Electron. J. Diff. Equ., Vol. 2014 (2014), No. 198, pp. 1-17.

Entropy solutions for nonlinear degenerate elliptic-parabolic-hyperbolic problems

Ning Su, Li Zhang

Abstract:
We consider the nonlinear degenerate elliptic-parabolic-hyperbolic equation
$$ \partial_t g (u) - \Delta b (u) - \hbox{div} \Phi(u) = f (g (u) ) \quad \text{in } (0,T) \times \Omega, $$
where g and b are nondecreasing continuous functions, $\Phi$ is vectorial and continuous, and f is Lipschitz continuous. We prove the existence, comparison and uniqueness of entropy solutions for the associated initial-boundary-value problem where $\Omega$ is a bounded domain in $\mathbb{R}^N$. For the associated initial-value problem where $\Omega= \mathbb{R}^N$, $N \geq 3$, the existence of entropy solutions is proved. Moreover, for the case when $\Phi \circ g^{-1}$ is locally Holder continuous of order $1- 1/N$, and $|b(u)| \leq \omega(|g(u)|)$, where $\omega$ is nondecreasing continuous with $\omega(0) = 0$, we can prove the $L^1$-contraction principle, and hence the uniqueness.

Submitted December 6, 2013. Published September 23, 2014.
Math Subject Classifications: 35J70, 35K65, 35L80.
Key Words: Nonlinear evolution equation; degenerate equation; entropy solution; existence; uniqueness.

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Ning Su
Department of Mathematical Sciences
Tsinghua University
Beijing 100084, China
email: nsu@math.tsinghua.edu.cn
Li Zhang
Department of Mathematical Sciences
Tsinghua University
Beijing 100084, China
email: zhli25@163.com

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