Combining linear and nonlinear diffusion

Combining linear and nonlinear diffusion Manuel Delgado1 Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla Calle Tarfia s/n 41012-Sevilla, Spain e-mail: [email protected] Juli´an L´opez-G´omez2 Departamento de Matem´atica Aplicada Universidad Complutense de Madrid 28040-Madrid, Spain e-mail: [email protected] Antonio Su´arez1 Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla Calle Tarfia s/n 41012-Sevilla, Spain e-mail: [email protected] 1Supported by the Spanish Ministry of Science and Technology under Grants BFM2000-0797 and BFM2003-06446. 2Supported by the Spanish Ministry of Science and Technology under Grants BFM2000-0797 and REN2003-00707. Abstract In this paper we study a generalized porous medium equation where the diffusion rate, say m(x) —spatially heterogeneous—, is assumed to be linear, m= 1, on a piece of the support domain, Ω1, and slow nonlinear, m(x)>1, in its complement, Ωm:= Ω \ ¯ Ω1. Most precisely, we characterize the existence of positive solutions and construct the corresponding global bifurcation diagram as one of the parameters of the model changes, showing that a continuous transition occurs between the diagrams of the completely linear case (Ω = Ω1) and of the completely nonlinear case (Ωm= Ω). As a result, the effect of a localized slow diffusion rate with varying support is completely characterized. Our analysis is imperative in order to design porous media multi-components systems with changing diffusion rates. Key Words. Nonlinear diffusion. Spatial heterogeneities. From linear to nonlinear diffusion. AMS Classification. 35B32, 35J25, 35J60, 35K57. 2M. Delgado, J. L´opez-G´omez and A. Su´arez 1 Introduction In this paper we study the positive solutions of the following boundary value problem    −∆¡wm(x)¢=λw in Ω , w= 0 on ∂Ω,(1.1) where Ω ⊂IRN,N≥1, is a bounded domain of class C2,λ∈IR , and m= 1 + p χΩm where Ωmis an open smooth subdomain of Ω such that ¯ Ωm⊂Ω and p∈ C(¯ Ωm) satisfies p(x)>0 for each x∈Ωm. Finally, we denote by Ω1:= Ω \¯ Ωm, the open set where m= 1. Given any measurable set M⊂Ω, χMstands for the characteristic function of M, i.e., χM(x) = 1 for each x∈M, and χM(x) = 0 for each x∈Ω\M. An admissible choice for pwould be taking a constant p > 0. Then, m=χΩ1+ (1 + p)χΩm. In Figure 1, we have represented an admissible configuration. The dark region stands for Ωm, where m>1, and the white region is the subdomain of Ω where m= 1. Ω Ω 1 m Linear diffusion Nonlinear diffusion Figure 1. An admissible configuration. In the special case Ωm=∅, (1.1) reduces to the classical linear eigenvalue problem for the Laplacian under Dirichlet boundary conditions in Ω. Subsequently, for any potential V∈L∞(Ω) we shall denote by σ[−∆ + V; Ω] the principal eigenvalue of −∆ + Vin Ω under homogeneous Dirichlet boundary conditions. According to Krein–Rutman theorem, (1.1) possesses a positive solution if, and only, if λ=σ[−∆; Ω]. Actually, in such case, all positive solutions are multiples of a principal eigenfunction. Combining linear and nonlinear diffusion 3 On the other hand, when Ωm= Ω and mis constant, (1.1) provides us with the classical porous medium equation, which generated a huge industry in Partial Differential Equations since the pioneering studies of G. I Barenblatt [2] and A. G. Aronson & L. A. Peletier [1]. In fact, one of the results of [1] establishes that (1.1) possesses a positive solution if, and only if, λ > 0, and that it is unique and asymptotically stable if it exists. Actually, if we denote it by wλit turns out that limλ↓0wλ= 0 and that λ7→ wλis increasing (cf. [4] for further details). In Figure 2, we have represented a bifurcation diagram scheme of the positive solutions of (1.1) in these extreme opposite cases. Figure 2(a) shows the bifurcation diagram for the linear eigenvalue problem, and Figure 2(b) represents the bifurcation diagram of positive solutions for the classical porous medium equation. In Figure 2(a) we have denoted σ1:= σ[−∆; Ω]. Our main interest in this paper is focused into the problem of analyzing how change these diagrams when diffusion is nonlinear in some piece of Ω, Ωm, whereas it is linear in the complement, Ω1, trying to ascertain all possible intermediate eventual transitions between the previous two limiting cases. Such an analysis is imperative in order to study the effect of local nonlinear diffusivities in the global dynamics of porous media. Consequently, we will throughout assume that Ωm, and so Ω1, are proper subdomains of Ω. It should be noted that, though the non-linearity is discontinuous, it is of Caratheodory in L∞and, hence, all solutions live in W1,p for all p > 1. The analysis of this problem fits into our general program of analyzing reaction diffusion equations in the presence of spatial heterogeneities; those heterogeneities might arise in nonlinear diffusion rates, of course. As it will become clear later the global nature of the corresponding bifurcation diagram of positive solutions of the general problem we are dealing with is rather different. ww 00 σ1 λ λ (a) (b) Figure 2. Bifurcation diagram in the limiting cases. Since the change of variable u=wm(x) transforms (1.1) into    −∆u=λu 1 m(x)in Ω , u= 0 on ∂Ω.(1.2) 4M. Delgado, J. L´opez-G´omez and A. Su´arez most of our attention will be focused into (1.2). By elliptic regularity theory, it is folklore that any weak non-negative solution u6= 0 is an strong solution almost everywhere twice differentiable and, as a result of the maximum principle, u(x)>0 for each x∈Ω and ∂u ∂n (x)<0 for each x∈∂Ω, where nstands for the outward normal vector-field of Ω. Therefore, a necessary condition for the existence of a positive weak solution is λ > 0. The following function will play a crucial role in our exposition µ(λ) := σ[−∆−λχΩ1; Ω] , λ ∈[0,∞).(1.3) It satisfies µ(0) >0, and, due to the monotonicity of the principal eigenvalue with respect to the potential, it is decreasing in λ. Actually, it satisfies µ0(λ)<0 for each λ > 0, since λ7→ µ(λ) is concave; by a celebrated theorem of P. Hess and T. Kato [6] (cf. [8] for further details). Moreover, by the monotonicity of the principal eigenvalue with respect to the domain, given a ball B⊂Ω1for each λ≥0 we have that µ(λ)< σ[−∆−λχΩ1;B] = σ[−∆; B]−λ . Thus, lim λ↑∞ µ(λ) = −∞ and, hence, there exists λ0=λ0(Ω1)∈(0, σ[−∆; Ω1]) such that µ−1(0) ∩[0,∞) = {λ0}.(1.4) Actually, λ0(Ω1)> σ[−∆; Ω] since µ(σ[−∆; Ω]) >0. Indeed, if we denote by ϕa principal eigenfunction associated with σ[−∆,Ω], then ¡−∆−σ[−∆; Ω]χΩ1¢ϕ=σ[−∆; Ω] ¡1−χΩ1¢ϕ > 0, and, thanks to [8, Theorem 2.5], it is apparent that µ(σ[−∆; Ω]) >0. Moreover, as a result of the classical theory of P. Hess and T. Kato, µ(λ) is real analytic and concave (e.g., [6] and [8]). Once introduced these notations, we can state our main results. The next one provides us with the bifurcation diagram of positive solutions. Theorem 1.1 Problem (1.2) possesses a positive solution if, and only if, 0< λ < λ0,(1.5) and it is unique if it exists. Moreover, if we denote it by θλ, then, for each α∈(0,1), the map λ7→ θλis increasing and of class C1((0, λ0); Cα 0(¯ Ω)). Further, lim λ↓0kθλkC1+α(¯ Ω) = 0 and lim λ↑λ0 kθλkC(K)=∞,(1.6) for any compact subset K⊂Ω. Combining linear and nonlinear diffusion 5 In Figure 3 we have represented the corresponding diagram of positive solutions of (1.2). 0λλ λ 0 u θ Figure 3. Bifurcation diagram in the general case. The bifurcation diagram consists of an increasing differentiable curve emanating from u= 0 at λ= 0 and blowing-up to infinity, everywhere in Ω, as λ↑λ0. It should be noted that the u-bifurcation diagrams for (1.2) look similar to those shown in Figure 2 for (1.1). The next result establishes the existence of an homotopy between the two limiting bifurcation diagrams of Figure 2 and the bifurcation diagram of Figure 3. The concept of domain convergence used in its formulation is the one introduced in [8], for which there is continuous dependence of the principal eigenvalue and of the normalized principal eigenfunction in W1,2 0. Theorem 1.2 Let m∈(1,∞)and {Ωε m}{0<ε≤1}a monotone family of C2subdomains of Ωsuch that Ω1 m= Ωmand Ωε 1:= Ω \¯ Ωε m,0< ε ≤1. Set mε=χΩε 1+m χΩε m,0< ε ≤1,(1.7) and denote by θ[λ,ε],0< λ < λ0(Ωε 1),0< ε ≤1, the unique positive solution of ½−∆u=λu 1 mε(x)in Ω, u= 0 on ∂Ω.(1.8) Then, the following assertions are true: (a) If limε↓0Ωε 1= Ω, then lim ε↓0λ0(Ωε 1) = σ[−∆; Ω] ,(1.9) and, for each λ∈(σ[−∆; Ω], λ0(Ω1)), there exists a unique ε0∈(0,1) such that λ0(Ωε0 1) = λ. Moreover, limε↓ε0θ[λ,ε]=∞uniformly on compact subsets of Ωand θ[λ,ε]=kθ[λ,ε]kC(¯ Ω)Φλ+o(kθ[λ,ε]kC(¯ Ω))as ε↓ε0in C1+α(¯ Ω) ,(1.10) where Φλstands for the principal eigenfunction of σ[−∆−λχΩε0 1 ; Ω] normalized so that kΦλkC(¯ Ω) = 1, while lim ε↓0kθ[λ,ε]kC(¯ Ω) = 0 (1.11) if λ∈(0, σ[−∆,Ω]). 6M. Delgado, J. L´opez-G´omez and A. Su´arez (b) If limε↓0Ωε m= Ω, then lim ε↓0λ0(Ωε 1) = ∞(1.12) and, for each λ∈(0,∞), lim ε↓0kθ[λ,ε]−ΘλkC(¯ Ω) = 0 (1.13) where Θλstands for the unique positive solution of the classical porous media equation ((1.2) with Ωm= Ω). The distribution of this paper is the following. In Section 2 we include the proof of Theorem 1.1 and analyze the asymptotic behavior of the positive solutions of the parabolic counterpart of (1.2). Finally, in Section 3 we prove Theorem 1.2. 2 Proof of Theorem 1.1 Subsequently, we denote by Pthe cone of positive functions of C1+α 0(¯ Ω); ◦ Pstanding for its interior. Given u,v∈ C1+α(¯ Ω), it is said that u>vif u−v∈P\ {0}, and uÀvif u−v∈◦ P. We already know that λ > 0 is necessary for the existence of a positive solution. Now, let ϕλÀ0 denote a principal eigenfunction associated to µ(λ) (cf. (1.3)) and asume that (1.2) possesses a positive solution, u. Then, multiplying (1.2) by ϕλ, and integrating in Ω it is apparent that µ(λ)ZΩ uϕλ=λZΩm u1 mϕλ. Thus, µ(λ)>0 and, therefore, λ<λ0(Ω1). Recall that µ(λ)>0 if and only if 0 <λ< λ0(Ω1). This shows that (1.5) is necessary for the existence. To show that (1.5) implies the existence of a positive solution we use the sharp version of the method of sub and supersolutions developed by P. Hess [5] which demands no regularity assumptions. Suppose (1.5) and consider ˜ ψ:=    ψin ¯ B , 0 in Ω \B , (2.1) where Bis a ball with ¯ B⊂Ωmand ψstands for the positive eigenfunction associated to σ[−∆; B] normalized so that kψkC(¯ B)= 1. It is routine to check that the function u:= ε˜ ψ is a weak subsolution of (1.2) if 0< ε ≤min (1,µλ σ[−∆; B]¶infBm infBm−1),(2.2) Combining linear and nonlinear diffusion 7 since ∂ψ ∂n <0 on ∂B, where nis the outward unit normal vector-field of B. It should be noted that inf B m>1. Actually, uprovides us with a subsolution for any λ > 0. Now, pick λ∈(0, λ0) and, for each sufficiently small δ > 0, consider Ωm,δ := {x∈Ωm: dist(x, ∂Ωm)> δ }, and µδ(λ) := σ[−∆−λχΩ1,δ ; Ω] , where Ω1,δ := Ω \¯ Ωm,δ . By the continuous dependence of the principal eigenvalue with respect to the potential, µδ(λ)>0 if δ > 0 is sufficiently small. Assume δhas been chosen in that way. Let ϕδ λ denote the positive eigenfunction associated to µδ(λ) normalized so that kϕδ λkC(¯ Ω) = 1. Then, the function ¯u:= Kϕδ λ provides us with a positive supersolution of (1.2) if K≥max          1,  λ µδ(λ)µinf Ωm,δ ϕδ λ¶ 1−supΩm,δ m supΩm,δ m  infΩm,δ m infΩm,δ m−1 ,µinf Ωm ϕδ λ¶−1         .(2.3) Note that infΩm,δ m>1. Finally, by choosing ε > 0 sufficiently small and K > 1 sufficiently large, it is clear that u≤¯uand, therefore, (1.2) possesses a weak positive solution in the interval [u, ¯u]; necessarily strong, by elliptic regularity. This concludes the proof of the existence. To prove the uniqueness we will adapt the argument given in the proof of [4, Theorem 3.2]. Suppose uis a positive solution of (1.2). Then,   ³−∆−λ u 1 m−1´u= 0 in Ω , u= 0 on ∂Ω, and, hence, by the uniqueness of the principal eigenvalue, we find that σ[−∆−λ u 1 m−1; Ω] = 0 .(2.4) Suppose (1.2) possesses a further positive solution v6=u. Then, −∆(u−v) = λ³u1 m−v1 m´=λ mZ1 0 [tu + (1 −t)v]1 m−1dt (u−v). 8M. Delgado, J. L´opez-G´omez and A. Su´arez Thus, setting W:= −λ mZ1 0 [tu + (1 −t)v]1 m−1dt , gives    (−∆ + W) (u−v) = 0 in Ω , u−v= 0 on ∂Ω.(2.5) In Ω1,W=−λ, while, in Ωm, Z1 0 [tu + (1 −t)v]1 m−1dt < u 1 m−1Z1 0 t1 m−1dt =mu1 m−1, and, hence, W > −λu 1 m−1. Thus, by the monotonicity of the principal eigenvalue with respect to the potential, we find from (2.4) that σ[−∆ + W; Ω] >0. As the principal eigenvalue is dominant, from (2.5) it is apparent that u=v. This contradiction ends the proof of the uniqueness. Subsequently, for each λ∈(0, λ0), we denote by θλthe unique positive solution of (1.2). The fact that the map (0, λ0)−→ Cα 0(¯ Ω) λ7→ θλ (2.6) is of class C1follows easily from the implicit function theorem applied to the operator (0, λ0)×◦ PT −→ Cα 0(¯ Ω) (λ, u)7→ u−λ(−∆)−1³u1 m´ whose zeros are in one-to-one correspondence with the positive solutions of (1.2). Tis of class C1and, for each λ∈(0, λ0), DuT(λ, θλ) : C1+α 0(¯ Ω) −→ Cα 0(¯ Ω) is the linear continuous compact operator defined by DuT(λ, θλ)u:= u−λ(−∆)−1µ1 mθ 1 m−1 λu¶, u ∈ C1+α 0(¯ Ω) . Combining linear and nonlinear diffusion 15 Fix ε < ε0. Thanks to the continuous dependence of the principal eigenvalue with respect to the domain, there exists δ0=δ(λ, ε)>0 such that µδ ε(λ) := σ[−∆−λχΩε 1; Ωδ]>0 if δ∈[0, δ0). Fix one of those δ’s and let ϕδdenote the principal eigenfunction of µδ ε(λ) normalized so that kϕδkC(¯ Ω) = 1. A direct calculation shows that the function ¯u:= Kϕδ provides us with a positive supersolution of (1.8) in Ω for each sufficiently small ε > 0 and sufficiently large K > 1, which can be chosen to be independent of ε. Notice that those supersolutions are bounded away from zero all over Ω. Also, thanks to (2.2), all the corresponding positive solutions are bounded bellow by a universal positive function —bellow the supersolution. By adapting the compactness argument of the proof of Part (a), one can easily see that Θλ:= limε↓0θ[λ,ε]À0 is well defined and that it provides us with a positive solution of the porous medium equation (i.e., (1.2) with Ωm= Ω). This concludes the proof. References [1] A. G. Aronson & L. A. Peletier, Large time behaviour of solutions of some porous medium equation in bounded domains, J. Diff. Eqns. 39 (1981), 378–412. [2] G. I. Barenblatt, On some unsteady motions of a liquid or a gas in a porous medium, Prikl. Mat. Meh. 16 (1952), 67–68. [3] H. Brezis & L. Oswald, Remarks on sublinear elliptic equations, Nonl. Anal. T.M.A. 10 (1986), 55–64. [4] M. Delgado, J. L´ opez-G´ omez & A. Su´ arez, Non-linear versus linear diffusion. From classical solutions to metasolutions, Adv. Diff. Eqns. 7(2002), 1101–1124. [5] P. Hess, On the solvability of nonlinear elliptic boundary value problems, Ind. Univ. Math. J.,25 (1976) 461-466. [6] P. Hess & T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Part. Diff. Eqns.,5(1980), 999-1030. [7] T. Kato,Perturbation Theory for Linear Operators, Classics in Mathematics, Springer, Berlin, 1995. [8] J. L´ opez-G´ omez, The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. Diff. Eqns.,127 (1996) 263294. 16 M. Delgado, J. L´opez-G´omez and A. Su´arez [9] D. Sattinger,Topics in Stability and Bifurcation Theory, Lectures Notes in Mathematics, 309, Springer, Berlin, 1973.