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This paper deals with the existence of weak periodic solutions for a model describing the electrical heating of a conductor taking into account the Joule-Thomson effect. The periodicity of solutions is established utilizing the classical Lax-Milgram theorem and the Schauder fixed point theorem. Keywords Nonlinear parabolic-elliptic system · Existence of periodic solutions · Joule-Thomson effect Mathematics Subject Classification (2000) 35B10 · 35D05 · 35K55 · 35J60

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We give sufficient conditions, being also necessary in many cases, for the existence of a periodic free boundary generated as the boundary of the support of the periodic solution of a general class of nonlinear parabolic equations. We show some qualitative properties of this free boundary. In some cases it may have some vertical shape linking the free boundaries of two stationary solutions , and, under the assumption of a strong absorption, it may have several periodic connected components. PHere T > 0, Ω ⊂ R N N 1 denotes an open bounded and regular set, Δ p u : div |∇u| p−2 ∇u , p > 1 is the so-called p-Laplacian operator, λ is a positive parameter, and