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The Debye system: Existence and large time behavior of solutions. (English) Zbl 0814.35054

The authors study the so-called Debye system: \[ u_ t = \nabla \cdot (\nabla u - u \nabla \varphi), \quad v_ t = \nabla \cdot (\nabla v + v \nabla \varphi) \] (coupled through \(\varphi\) satisfying \(\Delta \varphi = u - v)\) in a bounded domain \(\Omega \subset \mathbb{R}^ n\) with smooth boundary and no flux boundary conditions. Conditions for local and global existence and uniqueness of weak solutions are given in several space dimensions \(n\). For \(n = 2,3\) it is shown that a weak solution of the system above converges to the (unique) stationary solution if \(t\to\infty\).
Reviewer: R.Manthey (Jena)

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
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