##
Stability of Entropy Solutions to the Cauchy Problem
for a Class of Nonlinear Hyperbolic-Parabolic Equations

#### Author: Gui-Qiang Chen and Emmanuele DiBenedetto

#### Title: Stability of Entropy Solutions to the Cauchy Problem
for a Class of Nonlinear Hyperbolic-Parabolic Equations

**Abstract:**
Consider the Cauchy problem for the nonlinear hyperbolic-parabolic
equation:
$$
u_t + \frac{1}{2}\,\bl{a}\cdot\nabla_x u^2=\Delta u_+
\qquad \text{ for } t>0,
\tag *
$$
where $\bl{a}$ is a constant vector and $u_+=\max\{u,0\}$.
The equation is
hyperbolic in the region $[u<0]$ and parabolic in the region
$[u>0]$. It is shown that entropy solutions to (*), that grow
at most linearly as $|x|\to\infty$, are stable in a weighted
$L^1(\rn)$ space, which implies that the solutions are
unique.
The linear growth as $|x|\to\infty$ imposed on the
solutions is shown to be optimal for uniqueness to hold.
The same results hold if the Burgers nonlinearity $\frac{1}{2}\,\bl{a}u^2$
is replaced by a general flux function $\bl{f}(u)$, provided
$\bl{f}'(u(x,t)$ grows in $x$ at most linearly
as $|x|\to\infty$, and/or the degenerate term $u_+$ is replaced by
a non-decreasing, degenerate, Lipschitz continuous function $\beta(u)$
defined on $\rr$.
For more general $\beta(\cdot)$, the results continue to hold
for bounded solutions.

This paper was completed in 1998 with its preprint form in Jan. 1999;
and it has appeared in:

*SIAM J. Mathematical Analysis*, **vol. 33 (4)**,
pages 751-762 (2001)

This paper is available in the following formats:

A closely related paper is Change me.

**Author Address**
Department of Mathematics
Northwestern University
Evanston, IL 60208-2730
gqchen@math.nwu.edu