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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

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 and Continuous Dynamical Systems, 2005, 13 (3) : 659-682. doi: 10.3934/dcds.2005.13.659
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