On parabolic equations in one space dimension

TL;DR

This paper investigates the solvability of inhomogeneous second-order parabolic equations in one dimension, providing negative results and estimates for fundamental solutions under minimal regularity assumptions on the coefficients.

Contribution

It presents new negative solvability results and estimates for fundamental solutions of parabolic equations with bounded measurable coefficients in one dimension.

Findings

Negative solvability results in Sobolev classes for certain parabolic equations.

Upper and lower estimates for fundamental solutions.

Conditions on coefficients ensuring boundedness away from zero.

Abstract

Several negative results are presented concerning the solvability in Sobolev classes of the Cauchy problem for the inhomogeneous second-order uniformly parabolic equations without lower order terms in one space dimension. The main coefficient is assumed to be a bounded measurable function of $(t,x)$ bounded away from zero. We also discuss upper and lower estimates of certain kind on the fundamental solutions of such equations.