Blow up profiles for a quasilinear reaction-diffusion equation with weighted reaction

Abstract

We perform a thorough study of the blow up profiles associated to the following second order reaction-diffusion equation with non-homogeneous reaction: $$ \partial_tu=\partial_{xx}(u^m) + |x|^σu^p, $$ in the range of exponents $1<p<m$ and $σ>0$. We classify blow up solutions in self-similar form, that are likely to represent typical blow up patterns for general solutions. We thus show that the non-homogeneous coefficient $|x|^σ$ has a strong influence on the qualitative aspects related to the finite time blow up. More precisely, for $σ\sim0$, blow up profiles have similar behavior to the well-established profiles for the homogeneous case $σ=0$, and typically \emph{global blow up} occurs, while for $σ>0$ sufficiently large, there exist blow up profiles for which blow up \emph{occurs only at space infinity}, in strong contrast with the homogeneous case. This work is a part of a larger program of understanding the influence of unbounded weights on the blow up behavior for reaction-diffusion equations.

Figures (4)

• Figure 1: Evolution of good profiles with interface for $\sigma$ sufficiently small
• Figure 2: Evolution of good profiles with interface for $\sigma$ sufficiently large
• Figure 3: Local behavior of the system \ref{['interm50']} with one elliptic sector and one hyperbolic sector at the origin.
• Figure 4: Orbits from $P_2$ and $P_0$ for different values of $\sigma$

Theorems & Definitions (56)

• Definition 1.1
• Theorem 1.2: Existence of good profiles with interface
• Theorem 1.3: Good profiles with interface for $\sigma>0$ small
• Theorem 1.4: Good profiles with interface for $\sigma>0$ large
• Theorem 1.5
• Lemma 2.1: Analysis of the point $P_0=(0,0,0)$
• proof
• Lemma 2.2: Analysis of the point $P_1=\left(0,-\beta/m,0\right)$
• proof
• Lemma 2.3: Analysis of the point $P_2=((m-1)/2m(m+1),1/m(m+1),0)$