Asymptotic behavior of positive solutions for heat equation with nonlinear term

We study the nonlinear parabolic equation , in Rn×(0,∞) with boundary condition u(x,0)=u0(x), not necessarily bounded function. The nonlinearity φ((x,t),u) is required to satisfy some conditions related to the parabolic Kato class P∞(Rn) while allowing existence of positive solutions of the equation and continuity of such solutions. Our approach is based on potential theory tools.