Double Criticality for a Hardy-Rellich Biharmonic Heat Equation in an Exterior Domain
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915997822222336 |
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| author | Alhatlani, Hadeel Jleli, Mohamed Samet, Bessem |
| author_facet | Alhatlani, Hadeel Jleli, Mohamed Samet, Bessem |
| contents | We study the existence and nonexistence of weak solutions to an inhomogeneous semilinear biharmonic heat equation in an exterior domain, involving a singular Hardy--Rellich potential, a weighted nonlinearity $|x|^σ|u|^{p}$, and a positive source term $f(x)$. We identify two distinct critical regimes governing the behavior of solutions. More precisely, we first determine a Fujita-type critical exponent that separates nonexistence from existence. We then show that, in the supercritical range, a second critical exponent arises in terms of the decay exponent of the source, in the sense of Lee and Ni. Our results extend the recent work \cite{Tobakhanov} by considering a singular Hardy--Rellich potential and a weighted nonlinearity, leading to a different critical behavior. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09074 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Double Criticality for a Hardy-Rellich Biharmonic Heat Equation in an Exterior Domain Alhatlani, Hadeel Jleli, Mohamed Samet, Bessem Analysis of PDEs We study the existence and nonexistence of weak solutions to an inhomogeneous semilinear biharmonic heat equation in an exterior domain, involving a singular Hardy--Rellich potential, a weighted nonlinearity $|x|^σ|u|^{p}$, and a positive source term $f(x)$. We identify two distinct critical regimes governing the behavior of solutions. More precisely, we first determine a Fujita-type critical exponent that separates nonexistence from existence. We then show that, in the supercritical range, a second critical exponent arises in terms of the decay exponent of the source, in the sense of Lee and Ni. Our results extend the recent work \cite{Tobakhanov} by considering a singular Hardy--Rellich potential and a weighted nonlinearity, leading to a different critical behavior. |
| title | Double Criticality for a Hardy-Rellich Biharmonic Heat Equation in an Exterior Domain |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.09074 |