Research

You can find a list of my publications and presentations here.

Under the guidance of Hans Volkmer, I studied asymptotics and analysis of PDEs. The primary focus of my research was to find the asymptotic expansion of the L^2-norm of the weak solution of the strongly damped wave equation in R^n. The secondary focus was to further obtain the expansion of the L^2-norm of an asymptotic profile (Ikehata, 2014) of the weak solution. The upshot was to exhibit a cancellation of the leading terms in the asymptotic expansion the norm of the difference of the weak solution and its profile—this behavior is analogous to the diffusion phenomenon involving the heat and dissipative wave equations. With proper assumptions on the Cauchy data, I was able to compute the asymptotic expansions up to any desired accuracy. I've included some pictures below showing the convergence of the expansions to the actual L^2-norms.

Since formally publishing the work from my thesis I have published on the L^2-asymptotics on beam equations with strong damping. My long-term goal since completing my thesis, however, has been to derive the L^2-asymptotics of any time- or space-derivative of the weak solution of the strongly damped wave equation and its asymptotic profile. The idea resembles the work of Volkmer (2010) in which he does this very general expansion of arbitrary time- and space-derivatives as relating to the heat and dissipative wave equations.