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

#### 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)
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Author Address

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