### Nonhomogeneous Dirichlet Problems for Degenerate Parabolic-Hyperbolic Equations

Corrado Mascia, Alessio Porretta, and Andrea Terracina

Abstract:

The aim of the paper is to give a formulation for the initial boundary value problem of parabolic-hyperbolic type

  \left\{\begin{array}{l}
\partial _t u - \Delta b(u) +\dive \Phi (u) =g,\\
\end{array}\right.

in the case of nonhomogeneous boundary data $a_0$. Here $u=u(x,t)\in\R$, with $(x,t)\in Q=\Omega\times (0,T)$, where $\Omega$ is a bounded domain in $\R^N$ with smooth boundary and $T>0$. The function $b$ is assumed to be nondecreasing (allowing degeneration zones where $b$ is constant), $\Phi$ is Lipschitz continuous and $g\in L^\infty(\Omega\times (0,T))$. After introducing the definition of entropy solution to the above problem (in the spirit of Otto \cite{Ot_in}), we prove results of uniqueness of the solution in the proposed setting. Moreover we prove that the entropy solution previously defined can be obtained as limit of solutions of regularized equations of nondegenerate parabolic type (specifically the diffusion function $b$ is approximated by functions $b^\e$ that are strictly increasing).

The approach proposed for the hyperbolic-parabolic problem can be used to prove similar results for the class of hyperbolic-elliptic boundary value problems of the form

  u - \Delta b(u) +\dive \Phi (u) = g \qquad\qquad
u\vert_{\partial\Omega}=a_0.

again in the case of nonconstant boundary data $a_0$.

Paper:
Available as PostScript (384 Kbytes) or gzipped PostScript (139 Kbytes; uncompress using gunzip).
Author(s):