One-dimensional nonlinear heat equations defined by fractal Laplacians

We study the nonlinear heat equation defined by a class of one-dimensional self-similar measures with overlaps, such as the infinite Bernoulli convolution associated with the golden ratio and [Formula: see text]-fold convolution of the Cantor measure. By using a monotone iteration scheme and the maximum principle, we show that the nonlinear heat equation possesss a global solution for suitable nonlinear term and small initial data.