A MULTIPOINT IN-TIME PROBLEM FOR THE 2b-PARABOLIC EQUATION WITH DEGENERATION

Abstract

In recent decades, special attention has been paid to problems with nonlocal conditions for partial differential equations. Such interest in such problems is due to both the needs of the general therapy of boundary value problems and their rich practical application (the process of diffusion, oscillations, salt and moisture transport in soils, plasma physics, mathematical biology, etc.). A multipoint in-time problem for a nonuniformly 2b-parabolic equation with degeneracy is studied. The coefficients of the parabolic equation of order 2b allow for power singularities of arbitrary order both in the time and spatial variables at some set of points. Solutions of auxiliary problems with smooth coefficients are studied to solve the given problem. Using a priori estimates, inequalities are established for solving problems and their derivatives in special Hölder spaces. Using the theorems of Archel and Riess, a convergent sequence is distinguished from a compact sequence of solutions of auxiliary problems, the limiting value of which will be the solution of the given problem. Estimates of the solution of the multipoint time problem for the 2b-parabolic equation are established in Hölder spaces with power-law weights. The order of the power weight is determined by the order of degeneracy of the coefficients of the groups of higher terms and the power features of the coefficients of the lower terms of the parabolic equation. With certain restrictions on the right-hand side of the equation, an integral image of the solution to the given problem is obtained.