A priori estimates and Liouville-type theorems for the semilinear parabolic equations involving the nonlinear gradient source
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910557587636224 |
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| author | Liang, Wenguo Zhang, Zhengce |
| author_facet | Liang, Wenguo Zhang, Zhengce |
| contents | This paper is concerned with the local and global properties of nonnegative solutions for semilinear heat equation $u_t-Δu=u^p+M|\nabla u|^q$ in $Ω\times I\subset \R^N\times \R$, where $M>0$, and $p,q>1$. We first establish the local pointwise gradient estimates when $q$ is subcritical, critical and supercritical with respect to $p$. With these estimates, we can prove the parabolic Liouville-type theorems for time-decreasing ancient solutions. Next, we use Gidas-Spruck type integral methods to prove the Liouville-type theorem for the entire solutions when $q$ is critical. Finally, as an application of the Liouville-type theorem, we use the doubling lemma to derive universal priori estimates for local solutions of parabolic equations with general nonlinearities. Our approach relies on a parabolic differential inequality containing a suitable auxiliary function rather than Keller-Osserman type inequality, which allows us to generalize and extend the partial results of the elliptic equation (Bidaut-Véron, Garcia-Huidobro and Véron (2020) \cite{veron-sum}) to the parabolic case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02893 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A priori estimates and Liouville-type theorems for the semilinear parabolic equations involving the nonlinear gradient source Liang, Wenguo Zhang, Zhengce Analysis of PDEs This paper is concerned with the local and global properties of nonnegative solutions for semilinear heat equation $u_t-Δu=u^p+M|\nabla u|^q$ in $Ω\times I\subset \R^N\times \R$, where $M>0$, and $p,q>1$. We first establish the local pointwise gradient estimates when $q$ is subcritical, critical and supercritical with respect to $p$. With these estimates, we can prove the parabolic Liouville-type theorems for time-decreasing ancient solutions. Next, we use Gidas-Spruck type integral methods to prove the Liouville-type theorem for the entire solutions when $q$ is critical. Finally, as an application of the Liouville-type theorem, we use the doubling lemma to derive universal priori estimates for local solutions of parabolic equations with general nonlinearities. Our approach relies on a parabolic differential inequality containing a suitable auxiliary function rather than Keller-Osserman type inequality, which allows us to generalize and extend the partial results of the elliptic equation (Bidaut-Véron, Garcia-Huidobro and Véron (2020) \cite{veron-sum}) to the parabolic case. |
| title | A priori estimates and Liouville-type theorems for the semilinear parabolic equations involving the nonlinear gradient source |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.02893 |