Wellposedness for a parabolic-elliptic system

Giuseppe Maria Coclite; Helge Holden; Kenneth H. Karlsen

doi:10.3934/dcds.2005.13.659

Article Contents

2005, Volume 13, Issue 3: 659-682. Doi: 10.3934/dcds.2005.13.659

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Wellposedness for a parabolic-elliptic system

1.
Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO-0316 Oslo
2.
Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim
3.
Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo

Received: August 31, 2004

Revised: February 28, 2005

Published: March 2005

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Abstract

We show existence of a unique, regular global solution of the parabolic-elliptic system $u_t +f(t,x,u)_x+g(t,x,u)+P_x=(a(t,x) u_x)_x$ and $-P_{x x}+P=h(t,x,u,u_x)+k(t,x,u)$ with initial data $u|_{t=0} = u_0$. Here inf$_(t,x) a(t,x)>0$. Furthermore, we show that the solution is stable with respect to variation in the initial data $u_0$ and the functions $f$, $g$ etc. Explicit stability estimates are provided. The regularized generalized Camassa--Holm equation is a special case of the model we discuss.

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Mathematics Subject Classification: Primary: 35A05; Secondary: 35B30.

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