In the case of initial data belonging to a wide class of functions including distributions of Gelfand-Shilov type we establish the correct solvability of the Cauchy problem for a new class of Shilov parabolic systems of equations with partial derivatives with bounded smooth variable lower coefficients and nonnegative genus. We also investigate the conditions of local improvement of the convergence of a solution of this problem to its limiting value when the time variable tends to zero.

This paper addresses analytical investigations of degenerating PDE systems for phase separation and damage processes considered on nonsmooth time-dependent domains with mixed boundary conditions for the displacement field. The evolution of the system is described by a degenerating Cahn-Hilliard equation for the concentration, a doubly nonlinear differential inclusion for the damage variable and a quasi-static balance equation for the displacement field. The analysis is performed on a time-dependent...

We are concerned with the problem of differentiability of the derivatives of order $m+1$ of solutions to the “nonlinear basic systems” of the type $${(-1)}^m\sum _{\left|\alpha \right|=m}{D}^{\alpha }{A}^{\alpha }\left(D^{\left(m\right)}u\right)+\frac{\partial u}{\partial t}=0\text{in}Q.$$ We are able to show that $$D^{\alpha }u\in L^2(-a,0,H^{\vartheta }(B\left(\sigma \right),{ℝ}^N)),|\alpha |=m+1,$$ for $\vartheta \in (0,1/2)$ and this result suggests that more regularity is not expectable.