Nonlinear wave equations on a class of bounded fractal sets

The nonlinear wave equation ut t = Δu + f (u) with given initial data and zero boundary conditions on a class of bounded self-similar fractal sets is investigated. The Sobolev-type inequality is the starting point of this work, which holds for a class of fractals including the well-known Sierpínski gasket. We obtain the global existence of strong solutions for suitable f if the spectral dimension ds of the fractal satisfies ds < 2. The key is to construct the wave propagator and Hilbert spaces of functions on the fractal. The main difficulty in obtaining the global existence of a weak solution is establishing a priori estimates depending on a regularity property for f. The regularity property of a weak solution is obtained through a fine analysis in which the Sobolev-type inequality plays a crucial role.  2002 Elsevier Science (USA). All rights reserved.