Blow-up and finite time extinction for p(x, t)-curl systems arising in electromagnetism

Abstract

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 $\lambda=1$.