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Moussa, H.; Ortegón Gallego, F.; Rhoudaf, M.

Capacity solution to a coupled system of parabolic-elliptic equations in Orlicz-Sobolev spaces. (English) Zbl 1391.35233

NoDEA, Nonlinear Differ. Equ. Appl. 25, No. 2, Paper No. 14, 37 p. (2018).
Summary: The existence of a capacity solution to a coupled nonlinear parabolic-elliptic system is analyzed, the elliptic part in the parabolic equation being of the form \(-\,\operatorname{div}\, a(x,t,u,\nabla u)\). The growth and the coercivity conditions on the monotone vector field \(a\) are prescribed by an \(N\)-function, \(M\), which does not have to satisfy a \(\Delta _2\) condition. Therefore we work with Orlicz-Sobolev spaces which are not necessarily reflexive. We use Schauder’s fixed point theorem to prove the existence of a weak solution to certain approximate problems. Then we show that some subsequence of approximate solutions converges in a certain sense to a capacity solution.

Cited in 11 Documents

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35D30 Weak solutions to PDEs
35J70 Degenerate elliptic equations
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords:

capacity solution; coupled system; nonlinear parabolic-elliptic equations; weak solutions; Orlicz-Sobolev spaces; thermistor problem
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