On the Nonexistence of Global Solutions for Nonlocal Parabolic Equations with Forcing Terms
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866913733546082304 |
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| author | Belgacem, Rihab Ben Majdoub, Mohamed |
| author_facet | Belgacem, Rihab Ben Majdoub, Mohamed |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09738 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Nonexistence of Global Solutions for Nonlocal Parabolic Equations with Forcing Terms Belgacem, Rihab Ben Majdoub, Mohamed Analysis of PDEs 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. |
| title | On the Nonexistence of Global Solutions for Nonlocal Parabolic Equations with Forcing Terms |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.09738 |