Separate variable blow-up patterns for a reaction-diffusion equation with critical weighted reaction

Abstract

We study the separate variable blow-up patterns associated to the following second order reaction-diffusion equation: $$ \partial_tu=Δu^m + |x|^σu^m, $$ posed for $x\in\mathbb{R}^N$, $t\geq0$, where $m>1$, dimension $N\geq2$ and $σ>0$. A new and explicit critical exponent $$ σ_c=\frac{2(m-1)(N-1)}{3m+1} $$ is introduced and a classification of the blow-up profiles is given. The most interesting contribution of the paper is showing that existence and behavior of the blow-up patterns is split into different regimes by the critical exponent $σ_c$ and also depends strongly on whether the dimension $N\geq4$ or $N\in\{2,3\}$. These results extend previous works of the authors in dimension $N=1$.

Figures (3)

• Figure 1: Periodic orbits in the invariant plane $\{X=0\}$
• Figure 2: Trajectories in the phase space for different values of $\sigma\in(0,\sigma_c)$. Numerical experiment for $m=2$, $N=4$ and $\sigma=0.5$, respectively $\sigma=0.84$
• Figure 3: The phase space and the separatrix surface, in form of an elliptic paraboloid, for $\sigma$ sufficiently large. Numerical experiment for $m=2$, $N=4$ and $\sigma=5$

Theorems & Definitions (34)

• Definition 1.1
• Theorem 1.2: Existence of blow-up profiles in dimension $N\geq4$
• Theorem 1.3: Classification for $N\geq4$
• Theorem 1.4: Existence and classification for $N=2$ and $N=3$
• Theorem 1.5: Non-existence for $\sigma$ large
• 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: Local analysis near the point $P_3$