Global Cauchy problem for the complex NLKG and sinh-Gordon equations in super-critical spaces

By introducing a class of new function spaces B-p,q(sigma,s) as the resolution spaces, we study the Cauchy problem for the nonlinear Klein-Gordon (NLKG) and sinh-Gordon equations in all spatial dimensions d >= 1, partial derivative(2)(t)u+u-Delta u+f(u) = 0, (u, partial derivative(t) u)t_0 = (u(0), u(1)), where f(u) = (u1+ alpha) or f(u) =sinh u - u. We consider the initial data (u(o), u(1)) in supercritical function spaces E-sigma,E-s x E sigma-1s, for which their norms are defined by parallel to f parallel to(E sigma,s) = parallel to <xi >(sigma) 2s vertical bar vertical bar f()parallel to(L2), s < 0, sigma is an element of R. Any Sobolev space H-kappa can be embedded into E-sigma,E-s i.e., H-kappa subset of E-sigma,E-s for any kappa, sigma is an element of R and s < 0. We show the global existence and uniqueness of the solutions of NLKG if the initial data belong to some E-sigma,E-s x E-sigma-1,E-s (s < 0, sigma >= max(d/2-2/alpha, 1/2), alpha is an element of N, alpha >= 4/d) and their Fourier transforms are supported in the first octant, the smallness conditions on the initial data in E-sigma,E-s x E-sigma-1,E-s, are not required for the global solutions. Similar results hold for the sinh-Gordon equation if the spatial dimensions d >= 2. (C) 2024 Elsevier Inc, All rights reserved.