We prove the existence of solutions converging to a stationary state for abstract semilinear parabolic problems with a convolution kernel that is unknown (together with the solution). These solutions are suitable perturbations of stationary states. The main tools are maximal regularity results in an $L^1$ (time) setting. The abstract results are applied to a reaction-diffusion integrodifferential system.
"Convergence to a stationary state for solutions to parabolic inverse problems of reconstruction of convolution kernels." Differential Integral Equations 20 (9) 961 - 990, 2007.