Weak solvability of a boundary value problem for a parabolic equation with a global-in-time term that contains a weighted integral

TL;DR

This paper proves the weak solvability of a parabolic PDE with a nonlocal-in-time interaction term involving a weighted integral, without requiring solution continuity in time, using energy estimates.

Contribution

It establishes weak solvability for a nonlinear parabolic PDE with a global-in-time integral term under general conditions, without assuming solution time continuity.

Findings

Weak solvability is proven for the PDE.

The proof relies solely on energy estimates.

No continuity in time of the solution is required.

Abstract

This paper deals with a parabolic partial differential equation that includes a non-linear nonlocal in time term. This term is the product of a so-called interaction potential and the solution of the problem. The interaction potential depends on a weighted integral of the solution over the entire time interval, where the problem is considered, and satisfies fairly general conditions. Namely, it is assumed to be a continuous bounded from below function that can behave arbitrarily at infinity. This fact implies that the interaction term is not a lower order term in the equation. The weak solvability of the initial boundary value problem for this equation is proven. The proof does not use any continuity properties of the solution with respect to time and is based on the energy estimate only.