On the Fujita Phenomenon for a Forced Spatio-Temporal Fractional Diffusion Equation
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| Format: | Preprint |
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2025
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| author | Belgacem, Rihab Ben Majdoub, Mohamed |
| author_facet | Belgacem, Rihab Ben Majdoub, Mohamed |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_19424 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Fujita Phenomenon for a Forced Spatio-Temporal Fractional Diffusion Equation Belgacem, Rihab Ben Majdoub, Mohamed Analysis of PDEs 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. |
| title | On the Fujita Phenomenon for a Forced Spatio-Temporal Fractional Diffusion Equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.19424 |