Self-similar solutions preventing finite time blow-up for reaction-diffusion equations with singular potential

Abstract

We prove existence and uniqueness of a global in time self-similar solution growing up as $t\to\infty$ for the following reaction-diffusion equation with a singular potential $$ u_t=Δu^m+|x|^σu^p, $$ posed in dimension $N\geq2$, with $m>1$, $σ\in(-2,0)$ and $1<p<1-σ(m-1)/2$. For the special case of dimension $N=1$, the same holds true for $σ\in(-1,0)$ and similar ranges for $m$ and $p$. The existence of this global solution prevents finite time blow-up even with $m>1$ and $p>1$, showing an interesting effect induced by the singular potential $|x|^σ$. This result is also applied to reaction-diffusion equations with general potentials $V(x)$ to prevent finite time blow-up via comparison.

Figures (3)

• Figure 1: A shooting of profiles with local behavior \ref{['beh.Q1']}. Experiments for $m=3$, $N=3$, $p=1.2$ and $\sigma=-0.7$, respectively $p=1.5$ and $\sigma=-1.5$
• Figure 2: The isoclines and monotonicity regions of the phase plane associated to the system \ref{['systz=0']}
• Figure 3: The isoclines and monotonicity regions of the phase plane associated to the system \ref{['systw=0']}

Theorems & Definitions (33)

• Theorem 1.1
• Lemma 2.1: Local analysis near $P_0$
• proof
• Lemma 2.2: Local analysis near $P_1$
• proof
• Lemma 2.3
• proof
• Lemma 2.4: Analysis of the points $P_1^{\gamma}$ for $p=1$
• Lemma 3.1: Local analysis near $Q_1$ for $N\geq3$
• proof