On the first mixed problem for degenerate parabolic equations in stellar domains with Lyapunov boundary in Banach spaces. II

Abstract

This work continues the article «On the first mixed problem for degenerate parabolic equations in stellar domains with Lyapunov boundary in Banach spaces» and studies the behavior of a solution to a second-order parabolic equation with Tricomi degeneracy on the lateral boundary of a cylindrical domain Q^T, where Q is a stellar domain whose boundary ∂Q is an (n − 1)-dimensional closed surface without an edge of class C^1+λ, 0 < λ < 1.

We consider two ways to choose the boundary condition: 1) due to the fact that Q is stellar, 2) some direction orthogonal to the boundary is allocated and the continuity of the solution as a function of a special variable with values in L_p in this direction is asserted. To do this, by determining the boundary value, while mapping the boundary ∂Q, it is necessary to take a shift not along the normal at each point x ∈ ∂Q, but to take a sufficiently small covering of the boundary and shift each piece of this covering «parallel» along the normal at one fixed point of this piece x₀.

We also consider the question of the unambiguous solvability of the first mixed problem for an equation when the boundary and initial functions belong to spaces of type L_p, p > 1.