Convergence to a steady state for asymptotically autonomous semilinear heat equations on RNRN

We consider parabolic equations of the formut=Δu+f(u)+h(x,t),(x,t)∈RN×(0,∞), where f   is a C1C1 function with f(0)=0f(0)=0, f′(0)<0f′(0)<0, and h   is a suitable function on RN×[0,∞)RN×[0,∞) which decays to zero as t→∞t→∞ (hence the equation is asymptotically autonomous). We show that, as t→∞t→∞, each bounded localized solution u⩾0u⩾0 approaches a set of steady states of the limit autonomous equation ut=Δu+f(u)ut=Δu+f(u). Moreover, if the decay of h is exponential, then u converges to a single steady state. We also prove a convergence result for abstract asymptotically autonomous parabolic equations.