Abstract
We consider the Cauchy problem for semilinear wave equationsu tt−Δu=g(u) in 3+1 dimensions with smooth but possibly large data. Ifg isC 2,α and bounded from above everywhere and from below for negative arguments the existence of a global classical solution is shown. If moreoverg is nonpositive and vanishes at least of order 2+∈ at the origin and if the data decay sufficiently rapidly at infinity the scattering operator exists.
Similar content being viewed by others
References
Grillakis, M.G.Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity. Preprint
John, F.Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscripta math. 28, 235–268 (1979)
Jörgens, K.Das Anfangswertproblem im Großen für eine Klasse nichtlinearer Wellengleichungen. Math. Z. 77, 295–308 (1961)
Pecher, H.Scattering for semilinear wave equations with small data in three space dimensions. Math. Z. 198, 277–289 (1988)
Peral Alonso, I.Some remarks on semilinear wave equations in R n. In: J.I. Diaz, P.L. Lions (editors):Contributions to nonlinear partial differential equations, vol. II, Pitman Research Notes in Mathematics Series No. 155, 193–209 (1987)
Schaeffer, J.The equation u tt−Δu=|u|p for the critical value of p. Proc. Roy. Soc. Edinburgh 101A, 31–44 (1985)
Struwe, M.On the u 5—Klein Gordon equation. Preprint
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pecher, H. Global smooth solutions to a class of semilinear wave equaiions with strong nonlinearities. Manuscripta Math 69, 71–92 (1990). https://doi.org/10.1007/BF02567913
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02567913