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

Abstract 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.