Nonlinear Instability Of Steady-state Solution Of Reaction-Diffusion Equation

With the developing of science and technology, contralment of the biological and financial systems bring huge influence to the entire society's development. People always expect they can be controlled, so the researches about the stability of these systems especially receive more attentions. If the systems are stable, we expect that they have smaller error. And if the systems are instable, we hope to find the reasons, then we can change the parameter to control them easily.We know that many useful systems have corresponding mathematical models, for example physical, chemical, biological and financial systems can be described by partial differential equations. The equilibrium state of a system is a state independent with time which is one of the most important states, its stability directly reflect the stability of the whole system. An equilibrium state corresponds a steady-state solution of an equation, so the researches about the stability of the steady-state solutions have many practical significances.Chemotaxis is an important model in biomedicine. This article will discuss the nonlinear instability of the steady-state solution to this reaction-diffusion system. It is well known that the perturbation analysis is an effective method to proof the stability, so given a small perturbation we discuss the influence of the steady-state solution from this small perturbation. For this we linearize the equations, we can find a particular solution to the linearized equations and discuss the instability of the linearized equations firstly. Then we use the energy estimate to determine the influence of the nonlinear part of the equation to the whole system. Finally we use the spectral theory of linear operator and energy estimate to proof that linear instability always implies nonlinear instability. So we have the nonlinear instability of the steady-state solution.