i have the following semilinear parabolic problem $$ \partial_t u(x,t)- \Delta u +F(u)= f(x,t); x \in \mathbb{R}^n, t > 0 $$ $$ u(x,0)=0 $$ with periodic boundary conditions, and $F$ is non linear.
What are the conditions of the non linear term $F$ and the second member $f$ to have a unique classical solution to this problem? Or there exists an reference for this result?