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Dynamics of measured valued solutions to a backward-forward heat equation

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Abstract

This paper examines the asymptotic behavior of measure valued solutions to the initial value problem for the nonlinear heat conduction equation

$$\frac{{\partial u}}{{\partial t}} = \nabla \cdot q(\nabla u), x \in \Omega , t > 0$$

,xεΩ, t>0 in a bounded domainΩR N with boundary conditions of the form

$$u = 0 on \partial \Omega or q(\nabla u) \cdot n = 0 on \partial \Omega $$

In particular, use of the Young measure representation of composite weak limits allows proof of a general trend to equilibrium. No linearity or monotonicity is assumed forq; the only major restriction onq is that it satisfies the Fourier inequalityq(λλ⩾0 for allλε R N. Applications are given to problems whereq is not monotone.

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Slemrod, M. Dynamics of measured valued solutions to a backward-forward heat equation. J Dyn Diff Equat 3, 1–28 (1991). https://doi.org/10.1007/BF01049487

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