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Entropy solutions for nonlinear degenerate problems. (English) Zbl 0935.35056
The following problems are considered: \[ \begin{aligned} \frac{\partial g(u)}{\partial t}-\Delta b(u)+\text{div}(\phi(u))=f\qquad & \text{in }(0,T)\times \Omega,\\ g(u)=g_0\qquad & \text{on }\{0\}\times \Omega,\\ b(u)=0\qquad & \text{on }\{0,T\}\times \Gamma\end{aligned} \] and \[ \begin{aligned} g(u)-\Delta b(u)+\text{div}(\phi(u))=f\qquad & \text{in }\Omega,\\ b(u)=0\qquad & \text{on }\Gamma.\end{aligned} \] Here \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with a Lipschitz boundary \(\Gamma\); \(g,b:\mathbb{R}\to\mathbb{R}\) are continuous and nondecreasing with \(g(0)=b(0)=0\); \(\phi\in{\mathcal C}(\mathbb{R};\mathbb{R}^N)\), \(\phi_j(0)=0\), \(1\leq j\leq N\).
Existence of entropy solutions and comparison and uniqueness for such solutions to both problems are proved.

MSC:
35J70 Degenerate elliptic equations
35K65 Degenerate parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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