The weak well-posedness of strong damping wave equations de fined by fractal
Laplacians is proved by using Galerkin method. These fractal Laplacians are de fined by
self-similar measures with overlaps, such as the well-known infi nite 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 overlap are complicated and intractable. However, some
important information about the structure of the three measures above can be obtained.
We make use of these information to set up a framework for one-dimensional measures to
discretize the 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.