Global solutions of the non-linear problem describing Joule's heating in three space dimensions

Abstract

SUMMARYThe existence of global-in-time weak solutions to the Joule problem modelling heating or cooling in a current and heat conductive medium is proved via the Faedo -Galerkin method. The existence proof entails some a priori estimates that together with the monotonicity and compactness methods make up a main tool to prove the desired result. Under appropriate hypotheses on the data, it will be shown the boundedness in L∞(Q T ) of the absolute temperature of the medium and of the t-derivative of this temperature, which is achieved by means of the Gagliardo -Nirenberg theorem, the Sobolev embedding theorem and the method of Stampacchia. The paper is some extension of our investigation initiated in (Math. Meth. Appl. Sci. 1998; 23:1275-1291 PRELIMINARIES AND STATEMENT OF THE RESULTThe main aim of this paper is to prove existence of global-in-time weak solutions to the initial-boundary value problem describing Joule's heating in a heat and current conductive medium in three space dimensions; this problem reads:

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