Localization phenomena in degenerate logistic equation

Transcription

1 Localization phenomena in degenerate logistic equation José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal 1,2 1 Universidad Complutense de Madrid, Madrid, Spain 2 Instituto de Ciencias Matemáticas, Madrid, Spain Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems Northern Arizona University Flagsta, Arizona, USA José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

2 Localization phenomena in degenerate logistic equation José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal 1,2 1 Universidad Complutense de Madrid, Madrid, Spain 2 Instituto de Ciencias Matemáticas, Madrid, Spain Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems Northern Arizona University Flagsta, Arizona, USA José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

3 Outline José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 phenomena (UCMICMAT) June 6-9, / 35

4 Outline 1 Introduction 2 Behavior of the positive equilibria 3 Boundedness and unboundedness of solutions José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

5 Let us consider the following problem u t u = λu n(x)u ρ in Ω, t > 0 u = 0 on Ω, t > 0 u(0) = u 0 0 (1) in a bounded domain Ω R N, N 1, with ρ > 1, λ R and n(x) 0 in Ω is a continuous function not identically zero. We will assume that n(x) remains strictly positive near the boundary of Ω and therefore K 0 = {x Ω : n(x) = 0} (2) K 0 is a compact set. José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

6 Hypothesis (HK) K 0 = K 1 K 2 Ω, where K 1 = Ω 0, is the closure of a regular open set Ω 0, and K 2 is a regular d dimensional manifold, with d N 1, (Hn) as a consequence int K 2 = and K 2 = 0. where d 0 (x) := dist(x, K 0 ) n(x) (d 0 (x)) γ for some γ 0. José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

7 1 On the region where n(x) = 0, the growth rate is Malthusian, an exponential growth, and could be expected to be unbounded. 2 In the region where n(x) > n 0 > 0 it is a logistic growth, and could be expected to be bounded. José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

8 1 On the region where n(x) = 0, the growth rate is Malthusian, an exponential growth, and could be expected to be unbounded. 2 In the region where n(x) > n 0 > 0 it is a logistic growth, and could be expected to be bounded. What kind of behavior can be expected?. The answer is known when K 2 =, see [Ouyang, Gámez, G MeliánG ReñascoL G& Sabina 98, G MeliánLetelier& Sabina, L G] and references therein. Assume for instance, that K 0 = {x Ω : dist(x, Ω) η 0 }. In that case, it is known that the solution u λ in K 0 \ Ω as λ λ 1 (int(k 0 )). José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

9 The question in our situation is: Will u λ blow up on K 2 or not?. We are interested in analyzing possible interactions between n(x), the dimension of K 2 and the exponent ρ. Applications 1 in the nonlinear Schrodinger equation and 2 in the study of Bose - Einstein condensates. This phenomenon explains the fact that the ground state presents a strong localization in the spatial region in which n(x) = 0, [Pardo& P G]. José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

10 Theorem Assume K 0 satises (HK). Assume n(x) satises (Hn). Assume also that γ + 2 < (ρ 1)(N d). Then if λ λ 1 (Ω 0 ), any solution of (1) satisfy lim u(x, t) =, for all x Ω 0, (3) t and remains bounded on K 2 \ (K 1 K 2 ). José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

11 Lemma Assume K R N dimension is a compact set with zero Lebesgue measure and dim(k) = d and consider a function dened on a bounded neighborhood Ω of K of the form f (x) = (dist(x, K)) α for α > 0. If then rα < N d, f L r (Ω) with r 1. José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

12 Outline 1 Introduction 2 Behavior of the positive equilibria 3 Boundedness and unboundedness of solutions José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

13 Stationary solutions { u = λu n(x)u ρ in Ω, u = 0 on Ω. (1) Lemma Assume that ϕ 0 is a nonegative classical stationary solution of (1) for λ = λ 0. Then the following holds: (i) λ 0 (λ 1 (Ω), λ 1 (Ω 0 )) (ii) For each λ in a neighborhood of λ 0 there exists a unique nonnegative stationary solution of (1), ϕ λ which is moreover a smooth function of λ. (iii) The equilibria ϕ λ is an increasing function of λ. José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

14 Sketch of the proof (i) Assume that (λ 0, ϕ 0 ) is a non negative nontrivial stationary solution, then λ 0 = λ 1 ( + n(x)ϕ ρ 1 0, Ω), that is, λ 0 is the rst eigenvalue of the operator + n(x)ϕ ρ 1 0 in Ω, with Dirichlet boundary conditions. This and the monotonicity of the rst eigenvalue with respect to the potential implies that λ 0 > λ 1 (Ω). On the other side, the monotonicity with respect to the domain of this eigenvalue gives λ 0 < λ 1 ( + n(x)ϕ ρ 1 0, Ω 0 ), Also note that λ 1 ( + n(x)ϕ ρ 1 0, Ω 0 ) = λ 1 (, Ω 0 ). José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

15 (ii) F : (λ, u) u λu + n(x)u ρ (2) F : R C 2 0 (Ω) C(Ω) where C 2 0 (Ω) := {u C 2 (Ω) : u = 0, Ω}. We apply the implicit function theorem for (λ, u) = (λ 0, ϕ 0 ). 1 F is a continuously dierentiable map 2 F (λ 0, ϕ 0 ) = 0, 3 and λ 0 = λ 1 ( + n(x)ϕ ρ 1 0 ). (3) José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

16 Moreover, and D u F (λ 0, ϕ 0 ) = λ 0 + ρn(x)ϕ ρ 1 0, λ 1 ( λ 0 + ρn(x)ϕ ρ 1 0 ) > λ 1 ( λ 0 + n(x)ϕ ρ 1 0 ) = 0, (4) i.e. D u F (λ 0, ϕ 0 ) is an isomorphism. So, for each λ in a neighborhood of λ 0 1 there is a unique solution ϕ λ of (1) in a neighborhood of ϕ 0 2 and the map λ ϕ λ is continuously dierentiable with ϕ λ0 = ϕ 0. José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

17 (iii) Let v := dϕ λ dλ, taking derivatives with respect to λ in (1) we obtain { v = λv + ϕ ρn(x)ϕ ρ 1 v in Ω v = 0 on Ω. (5) Due to (3)-(4) and to the maximum principle, v := dϕ λ dλ > 0, therefore ϕ λ is an increasing function of λ. José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

18 Theorem For any λ (λ 1 (Ω), λ 1 (Ω 0 )) there exists a unique classical solution ϕ λ C0 2 (Ω) of (1) strictly positive. Moreover the following holds: (i) ϕ λ is globally asymptotically stable for nonnegative nontrivial solutions of (1). (ii) Even more, as λ λ 1 (Ω 0 ), we have ϕ λ L 1 (Ω). (6) Remark This result is know when n(x) is a smooth function, and int K 0 = Ω 0 is an open set with regular boundary, see [Ouyang, Theorem 2] and [Gámez, Theorem 3.2]. José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

19 Sketch of the proof We apply the Crandall-Rabinowitz's theorem, at (λ, u) = (λ 1 (Ω), 0). F (λ, 0) = 0 for every λ R. F is a continuously dierentiable map D u F (λ 1 (Ω), 0) = λ 1 (Ω) D λu F (λ 1 (Ω), 0) = I. and Fredholm alternative implies that D λu F (λ 1 (Ω), 0)Φ 1 = Φ 1 R( λ 1 (Ω)) where Φ 1 is the rst eigenfunction corresponding to the eigenvalue problem in Ω with Dirichlet boundary conditions. José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

20 Therefore, (λ 1 (Ω), 0) is a bifurcation point for positive solutions, i.e there exists a neighborhood of the bifurcation point and continuous functions (λ, u) : ( ε, ε) R C 2 0 (Ω) such that λ(0) = λ 1 (Ω), u(s) = s(φ 1 + v(s)) > 0 for s small enough, and (λ(s), u(s)) are the only nontrivial solutions of (1) in that neighborhood of (λ 1 (Ω), 0). José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

21 1 the local branch is composed of positive solutions. 2 any solution in the branch does not degenerate into a turning point So for each λ in the projection of the branch to the parameter space, there exists a unique nonnegative stationary solution of (1), ϕ λ. By the Rabinowitz's therorem, the local branch is also a global branch. Moreover, the branch is unbounded. Therefore, 1 either, there exists some µ λ 1 (Ω 0 ) < such that ϕ λ C 2 (Ω) as λ µ, 2 or λ 1 (Ω 0 ) =. If µ < λ 1 (Ω 0 ) by sub-supersolutions method, there is a bounded solution, which contradicts that λ > µ. José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

22 (i) See [R Bernal & Vidal]. (ii) By monotonicity, there exists the pointwise limit ϕ (x) = lim ϕ λ(x). (7) λ λ 1 (Ω 0 ) If ϕ L 1 (Ω), it can be proved, by a bootstrap argument, that ϕ L (Ω). Sobolev's Theorem ϕ λ ϕ in W 1, (Ω) C( Ω), therefore ϕ is a weak solution and, by a bootstrap argument, ϕ will be a classical solution with λ = λ 1 (Ω 0 ), which contradicts part (i). José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

23 Outline 1 Introduction 2 Behavior of the positive equilibria 3 Boundedness and unboundedness of solutions José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

24 What happens as λ λ 1 (Ω 0 )?, Where and how solutions become unbounded? Lemma Let {u λ } for λ (λ 1 (Ω), λ 1 (Ω 0 )) denote the family of positive solutions of (1). Then lim λ λ 1 (Ω 0 ) u λ(x) =, for all x Ω 0. José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

25 Remark In case K 1 = Ω 0 where Ω 0 is a smooth open set, and K 2 = this problem has been studied in [Ouyang] [FraileKochL G& Merino] [Gámez] [G MeliánG ReñascoL G& Sabina 98] [L G & Sabina] and further developments in [G Reñasco] [G Reñasco & L G] [L G] where the concept of metasolution is introduced as the limiting prole of the parabolic solutions as t. José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

26 To get upper bounds on the solutions we will use the following, see [G MeliánG ReñascoL G& Sabina 98], see also [Keller, Osserman]. Lemma Assume ρ > 1 and λ, β > 0 and consider a ball in R N of radius a > 0 and the following singular Dirichlet problem { z = λz βz ρ in B(0, a) Then z = on B(0, a). There exists a unique positive radial solution, z a (x). The solution satises ( ) 1 ( λ ρ 1 λ(ρ + 1) za (0) = inf β z a(x) + B ) 1 ρ 1 B(0,a) 2β βa 2 for some constant B > 0. José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

27 Proposition Let x 0 Ω \ K 0 and let u 0 0 be a bounded initial data for (1). Then for any given λ λ 1 (Ω 0 ) there exists b > 0 and M > 0 such that 0 u(t, x; u 0 ) M, x B(x 0, b), t 0. José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

28 Assume K 0 satises (HK) and has a decomposition in two separated components K 0 = K 1 K 2, where K 1 K 2 =. The following result shows that, for λ λ 1 (Ω 0 ), all solutions of (1) remain bounded in K 2, while all solutions of (1) start to grow up in K 1. Theorem Assume K 0 satises (HK). Assume also K 0 = K 1 K 2 where K 1 K 2 =. For any λ λ 1 (Ω 0 ) all solution of (1) are bounded on K 2. If λ < λ 1 (Ω 0 ) then all solution of (1) are bounded on K 1. If λ λ 1 (Ω 0 ) then the pointwise limit of the solutions of (1) is unbounded on K 1. José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

29 Now we want to discuss the behavior of solutions in parts of K 0 that can not be separated in components, but still have two parts, both glued together. First using Lemma 6 we prove the following universal bounds for solutions of (1). Lemma Assume that n(x) satises (Hn). For any solution of (1) there exists a constant A = A(u 0, λ) such that with d 0 (x) = dist(x, K 0 ). 0 u(t, x; u 0 ) h(x) = ( A ) γ+2 ρ 1 d 0 (x) José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

30 Main result Let K 0 = K 1 K 2 and K 1 K 2. Assume that K 0 satises hypothesis (HK). Then, we prove that solutions of (1) become unbounded on K 1 while remains bounded in K 2. Theorem Let K 0 = K 1 K 2 where K 1 K 2. Assume K 0 satises hypothesis (HK). Assume also that n(x) satises hypothesis (Hn). If γ + 2 < (ρ 1)(N d), then any solution of (1) remains bounded on K 2 \ K 1 although it is unbounded in K 1, for any λ λ 1 (Ω 0 ). José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

31 References I J. M. Arrieta, R. Pardo and A. Rodríguez-Bernal. Asymptotic behavior of degenerate logistic equations. In Preparation. V. M. Pérez-García and R. Pardo. Localization phenomena in nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities: theory and applications to Bose-Einstein condensates. Phys. D, 238(15): , J. M. Fraile, P. Koch Medina, J. López-Gómez, and S. Merino. Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation. J. Dierential Equations, 127(1):295319, José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

32 References II J. L. Gámez. Sub- and super-solutions in bifurcation problems. Nonlinear Anal., 28(4), , J. García-Melián, R. Gómez-Reñasco, J. López-Gómez, and J. C. Sabina de Lis. Pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from innity occurs. Arch. Ration. Mech. Anal., 145(3):261289, J. García-Melián, R. Letelier-Albornoz, and J. Sabina de Lis. Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up. Proc. Amer. Math. Soc., 129(12): (electronic), José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

33 References III R. Gómez-Reñasco. The eect of varying coecients in semilinear elliptic boundary value problems. From classical solutions to metasolutions, Ph D Dissertation, Universidad de La Laguna, Tenerife, March R. Gómez-Reñasco and J. López-Gómez, On the existence and numerical computation of classical and non-classical solutions for a family of elliptic boundary value problems Nonlinear Analysis: Theory, Methods & Applications, 48(4), J. B. Keller. On solutions of u = f (u). Comm. Pure Appl. Math., 10:503510, José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

34 References IV J. López-Gómez and J. C. Sabina de Lis. First variations of principal eigenvalues with respect to the domain and point-wise growth of positive solutions for problems where bifurcation from innity occurs. J. Dierential Equations, 148(1):4764, J. López-Gómez Metasolutions: Malthus versus Verhulst in population dynamics. A dream of Volterra, Stationary partial dierential equations. Vol. II, Handb. Dier. Equ., , Elsevier/North-Holland, Amsterdam, (2005), José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

35 References V T. Malthus. An Essay on the Principle of Population. Oxford World's Classics reprint. 1798: 1 a edición anónima. 1803: 2 a edición más extensa, y rmada. Ouyang, Tiancheng, On the positive solutions of semilinear equations u + λu hu p = 0 on the compact manifolds, Trans. Amer. Math. Soc., 331(2), , R. Osserman. On the inequality u f (u). Pacic J. Math., 7: , José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, / 35

36 References VI A. Rodríguez-Bernal and A. Vidal-López. Extremal equilibria for nonlinear parabolic equations in bounded domains and applications. Journal of Dierential Equations, 244: , Departamento de Matemática Aplicada, UCM, Preprint Series MA-UCM P. F. Verhulst. Notice sur la loi que la population poursuit dans son accroissement. Corresp. Math. Phys., 10, , P. F. Verhulst. Recherches mathématiques sur la loi d'accroissement de la population. Mémoires de l'académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique, 18, Bruselas, 142, José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 phenomena (UCMICMAT) June 6-9, / 35