The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients in a wedge

TL;DR

This paper establishes coercive estimates for non-divergence parabolic equations with time-measurable coefficients in a wedge, introducing the critical exponent concept and applying results to complex domains with edges or conical points.

Contribution

It introduces the critical exponent for such equations and proves its key properties, advancing understanding of parabolic problems with discontinuous coefficients in non-smooth domains.

Findings

Established coercive estimates in weighted $L_{p,q}$-spaces.

Introduced and analyzed the critical exponent concept.

Applied results to equations in domains with edges or conical points.

Abstract

We consider the Dirichlet problem in a wedge for parabolic equation whose coefficients are measurable function of t. We obtain coercive estimates in weighted $L_{p,q}$-spaces. The concept of "critical exponent" introduced in the paper plays here the crucial role. Various important properties of the critical exponent are proved. We give applications to the Dirichlet problem for linear and quasi-linear non-divergence parabolic equations with discontinuous in time coefficients in cylinders $\Omega\times(0,T)$, where $\Omega$ is a bounded domain with an edge or with a conical point.