NDLI: Young Measure Solutions for a Nonlinear Parabolic Equation of Forward-Backward Type

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Demoulini, Sophia
Copyright Year	1996
Abstract	The scope is to study the nonlinear parabolic evolution of forward-backward type \[u_t = \nabla \cdot q(\nabla u)\quad {\text{on }}Q_\infty \equiv \Omega \times \mathbb{R}^ + \] with initial data $u_0 $ given in $H_0^1 (\Omega )$, where $\Omega \subset \mathbb{R}^N $ is open, bounded, and $q \in C(\mathbb{R}^N ;\mathbb{R}^N )$, an analogue to heat flux, satisfies $q = \nabla \phi $ with $\phi \in C^1 (\mathbb{R}^N )$ of suitable growth. When $\phi $ is not convex classical solutions do not exist in general; the problem admits Young measure solutions. By that is meant a function u in a suitable Sobolev space and a gradient-generated family of probability measures $\nu = (\nu _{x,t} )_{(x,t) \in Q_\infty } $ related by $\nabla u = \langle {\nu ,id} \rangle $ almost everywhere (a.e.) (the identity integrated against $\nu $) and such that the equation can be interpreted distributionally in $H^{ - 1} :\int_0^{ + \infty } {\int_\Omega {\langle {\nu ,q} \rangle } } \cdot \nabla \zeta + u_t \zeta dxdt$ for all $\zeta \in H_0^1 (Q_\infty )$. The family $\nu $ is not unique, but through its first moment some of the classical properties are preserved: uniqueness of the function u is true; stability is reflected in a maximum principle and a comparison result. The asymptotic analysis yields, as time tends to infinity, a unique limit z and an associated Young measure $\nu ^\infty $ such that the pair $(z,\nu ^\infty )$ is a Young measure solution of the steady-state problem $\nabla \cdot q(\nabla z) = 0$. The relevant energy function is shown to be monotone decreasing and asymptotically tending to its minimum, globally and locally in space.
Starting Page	376
Ending Page	403
Page Count	28
File Format	PDF
ISSN	00361410
DOI	10.1137/S0036141094261847
e-ISSN	10957154
Journal	SIAM Journal on Mathematical Analysis (SJMAAH)
Issue Number	2
Volume Number	27
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	1996-03-01
Access Restriction	Subscribed
Subject Keyword	calculus of variations forward-backward heat equation Initial value problems for second-order parabolic equations Young measures weak convergence
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis Computational Mathematics

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