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

A. Tesei

Istituto per le Applicazioni del Calcolo "M. Picone", Consiglio Nazionale delle Ricerche, Via dei Taurini 19, I-00185 Rome, Italy

Abstract: 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.

UDC: 517.9