Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients.

*(English)*Zbl 1275.35129Summary: 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 |

##### Keywords:

functions of bounded variation; zero-flux boundary condition; weak convergence method; partial uniqueness
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\textit{H. Watanabe}, Discrete Contin. Dyn. Syst., Ser. S 7, No. 1, 177--189 (2014; Zbl 1275.35129)

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