Regularizing effect and decay results for a parabolic problem with repulsive superlinear first order terms
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arXiv
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| Format: | Preprint |
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2017
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| _version_ | 1866912198289260544 |
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| author | Magliocca, Martina |
| author_facet | Magliocca, Martina |
| contents | We want to analyse both regularizing effect and long, short time decay concerning parabolic Cauchy-Dirichlet problems of the type \begin{equation*} \begin{cases} \begin{array}{ll} u_t-\text{div} (A(t,x)|\nabla u|^{p-2}\nabla u)=γ|\nabla u|^q & \text{in}\,\,Q_T,\\ u=0 &\text{on}\,\,(0,T)\times\partialΩ,\\ u(0,x)=u_0(x) &\text{in}\,\, Ω. \end{array} \end{cases} \end{equation*} We assume that $A(t,x)$ is a coercive, bounded and measurable matrix, the growth rate $q$ of the gradient term is superlinear but still subnatural, $γ>0$, the initial datum $u_0$ is an unbounded function belonging to a well precise Lebesgue space $L^σ(Ω)$ for $σ=σ(q,p,N)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1712_09246 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Regularizing effect and decay results for a parabolic problem with repulsive superlinear first order terms Magliocca, Martina Analysis of PDEs We want to analyse both regularizing effect and long, short time decay concerning parabolic Cauchy-Dirichlet problems of the type \begin{equation*} \begin{cases} \begin{array}{ll} u_t-\text{div} (A(t,x)|\nabla u|^{p-2}\nabla u)=γ|\nabla u|^q & \text{in}\,\,Q_T,\\ u=0 &\text{on}\,\,(0,T)\times\partialΩ,\\ u(0,x)=u_0(x) &\text{in}\,\, Ω. \end{array} \end{cases} \end{equation*} We assume that $A(t,x)$ is a coercive, bounded and measurable matrix, the growth rate $q$ of the gradient term is superlinear but still subnatural, $γ>0$, the initial datum $u_0$ is an unbounded function belonging to a well precise Lebesgue space $L^σ(Ω)$ for $σ=σ(q,p,N)$. |
| title | Regularizing effect and decay results for a parabolic problem with repulsive superlinear first order terms |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1712.09246 |