Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients

Pages: 781 - 790, Issue special, November 2013

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Hiroshi Watanabe - Department of General Education, Salesian Polytechnic, 4-6-8 Oyamagaoka, Machida-city, Tokyo, 194-0215, Japan (email)

Abstract: In this paper, we consider the initial value problem for strongly degenerate parabolic equations with discontinuous coefficients. This equation has the both properties of parabolic equation and hyperbolic equation. Therefore, we should choose entropy solutions as generalized solutions to the equation. Moreover, entropy solutions to the equation may not belong to $BV$ in our setting. These are difficult points for this type of equations.
    In particular, we consider the case that coefficients are the functions of bounded variation with respect to the space variable $x$. Then, we prove the existence of Kružkov type entropy solutions. Moreover, we prove the uniqueness of the solution under additional conditions.

Keywords:  Strongly degenerate parabolic, discontinuous coefficients, functions of bounded variation.
Mathematics Subject Classification:  Primary: 35K65, 35K55; Secondary: 35L65, 35R05.

Received: September 2012;      Revised: February 2013;      Published: November 2013.