Uniqueness and root-Lipschitz regularity for a degenerate heat equation

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.