A priori estimate of the solution of the Cauchy problem in the Sobolev classes for discontinuous coefficients of degenerate heat equations

Abstract

Partial differential equations of the parabolic type with discontinuous coefficients and the heat equation degenerating in time, each separately, have been well studied by many authors. Conjugation problems for time-degenerate equations of the parabolic type with discontinuous coefficients are practically not studied. In this work, in an n-dimensional space, a conjugation problem is considered for a heat equation with discontinuous coefficients which degenerates at the initial moment of time. A fundamental solution to the set problem has been constructed and estimates of its derivatives have been found. With the help of these estimates, in the Sobolev classes, the estimate of the solution to the set problem was obtained.