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 $N\ge 2$, $σ<em>1,σ_2&gt;-2$, $\varrho&gt;-1$, $p&gt;1$, and $\mathbf{w}\in L<sup>1(\mathbb{R}<sup>N)$ is continuous. We establish critical exponents that sharply separate the regimes of global existence and finite-time blow-up. For $\varrho&gt;0$, we prove that there is no weak global solution for all $p&gt;1$. When $-1&lt;\varrho&lt;0$, 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 $\int\limits</em>{\mathbb{R}<sup>N}\mathbf{w}(x)\,dx&gt;0$. In the case $\varrho=0$, blow-up occurs for $p\le (N+σ<em>2)/(N-2)</em>+$ with $N\ge 2$. In contrast, for $p&gt;p<sup>*$ 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.