On initial boundary value problem for parabolic differential operator with non-coercive boundary conditions

TL;DR

This paper investigates the initial boundary value problem for a second-order parabolic differential operator with non-coercive boundary conditions, demonstrating existence and uniqueness of solutions despite reduced smoothness.

Contribution

It introduces a novel analysis of non-coercive boundary conditions for parabolic operators and proves solution existence and uniqueness in specialized Bochner spaces.

Findings

Unique solution existence proven using Faedo-Galerkin method

Reduced solution smoothness compared to coercive cases

Solution established in specific Bochner spaces

Abstract

We consider initial boundary value problem for uniformly 2-parabolic differential operator of second order in cylinder domain in ${\mathbb R}^n $ with non-coercive boundary conditions. In this case there is a loss of smoothness of the solution in Sobolev type spaces compared with the coercive situation. Using by Faedo-Galerkin method we prove that problem has unique solution in special Bochner space.