arXiv subjects

Rihab Ben Belgacem

Publications and source records attributed to Rihab Ben Belgacem.

3 recordsLinked to original sources

arXiv2026

Fujita Phenomenon for a Mixed Local-Nonlocal Hardy-Hénon Equation with Regularly Varying Time Weights

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.

math.AP↗

arXiv2026

On the Fujita Phenomenon for a Forced Spatio-Temporal Fractional Diffusion Equation

We investigate the Cauchy problem for a semilinear spatio--temporal fractional diffusion equation with a time-dependent forcing term: \[ \partial_t^αu + (-Δ)^{\mathsf{s}} u = |u|^p + t^σ\,\mathbf{w}(x), \quad (t,x) \in (0,\infty) \times \mathbb{R}^N, \] where $α,\mathsf{s}\in (0,1)$, $σ> -α$, and $\mathbf{w}$ is a given continuous function. Here $\partial_t^α$ denotes the Caputo fractional derivative. Our main results are threefold. First, we establish local-in-time existence of mild solutions and prove finite-time blow-up in the subcritical regime, under the positivity condition \[ \int\limits_{\mathbb{R}^N} \mathbf{w}(x)\,dx > 0. \] Second, in the supercritical case $-α< σ< 0$, we prove the global existence of solutions for sufficiently small initial data and forcing term, and we identify the corresponding critical exponent as \[ p_F=\frac{Nα-2\mathsf{s}σ}{Nα-2\mathsf{s}(α+σ)}. \] Finally, within this supercritical range, we obtain a more robust global existence result under weaker assumptions that require only local smallness and controlled growth of the data. To the best of our knowledge, a sharp Fujita-type threshold for fully spatio-temporal fractional diffusion equations with time-growing external forcing has not been previously established.

math.AP↗

arXiv2025

On the Nonexistence of Global Solutions for Nonlocal Parabolic Equations with Forcing Terms

The purpose of this work is to analyze the well-posedness and blow-up behavior of solutions to the nonlocal semilinear parabolic equation with a forcing term: \[ \partial_t u - Δu = \|u(t)\|_{q}^α|u|^p + t^{\varrho} \mathbf{w}(x) \quad \text{in} \quad \mathbb{R}^N \times (0, \infty), \] where $N \geq 1$, $p, q \geq 1$, $α\geq 0$, $\varrho > -1$, and $\mathbf{w}(x)$ is a suitably given continuous function. The novelty of this work, compared to previous studies, lies in considering a nonlocal nonlinearity $\|u(t)\|_{q}^α|u|^p$ and a forcing term $t^{\varrho} \mathbf{w}(x)$ that depend on both time and space variables. This combination introduces new challenges in understanding the interplay between the nonlocal structure of the equation and the spatio-temporal forcing term. Under appropriate assumptions, we establish the global existence of solutions for small initial data in Lebesgue spaces when the exponent $p$ exceeds a critical value. In contrast, we show that the global existence cannot hold for $p$ below this critical value, provided the additional condition $\int_{\mathbb{R}^N} \mathbf{w}(x) \, dx > 0$ is satisfied. The main challenge in this analysis lies in managing the complex interaction between the nonlocal nonlinearity and the forcing term, which significantly influences the behavior of solutions.

math.AP↗