Abstract: In this paper, a coupled elliptic-parabolic system modeling a class of engineering problems with thermal effect is studied. Existence of a weak solution is first established through a result of Meyers’ theorem and Schander fixed point theorem, where the coupled function sσ(s), k(s) are assumed to be bounded in the C(IR×(0,T)). If σ(s), k(s) are Lipschitz continuous we prove that solution is unique under some restriction on integrability of solution. The regularity of the solution in dimension n≤2 is then analyzed under the assumptions on σ(s)∈W1, ∞(Ω×(0,T)) and the boundedness ofσ′(s) andσ″(s). 

Key words: elliptic-parabolic system, existence, uniqueness, regularity