Seminar on nonlinear problems of PDE and mathematical physics on topic “Properties of solutions of degenerate parabolic equations with an inhomogeneous density”
Speaker: D.Sc., professor Tedeev A. F. (Vladikavkaz Scientiﬁc Center of the RAS, Southern Mathematical Institute-branch of VSC ,Vladikavkaz).
Title of the talk: Properties of solutions of degenerate parabolic equations with an inhomogeneous density.
The qualitative properties of solutions of parabolic equations with double non-linearity and with an inhomogeneous density are investigated. Under certain conditions of the decrease of the density function at inﬁnity the following properties of the solutions of the Cauchy problem are established:
- Optimal estimates in time of the solution
- The sharp bound of the radius of support of the solution
- The interface blow-up in ﬁnite time phenomena
- Universal bound in time estimates of solutions
The ﬁrst results of a qualitative study of the solution of the Cauchy problem for the equation of a porous medium with an inhomogeneous density were obtained in the works of S. Kamin, F. Rosenau in 1982-1983. In these works, non-standard properties of the solution were established with a suﬃciently rapid decrease density at inﬁnity. In 1993, the work of S. Kamin, R. Kersner was the ﬁrst to establish the phenomenon of the interface blow-up with a certain density behavior at inﬁnity