On prescribed change of profile for solutions of parabolic equations

TL;DR

This paper investigates parabolic equations with boundary conditions that specify a fixed change of the solution profile over time, establishing well-posedness, existence, regularity, and maximum principle analogs in an L2 framework.

Contribution

It introduces a novel boundary condition for parabolic equations requiring a prescribed profile change, and proves well-posedness and regularity results in the L2 setting.

Findings

The problem is well-posed in L2 space.

Existence and regularity of solutions are established.

An analog of the maximum principle is proved.

Abstract

Parabolic equations with homogeneous Dirichlet conditions on the boundary are studied in a setting where the solutions are required to have a prescribed change of the profile in fixed time, instead of a Cauchy condition. It is shown that this problem is well-posed in L_2-setting. Existence and regularity results are established, as well as an analog of the maximum principle.