Local and global solvability of the Grushin heat equation with mixed nonlinear memory and reaction terms

TL;DR

This paper studies the solvability of a degenerate heat equation with nonlinear reaction and memory terms, establishing conditions for local and global solutions and highlighting the impact of memory on solvability.

Contribution

It provides new local and global well-posedness results for the Grushin heat equation with complex nonlinearities, including memory effects.

Findings

Established local well-posedness in Lebesgue and continuous function spaces.

Derived sufficient conditions for global existence based on nonlinearities.

Showed how memory terms influence the global solvability of the equation.

Abstract

In this work, we investigate the solvability of a heat equation involving the Grushin operator. The equation is perturbed by two nonlinear reaction terms, one of which includes a memory component, introducing nonlocal effects in time. We first establish local well-posedness results in both Lebesgue spaces and in the space of continuous functions vanishing at infinity. Furthermore, we develop and apply comparison principles that allow us to derive sufficient conditions for the global existence of solutions, depending on the relative strength and nature of the nonlinearities involved. In particular, we highlight how the presence of the memory term influences global solvability. Our results contribute to the understanding of parabolic equations with degenerate operators and complex nonlinear interactions.