they are sub and supersolutions, respectively, of (1.3). Now, we are going to see for a xed and small and for all " small enough u, " u " and u + are ordered, for all " x. Applying Proposition 3.1, which states the L 1 convergence far from the transition points of the minimizers to the appropriate characteristic functions, we have thatu " goes to and u + " goes to +,as"!, far from their respective transition points. Moreover, by the maximum principles we have the apriori bounds <u " < 1 and <u+ " < 1+. Using these two facts and since <<+ we conclude the proof of the lemma. Considering now theevolution problem (1.1), the existence of these u + ", u " and the monotonicity properties of (1.1) we have the next result. Corollary 4.4 For a given > small, and for all " > small enough the region [u " u+ " ] is positive invariant under the ow dened by (1.1). Moreover we have the stability of proles that we assert in the previous Theorem 4.1 in some regions "independent, as we are going to see in the next proof. Proof of Theorem 4.1 For a given we have that u <u 3= <u <u <u + <u+ 3= <u+ : From the L 1 convergence of minimizers outside the transition points we have that there exists "() small enough such that, for any <"<"(), [u u+ ] [u " 3= u+ " 3= ] [u u+ ]: Since by Corollary 4.4 we have that [u " 3= u+ ]isinvariant, we obtain that for any " 3= function w " [u u+ ] its evolution remains in [u " 3= u+ ] and therefore it can not " 3= leave the region [u u+ ]. Remark 4.5 The theorem above shows that if we start with an initial condition w " [u u+ ] its evolution does not leave the region [u u+ ]. As a matter of fact, using maximum principles it is easy to see that its!-limit set for T " (t), <"<"(), needs to lie in the interval [u u+ ] \ [ 1]. 13

14 References [1] S. Angenent, J. Mallet-Paret, L. Peletier, \Stable Transition Layers in a Semilinear Boundary Value Problem", Journal of Dierential Equations 67, 1-4 (1987). [] J.M. Arrieta, A.N. Carvalho and A. Rodrguez-Bernal, \Parabolic Problems with Nonlinear Boundary Conditions and Critical Nonlinearities". Journal of Dierential Equations 156, (1999). [3] J.M. Arrieta, A.N. Carvalho and A. Rodrguez-Bernal, \Attractors of Parabolic Problems with Nonlinear Boundary Conditions. Uniform Bounds", Comm. in Partial Di. Equations 5, 1-37 (). [4] J.M. Arrieta, N. Consul, A. Rodrguez-Bernal, \Pattern Formation from Boundary Reaction, Fields Inst. Comm. 31, (). [5] N. Consul, \On Equilibrium Solutions of Diusion Equations with Nonlinear Boundary Conditions",. Angew. Math. Phys. 47, (1996). [6] N. Consul, J. Sola-Morales, \Stable Nonconstant Equilibria in Parabolic Equations with Nonlinear Boundary Conditions", C.R. Acad. Sci. Paris, T. 31, Serie I, (1995). [7] N. Consul, J. Sola-Morales, \Stability of Local Minima and Stable Nonconstant Equilibria", Journal of Dierential Equations 157, (1999). [8] A. do Nascimento, \Inner Transition Layers in a Elliptic Boundary Value Problem", Nonl. Anal.: Th. Meth. and Appl., (to appear). [9] R. Casten, C. Holland, \Stability properties of solutions to systems of reactiondiusion equations", SIAM J. Appl. Math. 33, (1977), 353{364 [1] H. Matano, \Asymptotic behavior and stability of solutions of semilinear diusion equations", Publ. Res. Inst. Math. Sci. 15 (1979), [11] C. Rocha, \Examples of Attractors in Scalar Reaction Diusion Equations", Journal of Dierential Equations 73, (1988). Jose M. Arrieta Departamento de Matematica Aplicada Universidad Complutense de Madrid 84 Madrid, Spain. arrieta@sunma4.mat.ucm.es Neus Consul Departament de Matematica Aplicada I 14

15 Universitat Politecnica de Catalunya 88 Barcelona, Spain Anibal Rodrguez-Bernal Departamento de Matematica Aplicada Universidad Complutense de Madrid 84 Madrid, Spain 15