Capacity solution to a coupled system of parabolic–elliptic equations in Orlicz–Sobolev spaces

Abstract

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 − div a (x, t, u, ∇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 Δ2 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.Mathematics Subject Classification. 35K60, 35D05, 35J70, 46E30.

Existence of a capacity solution to a nonlinear parabolic-elliptic coupled system in anisotropic Orlicz-Sobolev spaces

“…Then there exists a subsequence (u n(k) ) ⊂ (u n ) such that, for every ǫ > 0, there exists a constant value M = M(ǫ) and a function ψ ∈ L 1 (0, T ; W 1,1 (Ω)) satisfying the following properties: Proof. The proof of this result is almost identical to that of Lemma 3.6 in [29].…”

“…In order to avoid this difficulty, we will consider the function ρ(u)∇ϕ as a whole and then show that it belongs to L 2 (Q) d . This new formulation of (1.1) will lead us to the introduction of the concept of capacity solution, a notion first introduced by Xu ( [35,36]) in order to deal with the thermistor problem in Sobolev spaces and later on used by other authors in the resolution of certain generalizations of this problem in different frameworks ( [21,27,28,29,30,32]).…”

A pocket guide to nonlinear differential equations in Musielak–Orlicz spaces

“…The classical reference for existence in the reflexive isotropic Orlicz-Sobolev setting is the already mentioned paper [186] by Talenti. We refer for other results on existence and gradient estimates to [10,12,60,160,44,59,61], nonexistence [95,126], regularity [143,144,58] and then [37,73], and foundations of the potential theory to [19].…”