Fujita Phenomenon for Mixed Local--Nonlocal Diffusion Equations
Rihab Ben Belgacem, Mohamed Majdoub
TL;DR
This work analyzes a semilinear parabolic equation with a mixed local--nonlocal diffusion operator $\mathscr L=\Delta-(-\Delta)^{\mathsf s}$, time-dependent forcing $\mathsf h(t)|x|^{-b}|u|^p$, and a spatial forcing $t^{\varrho}\mathbf w(x)$. By allowing $\mathsf h$ to belong to the generalized class $\mathcal M(\gamma)$ of regularly varying-type functions, the authors derive sharp Fujita-type thresholds for blow-up vs global existence in the unforced case, and establish nonexistence results for the forced problem together with small-data global existence results via a fixed-point approach. The analysis relies on semigroup estimates for $\mathscr L$, test-function methods, and asymptotic properties of $\mathcal M(\gamma)$, providing a robust Fujita picture under time-dependent coefficients. Key contributions include extending Fujita theory to a broad class of temporal coefficients and delivering precise blow-up/global criteria for both forced and unforced settings, with potential connections to stochastic processes and nonlocal diffusion modeling in applications.
Abstract
We investigate the Cauchy problem for a semilinear parabolic equation driven by a mixed local-nonlocal diffusion operator of the form \[ \partial_t u - (Δ- (-Δ)^{\mathsf{s}})u = \mathsf{h}(t)|x|^{-b}|u|^p + t^\varrho \mathbf{w}(x), \qquad (x,t)\in \mathbb{R}^N\times (0,\infty), \] 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.