Uniqueness and root-Lipschitz regularity for a degenerate heat equation

TL;DR

This paper studies a degenerate quasilinear heat equation in one dimension, introducing a strong solution concept that guarantees uniqueness and regularity, with implications for stochastic heat equations.

Contribution

It defines a strong solution framework for the degenerate heat equation, ensuring uniqueness and regularity properties for solutions with suitable initial data.

Findings

Existence of a lower bound on the lifespan of strong solutions.

Global Lipschitz regularity of the square root of solutions.

Uniqueness of solutions under the strong solution framework.

Abstract

We consider nonnegative solutions of the quasilinear heat equation $\partial_t u = \tfrac{1}{2} u \partial_x^2 u$ in one dimension. Our solutions may vanish and may be unbounded. The equation is then degenerate, and weak solutions are generally nonunique. We introduce a notion of strong solution that ensures uniqueness. For suitable initial data, we prove a lower bound on the time for which a strong solution $u$ exists and $\sqrt{u}$ remains globally Lipschitz in space. In a companion paper, we show that this condition is important in the study of two-dimensional nonlinear stochastic heat equations.