Abstract
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 -,mathrm{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.
Highlights
There has been an increasing interest in the study of various mathematical problems involving the operators satisfying non-polynomial growth conditions instead of having the usual p-structure which employ the standard theory of monotone operators relying on the Sobolev space W 1,p(Ω), the origins of which can be traced back to the work of Orlicz in the 1930s
This paper deals with the existence of a capacity solution to a coupled system of parabolic–elliptic equations, whose unknowns are the temperature inside a semiconductor material, u, and the electric potential, φ, namely
The main goal of this paper is to prove the existence of a capacity solution of (1.1) in the sense of Definition 4.1 for a general N -function, M, along with the lack of reflexivity in this setting combined with the nonuniformly elliptic character of the elliptic equation
Summary
There has been an increasing interest in the study of various mathematical problems involving the operators satisfying non-polynomial growth conditions instead of having the usual p-structure which employ the standard theory of monotone operators relying on the Sobolev space W 1,p(Ω), the origins of which can be traced back to the work of Orlicz in the 1930s. Polish and Czechoslovak mathematicians investigated the modular function spaces (see, for example, Musielak [19] and Krasnoselskii and Rutickii [18]). Many properties of Sobolev spaces have been extended to Orlicz–Sobolev spaces, mainly by Dankert [7] Donaldson and Trudinger [9] and O’Neil [20] (see [1] for an excellent account of those works). The operators satisfying non-polynomial growth arouse much interest with the development of elastic mechanics, electro-rheological fluids as an important class
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