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Existence of blow-up self-similar solutions for the supercritical quasilinear reaction-diffusion equation

Razvan Gabriel Iagar, Ariel Sánchez

Abstract

We establish the existence of self-similar solutions presenting finite time blow-up to the quasilinear reaction-diffusion equation $$ u_t=Δu^m + u^p, $$ posed in dimension $N\geq3$, $m>1$. More precisely, we show that there is always at least one solution in backward self-similar form if $p>p_s=m(N+2)/(N-2)$. In particular, this establishes \emph{non-optimality of the Lepin critical exponent} introduced in \cite{Le90} in the semilinear case $m=1$ and extended for $m>1$ in \cite{GV97, GV02}, for the existence of self-similar blow-up solutions. We also prove that there are multiple solutions in the same range, provided $N$ is sufficiently large. This is in strong contrast with the semilinear case, where the Lepin critical exponent has been proved to be optimal.

Existence of blow-up self-similar solutions for the supercritical quasilinear reaction-diffusion equation

Abstract

We establish the existence of self-similar solutions presenting finite time blow-up to the quasilinear reaction-diffusion equation posed in dimension , . More precisely, we show that there is always at least one solution in backward self-similar form if . In particular, this establishes \emph{non-optimality of the Lepin critical exponent} introduced in \cite{Le90} in the semilinear case and extended for in \cite{GV97, GV02}, for the existence of self-similar blow-up solutions. We also prove that there are multiple solutions in the same range, provided is sufficiently large. This is in strong contrast with the semilinear case, where the Lepin critical exponent has been proved to be optimal.
Paper Structure (6 sections, 18 theorems, 148 equations, 2 figures)

This paper contains 6 sections, 18 theorems, 148 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A dynamical system. Analysis of critical points
  3. Local analysis
  4. Analysis at infinity
  5. Some preparatory results
  6. Proof of Theorem \ref{['th.1']}

Key Result

Theorem 1.1

Let $N\geq3$. (a) For any $p>m$, there exists at least one solution in self-similar form SSS to Eq. eq1, which moreover has a decreasing profile with local behavior decay for some $C>c_s$. (b) Given any natural number $K\geq2$, for any $p\in(m,(Km-1)/(K-1))$ there exist at least $K$ different soluti there are at least $K$ different solutions in the form SSS for any $p\in(p_L,(Km-1)/(K-1))$. The se

Figures (2)

  • Figure 1: Various trajectories in the sets $\mathcal{A}$ and $\mathcal{C}$, seen in the phase space and in profiles. Experiments for $m=2$, $N=20$, $p=10$, with $p_L=3.2$.
  • Figure 2: Various trajectories with oscillations, seen in the phase space and in profiles. Experiments for $m=2$, $N=100$, $p=2.2$, with $p_L\approx2.133$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Lemma 2.1: Local analysis near $P_0$
  • proof
  • Lemma 2.2: Local analysis near $P_1$
  • proof
  • Lemma 2.3: Local analysis near $P_2$
  • proof
  • Lemma 2.4: Local analysis near $P_3$
  • proof
  • Lemma 2.5: Local analysis near $Q_1$
  • ...and 26 more