Generalized fourier transforms and global small solutions to kirchhoff equations

Abstract

It is proved that the inverse of the generalized Fourier transform associated to V an appropriate compactly supported potential, maps into the space of rapidly decreasing functions. This is used for the study of wave equations with non-local nonlinearities of the type being an exterior domain in R3 with V = 0, assuming Dirichlet boundary conditions for u. For a class of smooth data we obtain global existence of small solutions, as well as a partial characterization of the asymptotic behavior as t→∞ . The existence of global large solutions generated by eigenvectors corresponding to negative eigenvalues of is also investigated.