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

Abstract

We classify the blow up self-similar profiles for the following reaction-diffusion equation with weighted reaction $$ u_t=(u^m)_{xx} + |x|^σu^m, $$ posed for $(x,t)\in\real\times(0,T)$, with $m>1$ and $σ>0$. In strong contrast with the well-studied equation without the weight (that is $σ=0$), on the one hand we show that for $σ>0$ sufficiently small there exist \emph{multiple self-similar profiles with interface} at a finite point, more precisely, given any positive integer $k$, there exists $δ_k>0$ such that for $σ\in(0,δ_k)$, there are at least $k$ different blow up profiles with compact support and interface at a positive point. On the other hand, we also show that for $σ$ sufficiently large, the blow up self-similar profiles with interface \emph{cease to exist}. This unexpected balance between existence of multiple solutions and non-existence of any, when $σ>0$ increases, is due to the effect of the presence of the weight $|x|^σ$, whose influence is the main goal of our study. We also show that for any $σ>0$, there are no blow up profiles supported in the whole space, that is with $u(x,t)>0$ for any $x\in\real$ and $t\in(0,T)$.

Figures (5)

• Figure 1: Some good profiles with interface for $\sigma$ small. Experiment for $m=2$, $p=2$ and $\sigma=0.1$
• Figure 2: A few profiles with interface and negative slope at the origin. Experiment for $m=2$, $p=2$ and $\sigma=0.5$
• Figure 3: Orbit passing through a point very close to $P_3$ and going out. Experiment for $m=2$, $p=2$ and $\sigma=1$ with plot in 3D and projection on the $(Y,Z)$ plane.
• Figure 4: The cylinder and sample orbits going out of $P_0$ and $P_2$, respectively entering $P_1$.
• Figure 5: Good and bad profiles with interface, with different slopes at $\xi=0$. Experiment for $m=2$, $p=2$ and $\sigma=0.1$

Theorems & Definitions (30)

• Definition 1.1
• Theorem 1.2: Existence of multiple good profiles with interface for $\sigma$ small
• Theorem 1.3: Non-existence of good profiles with interface for $\sigma$ large
• Theorem 1.4: Non-existence of positive good profiles
• Lemma 2.1: Analysis of the points $P_0$ and $P_1$
• proof
• Lemma 2.2: Analysis of the point $P_2$
• proof
• Lemma 2.3: Analysis of the point $Q_1$
• proof