Juan Luis Vázquez

A Strong Maximum Principle for some quasilinear elliptic equations

In its simplest form the Strong Maximum Principle says that a nonnegative superharmonic continuous function in a domain Ω ⊂ ℝn,n ⩾ 1, is in fact positive everywhere. Here we prove that the same conclusion is true for the weak solutions of − Δu + β(u) = f withβ a nondecreasing function ℝ → ℝ,β(0)=0, andf⩾0 a.e. in Ω if and only if the integral∫(β(s)s)−1/2ds diverges ats=0+. We extend the result to more general equations, in particular to − Δpu + β(u) =f where Δp(u) = div(|Du|p-2Du), 1 <p < ∞. Our main result characterizes the nonexistence of a dead core in some reaction-diffusion systems.

Continuation of blowup solutions of nonlinear heat equations in several space dimensions

The possible continuation of solutions of the nonlinear heat equation in R N R+ ut = u m + u p with m> 0 ;p > 1 ; after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p. Thus, for m +p 2 we find a phenomenon of nontrivial continuation where the regionfx : u(x;t )= 1g is bounded and propagates with finite speed. This we call incomplete blowup. For N 3 and p>m ( N +2 )= (N 2) we find solutions that blow up at finite t = T and then become bounded again for t>T . Otherwise, we find that blowup is complete for a wide class of initial data. In the analysis of the behavior for large p, a list of critical exponents appears whose role is described. We also discuss a number of related problems and equations. We apply the same technique of analysis to the problem of continuation after the onset of extinction, for example, for the equation ut = u m u p ;m > 0 :

Asymptotic behaviour for the porous medium equation posed in the whole space

This paper is devoted to present a detailed account of the asymptotic behaviour as t → ∞ of the solutions u(x, t) of the equation

Fundamental Solutions and Asymptotic Behaviour for the

{u_{{t = }}}\Delta ({u^{m}})

-Laplacian Equation

(0.1) with exponent m > 1, a range in which it is known as the porous medium equation written here PME for short. The study extends the well-known theory of the classical heat equation (HE, the case m = 1) into a nonlinear situation, which needs a whole set of new tools. The space dimension can be any integer n ≥ 1. We will also present the extension of the results to exponents m < 1 (fast-diffusion equation, Fde). For definiteness we consider the Cauchy Problem posed in Q = ℝ n x ℝ+ with initial data

A Nonlinear heat equation with singular diffusivity

u(x,0) = {u_{0}}(x), x \in {\mathbb{R}^{n}}