On some parabolic equations involving superlinear singular gradient terms
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866915114384359424 |
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| author | Magliocca, Martina Oliva, Francescantonio |
| author_facet | Magliocca, Martina Oliva, Francescantonio |
| contents | In this paper we prove existence of nonnegative solutions to parabolic Cauchy-Dirichlet problems with superlinear gradient terms which are possibly singular. The model equation is \[
u_t - Δ_pu=g(u)|\nabla u|^q+h(u)f(t,x)\qquad \text{in }(0,T)\timesΩ, \] where $Ω$ is an open bounded subset of $\mathbb{R}^N$ with $N>2$, $0<T<+\infty$, $1<p<N$, and $q<p$ is superlinear. The functions $g,\,h$ are continuous and possibly satisfying $g(0) = +\infty$ and/or $h(0)= +\infty$, with different rates. Finally, $f$ is nonnegative and it belongs to a suitable Lebesgue space. We investigate the relation among the superlinear threshold of $q$, the regularity of the initial datum and the forcing term, and the decay rates of $g,\,h$ at infinity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2101_05196 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On some parabolic equations involving superlinear singular gradient terms Magliocca, Martina Oliva, Francescantonio Analysis of PDEs In this paper we prove existence of nonnegative solutions to parabolic Cauchy-Dirichlet problems with superlinear gradient terms which are possibly singular. The model equation is \[ u_t - Δ_pu=g(u)|\nabla u|^q+h(u)f(t,x)\qquad \text{in }(0,T)\timesΩ, \] where $Ω$ is an open bounded subset of $\mathbb{R}^N$ with $N>2$, $0<T<+\infty$, $1<p<N$, and $q<p$ is superlinear. The functions $g,\,h$ are continuous and possibly satisfying $g(0) = +\infty$ and/or $h(0)= +\infty$, with different rates. Finally, $f$ is nonnegative and it belongs to a suitable Lebesgue space. We investigate the relation among the superlinear threshold of $q$, the regularity of the initial datum and the forcing term, and the decay rates of $g,\,h$ at infinity. |
| title | On some parabolic equations involving superlinear singular gradient terms |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2101.05196 |