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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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