Speaker: professor A.E.Shishkov (RUDN University).
Title: Localized and non-localized regimes with singular peaking for quasi-linear parabolic PDE
Abstract: Localization properties of boundary regimes with infinite peaking as t tends to T<∞ for some parabolic equations, admitting barrier technique, were studied since 60-th of 20 century by A.A.Samarskii,I.M.Sobol', S.P.Kurdyunov,V.A.Galaktionov,B.H.Gilding, M.A.Herrero, A.S.Kalashnikov,,C.Cortazar,M.Elgueta and other. In 1999 A.E Shishkov proposed new approach to the study of mentioned problem, which do not use any variant of barrier technique (any comparison theorems) and is based on some adaptation of local energy estimates method. In the series of papers of V.A.Galaktionov and A.E. Shishkov mentioned approach was adapted for obtaining of sharp localization conditions of boundary regimes with strong peaking for higher order quasi-linear parabolic PDE. In present talk we will discuss new results about sharp upper estimates of final profile of solutions of parabolic PDE near to the blow-up time of boundary data, which generate localized peaking regime.