The Stability Of Travelling Waves For Generalized Fisher Equation(Systems) And Viscous Balance Law

This thesis is composed of three parts.In the first part, the asymptotic stability of travelling front solutions for a class of generalized Fisher equation is studied. By detailed spectral analysis, semigroup theory, and comparison principle, we first prove that each wave with non-critical and critical speed is locally and globally exponentially stable in some weighted spaces, respectively. Further by Evans function method and detailed semigroup estimates, we prove that each wave with non-critical or critical speed is locally algebraically stable in some weighted spaces, which further explains some asymptotic phenomena obtained by numerical simulation. It's remarked that, when analyzing the asymptotic behavior of solutions of the eigenvalue problem for the linearized operator around the wave with non-critical speed, the standard asymptotic theory of ODEs can't be applied directly, which is due to the slower algebraic decay of the wave at the end x = +∞. However, by using more general asymptotic theory of ODEs, we obtain the detailed description of the behavior of solutions at +∞ to the eigenvalue problem and hence give the definition of the Evans function. Furthermore, we show that the Evans function has the same properties as its previous definitions, such as the analyticity in A, and the correspondence to the eigenvalue of the linearized operator. It is in this sense that we extend the definition of the Evans function to more general case. In addition, in the proof of the algebraic stability of waves with critical speeds, somekey properties of the Evans function D(λ) are obtained, in particular, D(0) = 0, but D_λ(0) ≠ 0, which is useful for semigroup estimates in the polynomially weighted spaces.In the second part, we investigate the existence and asymptotic stability of travelling wave solutions for a class of scalar viscous balance law. First, by using detailed spectral analysis, comparison principle, and the properties of the ω—limit set, we show that each wave connecting two adjacent saddles for general viscous balance law is globally exponentially stable in L~∞ spaces. Also, by detailed phase plane analysis, we obtain the existence of travelling fronts for a class of degenerate viscous balance law, then based on which, by similar methods to the first part, we obtain the locally exponential and algebraical stability of each wave front.In the third part, we study the stability of travelling waves for some autocatalytic chemical reaction systems. By detailed spectral analysis and semigroup theory, we prove that for the diffusion rate d = 1 case, each wave with critical or non-critical speed is locally exponentially stable in some weighted spaces. In addition, for the diffusion rate d ≠ 1 case, we obtain some elementary spectral properties of the linearized operator around the wave.