Radial Equivalence of Nonhomogeneous Nonlinear Diffusion Equations
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Radial Equivalence of Nonhomogeneous Nonlinear Diffusion Equations Razvan Gabriel Iagar · Guillermo Reyes · Ariel Sánchez
Received: 7 September 2011 / Accepted: 2 April 2012 / Published online: 6 June 2012 © Springer Science+Business Media B.V. 2012
Abstract We establish one-to-one transformations and self-maps between nonlinear diffusion equations in nonhomogeneous media, where the density function is given by a power. We use these transformations to deduce new interesting self-similar, radially symmetric solutions of the equations. In particular, Barenblatt, dipole and focusing Aronson-Graveleau type solutions are deduced, and some equations with singular potentials are studied. The new solutions are example of interesting or unexpected mathematical features of these equations, providing also natural candidates for the asymptotic behavior.
Keywords Non-homogeneous porous media · Self-similar solutions · Self-maps · Radially-symmetric solutions · Critical exponents · Non-homogeneous p-Laplacian equation
Mathematics Subject Classification 35C06 · 35K10 · 35K55 · 35K65
R.G. Iagar () Institute de Mathematiques de Toulouse, CNRS-UMR 5219, Route de Narbonne, 31062, Toulouse Cedex 9, France e-mail: [email protected] R.G. Iagar Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700, Bucharest, Romania G. Reyes Departamento de Matemática, E.T.S.I. de Caminos, Canales y Puertos, Universidad Politécnica de Madrid, 28040, Madrid, Spain A. Sánchez Departamento de Matemática Aplicada, Universidad Rey Juan Carlos, Móstoles, 28933, Madrid, Spain
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1 Introduction In a series of papers, Kamin and Rosenau [11–13] study various initial/boundary problems for the equation (1.1) (x)ut = div c(x)∇um + k(x)uq , which models thermal propagation by radiation in non-homogeneous plasma. Here, u stands for the temperature, while (x) represents the particle density, and the diffusivity mc(x)um−1 depends on the temperature as well as on the spatial variable. Yet another application of (1.1) arises from phenomena in biology, see Murray [18]. Of particular interest is the initial value problem when the density decays to zero as |x| → ∞. A natural assumption is that the decay is power-like, a typical choice of the density being 0 , γ > 0. (1.2) (x) = (1 + |x|)γ Starting with the paper [11], where the purely diffusive one-dimensional problem ⎧ ⎨(x)ut = (um )xx , (x, t) ∈ R × (0, ∞), ⎩u(x, 0) = u (x), 0
x∈R
(1.3)
with as in (1.2) was considered, it has been noticed the important role played by the associated singular equation |x|−γ ut = um
(1.4)
in the description of generic solutions for large time. Indeed, in [11] the authors prove that if γ ∈ (0, 1), generic solutions to (1.3) in the class L∞ ∩ L1 (ρ(x) dx) behave as t → ∞ as a fundamental solution to (1.4), in dimension n = 1. Here, fundamental means that |x|−γ u(x, t) → Eδ(x) as t ↓ 0, and E has to be chosen equal u0 L1 (ρ(x) dx) . These results have been further extended to the Cauchy problem ⎧ ⎨(x)ut = um , (x, t) ∈ Rn × (0,
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