UNIQUENESS AND ROOT-LIPSCHITZ REGULARITY FOR A DEGENERATE HEAT EQUATION

We consider nonnegative solutions of the quasilinear heat equation 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 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. (Formula presented)