Blow-up problem for semilinear heat equation with absorption and a nonlocal boundary condition

In this paper we consider a semilinear parabolic equation ut=Δu−c(x,t)uput=Δu−c(x,t)up for (x,t)∈Ω×(0,∞)(x,t)∈Ω×(0,∞) with nonlinear and nonlocal boundary condition u∣∂Ω×(0,∞)=∫Ωk(x,y,t)uldyu∣∂Ω×(0,∞)=∫Ωk(x,y,t)uldy and nonnegative initial data where p>0p>0 and l>0l>0. We prove some global existence results. Criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data are also given.