Convergence to a steady state for asymptotically autonomous semilinear heat equations on R{double-struck}^N

We consider parabolic equations of the form. u_t=δu+f(u)+h(x,t),(x,t)∈R{double-struck}^N×(0,∞), where f is a C1 function with f(0)=0, f<(0)<0, and h is a suitable function on RN×[0,∈) which decays to zero as t→∈ (hence the equation is asymptotically autonomous). We show that, as t→∈, each bounded localized solution u≥0 approaches a set of steady states of the limit autonomous equation u_t=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.