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.


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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