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

We study a class of p(x,t)p(x,t)-curl systems arising in electromagnetism, with a nonlinear source term. Denoting by h   the magnetic field, the source term considered is of the form λh(∫Ω|h|2)σ−22 where λ∈{−1,0,1}λ∈{−1,0,1}: when λ∈{−1,0}λ∈{−1,0} we consider 0<σ≤20<σ≤2 and for λ=1λ=1 we have σ≥1σ≥1. 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 λ and σ  . We study the finite time extinction or the stabilization towards zero of the solutions when λ∈{−1,0}λ∈{−1,0} and the blow-up of local solutions when λ=1λ=1.