Pseudo-parabolic regularization of forward-backward parabolic equations with bounded nonlinearities

A. Tesei

Contemporary Mathematics. Fundamental Directions, Proceedings of the Seventh International Conference on Differential and Functional-Differential Equations (Moscow, August 22–29, 2014). Part 3, Tome 60 (2016), pp. 164-183

Résumé

We study the initial-boundary value problem $$ \left\{\begin{array}{ll}u_t=[\varphi(u)]_{xx}+\varepsilon[\psi(u)]_{txx}\quad\text{in}~\Omega\times(0,T],\\ \varphi(u)+\varepsilon[\psi(u)]_t=0 \quad\text{in}~\partial\Omega\times(0,T],\\ u=u_0\ge0\quad\text{in}~\Omega\times\{0\}, \end{array}\right. $$ with Radon measure-valued initial data, by assuming that the regularizing term $\psi$ is increasing and bounded (the cases of power-type or logarithmic $\psi$ were dealt with in [2,3] in any space dimension). The function $\varphi$ is nonmonotone and bounded, and either (i) decreasing and vanishing at infinity, or (ii) increasing at infinity. Existence of solutions in a space of positive Radon measures is proven in both cases. Moreover, a general result proving spontaneous appearance of singularities in case (i) is given. The case of a cubic-like $\varphi$ is also discussed, to point out the influence of the behavior at infinity of $\varphi$ on the regularity of solutions.