  The Cauchy problem is studied for very general systems of evolution
equations, where the time derivative of solution is written by Fourier
multipliers in space and analytic nonlinearity, with no other structural
requirement. We construct a function space for the Fourier transform embedded
in the space of distributions, and establish the global wellposedness with no
size restriction. The major restriction on the initial data is that the Fourier
transform is supported on the half space, decaying at the boundary in the sense
of measure. We also require uniform integrability for the orthogonal directions
in the distribution sense, but no other condition. In particular, the initial
data may be much more rough than the tempered distributions, and may grow
polynomially at the spatial infinity. A simpler argument is also presented for
the solutions locally integrable in the frequency. When the Fourier support is
slightly more restricted to a conical region, the generality of equations is
extremely wide, including those that are even locally illposed in the standard
function spaces, such as the backward heat equations, as well as those with
infinite derivatives and beyond the natural boundary of the analytic
nonlinearity. As more classical examples, our results may be applied to the
incompressible and compressible Navier-Stokes and Euler equations, the
nonlinear diffusion and wave equations, and so on. The major drawback of the
Fourier support restriction is that the solutions cannot be real valued.
