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Solutions in Morrey spaces of some semilinear heat equations with time-dependent external forces. (English) Zbl 1053.35062

It is considered the Cauchy problem for a semilinear heat equation with time-dependent external forces \[ {\partial v\over \partial t}(t,x)=\Delta v(t,x)+v(t,x)^\nu+f(t,x)\quad\text{in}\quad (0,\infty)\times {\mathbb R}^n, \]
\[ v(0,x)=a(x)\quad\text{on}\quad {\mathbb R}^n \] with \(\nu\geq 3,\) \(\nu\in {\mathbb Z}.\) Both the external force and the initial data are assumed to be small in some Morrey spaces. It is proved unique existence of a small time-global solution. The stability of that solution follows by the time-global solvability of perturbation problems.

MSC:

35K55 Nonlinear parabolic equations
35B35 Stability in context of PDEs
35K15 Initial value problems for second-order parabolic equations
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