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Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. (English) Zbl 1275.35129
Summary: We consider the initial boundary value problem for strongly degenerate parabolic equations with discontinuous coefficients. This equation has the both properties of parabolic equation and hyperbolic equation. Moreover, approximate solutions for this equation may not belong to $$BV$$. These are difficult points for this type of equations. We consider the type of equations under the zero-flux boundary conditions. In particular, we prove the existence and partial uniqueness of weak solutions to such problems. Our proof use the compactness theorem derived by E. Yu. Panov [Arch. Ration. Mech. Anal. 195, No. 2, 643–673 (2010; Zbl 1191.35102)] and the estimate of degenerate diffusion term derived by Karlsen-Risebro-Towers [K. H. Karlsen et al., Electron. J. Differ. Equ. 2002, Paper No. 93, 23 p. (2002; Zbl 1015.35049)].
##### MSC:
 35K65 Degenerate parabolic equations 35K55 Nonlinear parabolic equations 35L65 Hyperbolic conservation laws 35R05 PDEs with low regular coefficients and/or low regular data 35K20 Initial-boundary value problems for second-order parabolic equations
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