The weak well-posedness of strong damping wave equations defined by fractal
Laplacians is proved by using 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 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.
We study linear and nonlinear Schrodinger equations defined by fractal measures. Under
the assumption that the Laplacian has compact resolvent, we prove that there exists a uniqueness
weak solution for a linear Schrodinger equation, and then use it to obtain the existence and
uniqueness of weak solution of a nonlinear Schrodinger equation. We prove that for a class of self-similar
measures on R with overlaps, the Schrodinger equations can be discretized so that the finite
element method can be applied to obtain approximate solutions. We also prove that the numerical
solutions converge to the actual solution and obtain the rate of convergence.
We study an asymptotic estimate on the number of negative eigenvalues of the Schr\"odinger operators on unbounded fractal spaces which admit a cellular decomposition. We first give some sufficient conditions for Weyl-type asymptotic formula to hold. Second, we verify these conditions for the infinite blowup of Sierpi\'nski gasket and unbounded generalized Sierpi\'nski carpets. Final, we demonstrate how the result can be applied to the infinite blowup of certain fractals associated with iterated function systems with overlaps, including those defining the classical infinite Bernoulli convolution with golden ratio.