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Self-similar shrinking of supports and non-extinction for a nonlinear diffusion equation with spatially inhomogeneous strong absorption

Departamento de Matemática Aplicada, Ciencia e Ingenieria de los Materiales y Tecnologia Electrónica, Universidad Rey Juan Carlos, Móstoles, 28933 Madrid, Spain

E-mail Address: [email protected]

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Philippe Laurençot

Institut de Mathématiques de Toulouse, CNRS UMR 5219, Université Paul Sabatier, F-31062 Toulouse Cedex 9, France

E-mail Address: [email protected]

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, and

Ariel Sánchez

Departamento de Matemática Aplicada, Ciencia e Ingenieria de los Materiales y Tecnologia Electrónica, Universidad Rey Juan Carlos, Móstoles, 28933 Madrid, Spain

E-mail Address: [email protected]

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https://doi.org/10.1142/S0219199723500281Cited by:3 (Source: Crossref)

Abstract

We study the dynamics of the following porous medium equation with strong absorption

\partial_{t} u = Δ u^{m} - | x |^{σ} u^{q},

posed for

(t, x) \in (0, \infty) \times ℝ^{N}

, with

m > 1

q \in (0, 1)

and

σ > 2 (1 - q) / (m - 1)

. Considering the Cauchy problem with non-negative initial condition

u_{0} \in L^{\infty} (ℝ^{N})

, instantaneous shrinking and localization of supports for the solution

u (t)

at any

t > 0

are established. With the help of this property, existence and uniqueness of a non-negative compactly supported and radially symmetric forward self-similar solution with algebraic decay in time are proven. Finally, it is shown that finite time extinction does not occur for a wide class of initial conditions and this unique self-similar solution is the pattern for large time behavior of these general solutions.

Keywords:

AMSC: 35B40, 35K65, 35K10, 34D05, 35A24

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