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 $$|x|^{\sigma_1}u_t = \Delta u + |x|^{\sigma_2}|u|^p + t^\varrho \mathbf{w}(x), \quad (t,x)\in(0,\infty)\times\mathbb{R}^N,$$ 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 $$p < p^*:=\frac{N+\sigma_2-\varrho(2+\sigma_1)}{N-2-\varrho(2+\sigma_1)},$$ 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 $p\le (N+\sigma_2)/(N-2)_+$ 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.