Semilinear heat equations with distributions in Morrey spaces as initial data (Q5957951)

The authors consider the following Cauchy problem NEWLINE\[NEWLINE{\partial v\over \partial t}(t,x)=\Delta v(t,x) +v(t,x)|v(t,x)|^{\nu-1}+f(x)\quad\text{in } (0,\infty)\times {\mathbb R}^n,NEWLINE\]NEWLINE NEWLINE\[NEWLINEv(0,x)=a(x)\quad \text{on }{\mathbb R}^n,NEWLINE\]NEWLINE where \(\nu>n/(n-2).\) Both the external force \(f\) and initial data \(a\) belong to suitable Morrey spaces. The unique existence of a small solution to the corresponding stationary problem NEWLINE\[NEWLINE -\Delta w(x)= w(x) |w(x)|^{\nu-1} +f(x) \quad \text{ on}\quad {\mathbb R}^n NEWLINE\]NEWLINE is proved when the norm of the external force is small. When the initial data \(a\) is close enough to the solution \(w\) of the stationary problem, the Cauchy problem possesses time-global solution, which leads also to stability of the small stationary solution.