Asymptotic behaviors of solutions to a reaction-diffusion equation with isochronous nonlinearity

We study the initial boundary value problem for the reaction-diffusion equation with isochronous nonlinearity. We prove that small solutions become spatially homogeneous and is subject to the ODE part asymptotically. We also discuss blow-up of an parabolic system with quadratic nonlinearity having the origin as an uniform isochronous center.