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On nonlinear diffusion problems with strong degeneracy. (English) Zbl 1157.35058

Let \(\Omega\) be a \(C^{1,1}\) bounded open domain of \(\mathbb R^N\), \(N > 1\) with a regular boundary \(\partial\Omega\). The author studies the following initial-boundary value problem of parabolic-hyperbolic type: \[ \begin{aligned} & b(v)_t - \Delta g(v) + \text{div }\Phi(v) = f \qquad \text{on } Q = (0,T) \times \Omega,\\ & g(v) = g(a) \qquad \text{on some part of } \Sigma = (0,T)\times\partial\Omega, \\ & b(v)(0,\cdot) = u_0 = b(v_0) \qquad \text{on } \Omega, \end{aligned} \] where \(\Phi\,:\, R \to \mathbb R^N\) is a continuous vector field, \(b\), \(g\,:\, \mathbb R^1\to \mathbb R^1\) are nondecreasing, locally Lipschitz continuous functions such that \(b(0)=g(0)=0\) and satisfy the range condition \(R(b+g)= \mathbb R^1\). It is assumed that \(v_0\in L^1(\Omega)\) with \(b(v_0) \in L^1(\Omega)\), \(f\in L^1(Q)\) and nonstationary boundary data \(a \in C(\Sigma)\) is a trace of a function \(\tilde a \in C(Q)\) with \(g(\tilde a) \in L^2\left(0,T;H^1(\Omega)\right)\), \(\text{div } \Phi(\tilde a) \in L^1(Q)\), \(\Delta g(\tilde a) \in L^1(Q)\) and \(\tilde a_t \in L^1(Q)\). Equations of this type arise in certain models of fluids flows through porous media, Stefan-type problem, and so on. In particular when \(g(u) = u\), the problem is of elliptic-parabolic type and when \(g(u) = 0\), \(b(u) = u\), it is of hyperbolic type. Using monotonicity and penalization methods, the author proves existence of the so-called weak renormalized entropy solution \(v\). The uniqueness result follows as a consequence of the \(L^1\)-comparison principle.

MSC:

35K65 Degenerate parabolic equations
35F30 Boundary value problems for nonlinear first-order PDEs
35K35 Initial-boundary value problems for higher-order parabolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35M10 PDEs of mixed type
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