Wellposedness for a parabolic-elliptic system

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April 2005, 13(3): 659-682. doi: 10.3934/dcds.2005.13.659

Wellposedness for a parabolic-elliptic system

Giuseppe Maria Coclite ^1, , Helge Holden ^2, and Kenneth H. Karlsen ^3,

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

Received September 2004 Revised March 2005 Published May 2005

Abstract
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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.

Keywords: wellposedness, Camassa-Holm equation., Parabolic-elliptic system.

Mathematics Subject Classification: Primary: 35A05; Secondary: 35B3.

Citation: Giuseppe Maria Coclite, Helge Holden, Kenneth H. Karlsen. Wellposedness for a parabolic-elliptic system. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 659-682. doi: 10.3934/dcds.2005.13.659

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