Wellposedness for a parabolic-elliptic system

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 $-P_{xx}+P=h(t,x,u,u_x)+k(t,x,u)$ 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.