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

We study the degenerate and singular parabolic equation with a forcing term $$\mathrm{\mid }x{\mathrm{\mid }}^{{\sigma }_1}{u}_t=\mathrm{\Delta }u+\mathrm{\mid }x{\mathrm{\mid }}^{{\sigma }_2}\mathrm{\mid }u{\mathrm{\mid }}^p+t^{\varrho }\mathbf{w}(x),(t,x)\in (0,\mathrm{\infty })\times {\mathbb{R}}^N,$$ where $N\ge 2$, ${\sigma }_1,{\sigma }_2>-2$, $\varrho >-1$, $p>1$, 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 $$p<p^{\ast }:=\frac{N+{\sigma }_2-\varrho (2+{\sigma }_1)}{N-2-\varrho (2+{\sigma }_1)},$$ then every weak solution blows up in finite time, provided $\underset{{\mathbb{R}}^N}{\int }\mathbf{w}(x)dx>0$. In the case $\varrho =0$, blow-up occurs for $p\le (N+{\sigma }_2)\mathrm{/}(N-2{)}_{+}$ with $N\ge 2$. In contrast, for $p>p^{\ast }$ 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.