Pseudo-Parabolic Regularization of Forward-Backward Parabolic Equations with Bounded Nonlinearities

Abstract

We study the initial-boundary value problem $$ \Big\{{\displaystyle \begin{array}{l} ut={\left[\varphi (u)\right]}_{xx}+\varepsilon {\left[\psi (u)\right]}_{txx}\kern1em \mathrm{in}\;\varOmega \times \left(0,T\right]\\ {}\varphi (u)+\varepsilon {\left[\psi (u)\right]}_t=0\kern3em \mathrm{in}\;\partial \varOmega \times \left(0,T\right]\\ {}u={u}_0\ge 0\kern7em \mathrm{in}\;\varOmega \times \left\{0\right\},\end{array}} $$ with Radon measure-valued initial data, by assuming that the regularizing term ψ is bounded and increasing (the cases of power-type or logarithmic ψ were examined in [2, 3] for spaces on any dimension). The function 𝜑 is nonmonotone and bounded, and either (i) decreases and vanishes at infinity, or (ii) increases at infinity. The existence of solutions in a space of positive Radon measures is proved in both cases. Moreover, a general result on the spontaneous appearance of singularities in he case (i) is presented. The case of a cubic-like 𝜑 is also discussed to point out the influence of the behavior at infinity of 𝜑 on the regularity of solutions.