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

Source 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.