Forcing Effects on Finite-Time Blow-Up in Degenerate and Singular Parabolic Equations

Abstract

We study the degenerate and singular parabolic equation with a forcing term \[ |x|^{σ_1}u_t = Δu + |x|^{σ_2}|u|^p + t^\varrho \mathbf{w}(x), \quad (t,x)\in(0,\infty)\times\mathbb{R}^N, \] where $N\ge 2$, $σ_1,σ_2>-2$, $\varrho>-1$, $p>1$, and $\mathbf{w}\in L^1(\mathbb{R}^N)$ is continuous. We establish critical exponents that sharply separate the regimes of global existence and finite-time blow-up. For $\varrho>0$, we prove that there is no weak global solution for all $p>1$. When $-1<\varrho<0$, we show that if \[ p < p^*:=\frac{N+σ_2-\varrho(2+σ_1)}{N-2-\varrho(2+σ_1)}, \] then every weak solution blows up in finite time, provided $\int\limits_{\mathbb{R}^N}\mathbf{w}(x)\,dx>0$. In the case $\varrho=0$, blow-up occurs for $p\le (N+σ_2)/(N-2)_+$ with $N\ge 2$. In contrast, for $p>p^*$ and under smallness conditions on the initial data and forcing term, we prove the existence of a unique global mild solution. The analysis relies on scaling transformations, semigroup estimates for degenerate operators, and a fixed-point argument in weighted-in-time Lebesgue spaces.

Theorems & Definitions (16)

• Definition 1.1
• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Remark 1.2
• Theorem 1.3
• Remark 1.3
• Theorem 2.1: $L^a$–$L^b$ estimates for degenerate heat semigroups
• Remark 2.1
• Proposition 2.1