Global solutions for semilinear parabolic evolution problems with Hölder continuous nonlinearities

It is shown that semilinear parabolic evolution equations $u^{\mathrm{\prime }}=A+f(t,u)$ featuring Hölder continuous nonlinearities $f=f(t,u)$ with at most linear growth possess global strong solutions for a general class of initial data. The abstract results are applied to a recent model describing front propagation in bushfires and in the context of a reaction-diffusion system.