Weak damped wave equations defined by a class of self-similar measures with overlaps

Abstract

We study weak damped wave equations defined by fractal Laplacians. These Laplacians are defined by self-similar measures with overlaps, such as the well-known infinite Bernoulli convolution associated with the golden ratio, the three-fold convolution of the Cantor measure, and a class of self-similar measures that we call essentially of finite type. First, we prove the weak well-posedness of these wave equations and some regularity results. Second, by using some important information about the structure of the three measures above, we set up a framework for one-dimensional measures to discretize the wave equations. Then we use the finite element and central difference methods to obtain numerical approximations of the weak solutions for such equations. Finally, we show that the numerical solutions converge to the actual weak solution and obtain the rate of convergence.