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Wellposedness for a parabolic-elliptic system. (English) Zbl 1082.35056

The authors study a system that constitutes a generalized and regularized Camassa-Holm equation, that is

$\begin{array}{cc}& {u}_{t}+{\left(f\left(t,x,u\right)\right)}_{x}+g\left(t,x,u\right)+{P}_{x}={\left(a\left(t,x\right){u}_{x}\right)}_{x},\hfill \\ & -{P}_{xx}+P=h\left(t,x,u,{u}_{x}\right)+\kappa \left(t,x,u\right){,\phantom{\rule{1.em}{0ex}}u|}_{t=0}={u}_{0}\left(x\right),\hfill \end{array}\phantom{\rule{2.em}{0ex}}\left(1\right)$

on the domain $\left(t,x\right)\in {Q}_{T}:=\left[0,t\right]×ℝ$. The authors address the question of wellposedness of the system (1). In particular, they focus on stability of solutions with respect to variation not only in the initial data, but also variation with respect to the functions $f,a$. Moreover, the authors are interested in the vanishing viscosity limit of (1), that is, when $a\to 0$.

MSC:
 35G25 Initial value problems for nonlinear higher-order PDE 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B30 Dependence of solutions of PDE on initial and boundary data, parameters