Young measure solutions for a nonlinear parabolic equation of forward-backward type

Abstract: "The scope is to study the nonlinear parabolic problem of forward-backward type u[subscript t] = [delta]·q([delta]u) on Q[subscript infinity] = [omega] x R⁺ with initial data u₀ given in H¹₀([omega]). Here [omega] [contained within] R[superscript N] is open, bounded with mildly smooth boundary and q [element of] C(R[superscript N];R[superscript N]), an analogue to heat flux, satisfies q = [delta phi] with [phi element of] C¹ (R[superscript 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 [element of] H¹[subscript loc](Q[subscript infinity]) [intersection of] L[superscript infinity] (R⁺; H¹₀(omega)) and a parameterized family of probability measures v = (v[subscript x,t])(x,t)[element of]Q[subscript infinity] related to u by [delta]u = fR[superscript N][lambda v][d[lambda]) a.e. in Q[subscript infinity]; via v the nonlinearity q([delta]u)is replaced by the moment = fR[superscript N]q(lambda])v(d[lambda]) a.e. in Q[subscript infinity] and the equation is then interpreted in H[superscript -1]. The family v is generated by the gradients of a sequence in H¹[subscript loc](Q[subscript infinity]), is non-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 v[superscript infinity] such that the pair (Z,v[superscript infinity]) is a Young measure solution of the steady-state problem [delta] · q([delta]z) = 0. The relevant energy function is shown to be monotone decreasing and asymptotically tending to its minimum, globally and locally in space."