Localization phenomena in degenerate logistic equation

Localization phenomena in degenerate logistic equation José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal 1,2 rpardo@mat.ucm.es 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, 2012 1 / 35

Outline José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, 2012 3 / 35

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, 2012 4 / 35

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, 2012 5 / 35

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, 2012 6 / 35

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, 2012 7 / 35

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, 2012 7 / 35

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, 2012 8 / 35

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, 2012 9 / 35

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, 2012 10 / 35

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, 2012 12 / 35

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, 2012 13 / 35

(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, 2012 14 / 35

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, 2012 15 / 35

(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, 2012 16 / 35

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, 2012 17 / 35

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, 2012 18 / 35

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, 2012 19 / 35

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, 2012 20 / 35

(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, 2012 21 / 35

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, 2012 23 / 35

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, 2012 24 / 35

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, 2012 25 / 35

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, 2012 26 / 35

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, 2012 27 / 35

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, 2012 28 / 35

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, 2012 29 / 35

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):13521361, 2009. 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, 1996. José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, 2012 30 / 35

References II J. L. Gámez. Sub- and super-solutions in bifurcation problems. Nonlinear Anal., 28(4), 625632, 1997. http://dx.doi.org/10.1016/0362-546x(95)00174-t 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, 1998. 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):35933602 (electronic), 2001. José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, 2012 31 / 35

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 1999. 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), 2002. J. B. Keller. On solutions of u = f (u). Comm. Pure Appl. Math., 10:503510, 1957. José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, 2012 32 / 35

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, 1998. J. López-Gómez Metasolutions: Malthus versus Verhulst in population dynamics. A dream of Volterra, Stationary partial dierential equations. Vol. II, Handb. Dier. Equ., 211309, Elsevier/North-Holland, Amsterdam, (2005), http://dx.doi.org/10.1016/s1874-5733(05)80012-9 José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, 2012 33 / 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), 503527, 1992 http://dx.doi.org/10.2307/2154124, R. Osserman. On the inequality u f (u). Pacic J. Math., 7:16411647, 1957. José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, 2012 34 / 35

References VI A. Rodríguez-Bernal and A. Vidal-López. Extremal equilibria for nonlinear parabolic equations in bounded domains and applications. Journal of Dierential Equations, 244:29833030, 2008. Departamento de Matemática Aplicada, UCM, Preprint Series MA-UCM-2006-06. P. F. Verhulst. Notice sur la loi que la population poursuit dans son accroissement. Corresp. Math. Phys., 10, 113121, 1838. 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, 1845. José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal Localization 1,2 rpardo@mat.ucm.es phenomena (UCMICMAT) June 6-9, 2012 35 / 35