Dynamic transitions and bifurcations of 1D reaction-diffusion equations: The non-self-adjoint case

Abstract

The main goal of this paper is to classify the first transitions of a class of 1D second order reaction diffusion equation with a non-self-adjoint linear part and semilinear nonlinearity on a bounded interval subject to homogeneous boundary conditions. The emphasis is on the interaction of the nonlinear terms, the boundary conditions and the strength of the first order derivative term which makes the linear operator non-self-adjoint. The equations admit a trivial steady state solution which loses stability as a control parameter exceeds a threshold. According to our results, the first transitions of the system are either continuous, jump or mixed type. We compare our results with a recent study where only self adjoint linear operator was considered. In the Dirichlet, Neumann and periodic boundary condition cases, the first transition occurs via a transcritical/pitchfork bifurcation and the dynamics remain unchanged between the self-adjoint and non-self-adjoint cases. However, for the periodic case, with imposed zero mean condition, in the non-self adjoint case, the first transition is always accompanied by a Hopf bifurcation with a stable/unstable bifurcated limit cycle which is different than the self-adjoint case. Finally, we apply our results to determine the first transitions of some well-known reaction diffusion equations such as the Kolmogorov-Fisher, Chaffee-Infante equation and the Burger\"s equation.