The Cauchy Problem for Nonlinear Klein-Gordon Equations in the Sobolev Spaces

Abstract

The local and global well-posedness for the Cauchy problem for a class of nonlinear Klein-Gordon equations is studied in the Sobolev space Hs = Hs(Rn) with s ≥ n/2. The global well-posedness of the problem is proved under the following assumptions: (1) Concerning the nonlinearity f, f(u) behaves as a power u1+4/n near zero. At infinity f(u) has an exponential growth rate such as exp(κ|u|ν) with κ > 0 and 0 n/2. (2) Concerning the Cauchy data (ϕ,Ψ) e Hs = Hs ⊕ HS-1, ‖(ϕ,Ψ); H1/2‖ is relatively small with respect to ‖(ϕ,Ψ); Hs∗‖, where s∗ is a number with s∗ = n/2 if s = n/2, n/2 < s∗ ≤ s if s > n/2, and the smallness of ‖(ϕ,Ψ); Hn/2‖ is also needed when s = n/2 and ν = 2. © 2001, Research Institute for Mathematical Sciences. All rights reserved.