PDE Seminar (2016/4/22): Asymptotics for Linear Dissipative Wave Equations, Borislav Yordanov (Hokkaido University)

2016-4-22 16:30 - 2016-4-22 17:30
Faculty of Science Building #3, Room 309
Borislav Yordanov (Hokkaido University)
The solutions of dissipative wave equations have relatively simple behavior as time goes to infinity: the leading terms are proportional to a Gaussian or, more generally, to a solution of the corresponding diffusion equation.
That second-order time derivatives are negligible in the long-time asymptotics of such equations is known as the ``diffusion phenomenon" for damped waves. We discuss two methods for showing the diffusion phenomenon: weighted energy method and spectral method.

We also give applications to $L^p$-estimates for dissipative wave equations with variable coefficients generalizing the classical estimates of A. Matsumura in
``On the asymptotic behavior of solutions of semi-linear wave equations", Publ. Res. Inst. Math. Sci., Kyoto Univ., 12 (1976), 169-189.