Preprint 2005-007

Wellposedness for a parabolic-elliptic system

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)_x+P_x=(a(t,x)u_x)_x and −P_xx+P=h(t,x,u,u_x)+k(t,x,u) with initial data u|_t=0=u₀. Here inf_(t,x)a(t,x)>0. Furthermore, we show that the solution is stable with respect to variation in the initial data u₀ 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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Author(s):

Giuseppe M. Coclite <giusepc@math.uio.no>

Helge Holden, <holden@math.ntnu.no>

Kenneth H. Karlsen, <kennethk@math.uio.no>

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<giusepc@math.uio.no> January 28 2005.