Global solutions of semilinear evolution equations satisfying an energy inequality

AbstractWe prove the global existence (in time) for any solution of an abstract semilinear evolution equation in Hilbert space provided the solution satisfies an energy inequality and the nonlinearity does not exceed a certain growth rate. When applied to semilinear parabolic initial-boundary-value problems the result admits also the limiting growth rates which were given by Sobolevskii and Friedman, but which where not permitted in their theorem. The Navier-Stokes system in two dimensions is a special case of our general result. The method is based on the theories of semigroups and fractional powers of regularly accretive linear operators and on a nonlinear integral inequality which gives the crucial a-priori estimate for global existence