Positive solutions with a time-independent boundary singularity of semilinear heat equations in bounded Lipschitz domains

We study time-global positive solutions of semilinear heat equations of the form ut−Δu=f(x,u)ut−Δu=f(x,u) in a bounded Lipschitz domain ΩΩ in RnRn. In particular, we show the existence of a positive solution with a time-independent singularity at a boundary point ξξ of ΩΩ which converges to a positive solution, with the behavior like the Martin kernel at ξξ, of the corresponding elliptic equation at time infinity. A nonlinear term ff is conditioned in terms of a certain Lipschitz continuity with respect to the second variable and a generalized Kato class associated with the Martin kernel at ξξ, and admits not only usual one V(x)up(log(1+u))qV(x)up(log(1+u))q, but also one with variable exponents.