Jan Wladyslaw Cholewa | University of Silesia - Polônia

Resumo

The proposed research concerns continuation of solutions, which embeds into the fundamental problems of the theory of differential equations. Although the existence of the solutions on a small time interval has been successfully treated for a large class of evolutionary equations then, not rarely, their long time existence remains unknown. This can be observed even for some widely known equations, that are investigated by top specialists in the field in many scientific centers all over the world. A crown example here is the Navier-Stokes system, falling into a group of millennium problems announced by the Clay Mathematics Institute (CMI, Cambridge, Massachusetts), for which the existence of global regular solutions remains unknown. The latter shows the importance of the proposed research for the development of Science. Mentioned should be made that in the qualitative theory of differential equations it is of central importance to establish that solutions exist globally in time. Further questions concerning asymptotic properties of the solutions and the existence of a suitable notion of attractor can only be addressed after the global existence is primarily established. During the past cooperation the authors already discussed continuation properties of the solutions. The continuation after the blow up will also be discussed during this visit for parabolic type problems with critical nonlinearities which includes the strongly damped wave equations. (AU)