Publicação
Blow-up and finite time extinction for p(x, t)-curl systems arising in electromagnetism
| Resumo: | We study a class of $p(x,t)$-curl systems arising in electromagnetism, with a nonlinear source term. Denoting by $\boldsymbol{h}$ the magnetic field, the source term considered is of the form $\lambda\boldsymbol{h}\left( \int_{\Omega}|\boldsymbol{h}|^{2}\right)^{\frac{\sigma-2}{2}}$ where $\lambda\in\{-1,0,1\}$: when $\lambda\in\{-1,0\}$ we consider $0<\sigma\leq2$ and for $\lambda=1$ we have $\sigma\geq1$. We introduce a suitable functional framework and a convenient basis that allow us to apply the Galerkin's method and prove existence of local or global solutions, depending on the values of $\lambda$ and $\sigma$. We study the finite time extinction or the stabilization towards zero of the solutions when $\lambda\in\{-1,0\}$ and the blow-up of local solutions when $\lambda=1$. |
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| Autores principais: | Antontsev, Stanislav |
| Outros Autores: | Miranda, Fernando; Santos, Lisa |
| Assunto: | Electromagnetic problems p(x,t)-curl systems Variable exponents Blow-up Extinction in time |
| Ano: | 2016 |
| País: | Portugal |
| Tipo de documento: | artigo |
| Tipo de acesso: | acesso aberto |
| Instituição associada: | Universidade do Minho |
| Idioma: | inglês |
| Origem: | RepositóriUM - Universidade do Minho |
| Resumo: | We study a class of $p(x,t)$-curl systems arising in electromagnetism, with a nonlinear source term. Denoting by $\boldsymbol{h}$ the magnetic field, the source term considered is of the form $\lambda\boldsymbol{h}\left( \int_{\Omega}|\boldsymbol{h}|^{2}\right)^{\frac{\sigma-2}{2}}$ where $\lambda\in\{-1,0,1\}$: when $\lambda\in\{-1,0\}$ we consider $0<\sigma\leq2$ and for $\lambda=1$ we have $\sigma\geq1$. We introduce a suitable functional framework and a convenient basis that allow us to apply the Galerkin's method and prove existence of local or global solutions, depending on the values of $\lambda$ and $\sigma$. We study the finite time extinction or the stabilization towards zero of the solutions when $\lambda\in\{-1,0\}$ and the blow-up of local solutions when $\lambda=1$. |
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