On Maxwell's Equations with a Temperature Effect, II
Abstract
In this paper we study Maxwell's equations with a thermal effect. This system models an induction heating process where the electric conductivity σ strongly depends on the temperature u. We focus on a special onedimensional case where the electromagnetic wave is assumed to be parallel to the yaxis. It is shown that the resulting hyperbolicparabolic system has a global smooth solution if the electrical conductivity σ(u) grows like u^{q} with 0 ≤q<8+4 √ 3. A fundamental element in this paper is the establishment of a maximum principle for wave equations with damping. This maximum principle provides an a priori bound for the first derivative with respect to both x and t of the solution without the imposition of any differentiability assumptions nor bounds on the coefficient of the damping term. The use of a nonlinear multiplier then permits (via a bootstrap procedure) the estimation of successively higher L^{p}norms of the temperature function u.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 1998
 DOI:
 10.1007/s002200050361
 Bibcode:
 1998CMaPh.194..343G