Sobolev regularization of a class of forward–backward parabolic equations

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We address existence and asymptotic behaviour for large time of Young measure solutions of the Dirichlet initial–boundary value problem for the equation ut=∇⋅[φ(∇u)], where the function φ need not satisfy monotonicity conditions. Under suitable growth conditions on φ, these solutions are obtained by a “vanishing viscosity” method from solutions of the corresponding problem for the regularized equation ut=∇⋅[φ(∇u)]+ϵΔut. The asymptotic behaviour as t→∞ of Young measure solutions of the original problem is studied by ω-limit set techniques, relying on the tightness of sequences of time translates of the limiting Young measure. When N=1 this measure is characterized as a linear combination of Dirac measures with support on the branches of the graph of φ.