Strong damping wave equations defined by a class of self-similar measures with overlaps

The weak well-posedness of strong damping wave equations defined by fractal Laplacians is proved by using the Galerkin method. These fractal 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. In general, the structure of self-similar measures with overlaps is complicated and intractable. However, some important information about the structure of the above three measures can be obtained. We make use of this information to set up a framework for one-dimensional measures to discretize the associated strong damping wave equations, and use the finite element and central difference methods to obtain numerical approximations of the weak solutions. We also show that the numerical solutions converge to the actual solution and obtain the rate of convergence.