Advances in Differential Equations

$BV$-entropy solutions to strongly degenerate parabolic equations

Shinnosuke Oharu and Hiroshi Watanabe

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In this paper a new notion of generalized solution to the initial-boundary-value problem for a strongly degenerate parabolic equation of the form $u_{t}+\nabla \cdot A(x,t,u)+B(x,t,u)={\varDelta} \beta(u)$ is treated. This type of solution is called a $BV$-entropy solution. Since equations of this form are linear combinations of time-dependent conservation laws and porous medium type equations, it is interesting to investigate interactions between singularities of solutions associated with the two different kinds of nonlinearities. However either of the part of conservation laws and that of porous medium type diffusion term may not be treated as a perturbation of the other. This observation leads us to the new notion of $BV$-entropy solution. Our objective here is to establish unique existence of such $BV$-entropy solutions under the homogeneous Neumann boundary conditions.

Article information

Adv. Differential Equations Volume 15, Number 7/8 (2010), 757-800.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K65: Degenerate parabolic equations 35K60: Nonlinear initial value problems for linear parabolic equations 47H20: Semigroups of nonlinear operators [See also 37L05, 47J35, 54H15, 58D07] 47J35: Nonlinear evolution equations [See also 34G20, 35K90, 35L90, 35Qxx, 35R20, 37Kxx, 37Lxx, 47H20, 58D25]


Watanabe, Hiroshi; Oharu, Shinnosuke. $BV$-entropy solutions to strongly degenerate parabolic equations. Adv. Differential Equations 15 (2010), no. 7/8, 757--800.

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