Blow-up for a Nonlocal Diffusion Equation with Time Regularly Varying Nonlinearity and Forcing

Abstract: We investigate the Cauchy problem for a semilinear parabolic equation driven by a mixed local-nonlocal diffusion operator of the form [ \partial_t u - (\Delta - (-\Delta)^{\mathsf{s}})u = \mathsf{h}(t)|x|^{-b}|u|^p + t^\varrho \mathbf{w}(x), \qquad (x,t)\in \mathbb{R}^N\times (0,\infty), ] where $\mathsf{s}\in (0,1)$, $p>1$, $b\geq 0$, and $\varrho>-1$. The function $\mathsf{h}(t)$ is assumed to belong to the generalized class of regularly varying functions, while $\mathbf{w}$ is a prescribed spatial source. We first revisit the unforced case and establish sharp blow-up and global existence criteria in terms of the critical Fujita exponent, thereby extending earlier results to the wider class of time-dependent coefficients. For the forced problem, we derive nonexistence of global weak solutions under suitable growth conditions on $\mathsf{h}$ and integrability assumptions on $\mathbf{w}$. Furthermore, we provide sufficient smallness conditions on the initial data and the forcing term ensuring global-in-time mild solutions. Our analysis combines semigroup estimates for the mixed operator, test function methods, and asymptotic properties of regularly varying functions. To our knowledge, this is the first study addressing blow-up phenomena for nonlinear diffusion equations with such a class of time-dependent coefficients.