Two-phase Entropy Solutions of a Forward–Backward Parabolic Equation

Long-time behaviour of two-phase solutions to a class of forward-backward parabolic equations

“…In this paper we obtain global existence of two-phase solutions to the Neumann problem associated to equation (1.1) in the case of cubic-like response functions φ and initial data functions u 0 subject to the constraint a u 0 b in (ω 1 , 0) and c u 0 d in (0, ω 2 ); such a result can be regarded as the counterpart of the one obtained in [MTT2] for the Cauchy problem associated to equation (1.1) in the case of piecewise nonlinearities φ. In particular, we will prove that global two-phase solutions (in the sense of Definition 2.1 below) can be obtained as limiting points of the solutions (u ε , φ(u ε )) to the Neumann initial-boundary value problems associated to the pseudoparabolic regularization (1.7) of equation (1.1).…”

“…( [EP,MTT1]). That is, jumps between the stable phases S 1 and S 2 occur only at the points (x, t) where the function v(x, t) takes the value A (jumps from S 2 to S 1 ) or B (jumps from S 1 to S 2 ).…”

“…Therefore, following the terminology in [MTT2], problem (2.1) can be regarded as a steady boundary problem, since ξ (t) = 0 for any t 0.…”

“…On the other hand, a natural question is whether uniqueness can be recovered by introducing some additional constraints. For this purpose, two-phase solutions have been introduced in [EP] and investigated in [MTT1,MTT2,T]. Roughly speaking, a two-phase solution to the Neumann initial-boundary value problem associated to equation (1.1) in Q T = Ω × (0, T ) is a weak entropy measure-valued solution (u, v) (in the sense of [Pl1]) which describes transitions only between stable phases.…”

“…Local existence and uniqueness of smooth two-phase solutions to the Cauchy problem associated to equation (1.1) was studied in [MTT2] for piecewise response functions φ. Actually, global existence of such solutions is proven to hold for initial data functions u 0 satisfying the condition a u 0 d (see Figure 1), whereas it is still unknown in the general case.…”