Non‐linear parabolic problem associated with the penetration of a magnetic field into a substance

Abstract

Abstract We study the following initial and boundary value problem: equation image In section 1, with u 0 in L 2 (Ω), f continuous such that f (u) + ϵ non‐decreasing for ϵ positive, we prove the existence of a unique solution on (0, T ), for each T > 0. In section 2 it is proved that the unique soluition u belongs to L 2 (0, T ; H ∩ H 2 ) ∩ L ∞ (0, T; H ) if we assume u 0 in H and f in C 1 (ℝ,ℝ). Numerical results are given for these two cases.