Solutions in Morrey spaces of some semilinear heat equations with time-dependent external forces

In this paper, we consider the Cauchy problem for some semilinear heat equations with time-dependent external forces. Both the external force and the initial data are assumed to be small in some Morrey spaces. We first prove the unique existence of a small time-global solution. We next show the stability of that solution by proving the time-global sovability of perturbation problems.