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

We consider parabolic equations of the form ut = ∆u + f(u) + h(x, t), (x, t) ∈ R × (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 ut = ∆u + 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.