Interpolation inequality at one time point for parabolic equations with time-independent coefficients and applications

TL;DR

This paper establishes a new interpolation inequality and observability results for parabolic equations with unbounded or electric potentials, using a novel reduction method and stability estimates, applicable in bounded domains with Neumann boundary conditions.

Contribution

It introduces a new approach for deriving interpolation and observability inequalities for parabolic equations with complex potentials, expanding their applicability.

Findings

Established H"older-type interpolation inequalities for parabolic equations.

Proved observability inequalities from measurable sets in time.

Applied the method to equations with unbounded and electric potentials.

Abstract

In this paper, we study the H\"older-type interpolation inequality and observability inequality from measurable sets in time for parabolic equations either with L^p unbounded potentials or with electric potentials. The parabolic equations under consideration evolve in bounded C^{1,1} domains of R^N (N\geq3) with homogeneous Neumann boundary conditions. The approach for the interpolation inequality is based on a modified reduction method and some stability estimates for the corresponding elliptic operator.