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

Capacity solution to a coupled system of parabolic–elliptic equations in Orlicz–Sobolev spaces 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 $$-\,\mathrm{div}\, a(x,t,u,\nabla u)$$ - 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 $$\Delta _2$$ Δ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Nonlinear Differential Equations and Applications NoDEA Springer Journals

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

, Volume 25 (2) – Mar 12, 2018
37 pages

/lp/springer_journal/capacity-solution-to-a-coupled-system-of-parabolic-elliptic-equations-M8wpUsYTm2
Publisher
Springer Journals
Subject
Mathematics; Analysis
ISSN
1021-9722
eISSN
1420-9004
D.O.I.
10.1007/s00030-018-0505-y
Publisher site
See Article on Publisher Site

### 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 $$-\,\mathrm{div}\, a(x,t,u,\nabla u)$$ - 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 $$\Delta _2$$ Δ 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.

### Journal

Nonlinear Differential Equations and Applications NoDEASpringer Journals

Published: Mar 12, 2018

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