一类反应扩散方程显示波前解的稳定性

The Stability of an Explicit Traveling Front for a Reaction-Diffusion Equation

【目的】对于反应扩散方程■u/■t=■^2u/■x^2+u^q+1(1-u^q),研究关于它的波前解的渐近指数稳定性。【方法】将方程在显示波前解处线性化,利用谱方法得到线性化算子在指数加权空间中的本质谱和除0以外的具有有限代数重数的孤立特征值有一致负上界,因此由经典解析半群理论可得显示波前解的指数稳定性。【结果】证明了该方程的显示波前解在指数加权空间中是带平移局部渐近指数稳定的。【结论】得到了此类反应扩散方程行波解的渐近指数稳定性。

[Purposes]For the reaction-diffusion equation ■u/■t=■^2u/■x^2+u^q+1(1-u^q),the asymptotic exponential stability of its traveling front is studied.[Methods]Linearizing the equation at the given traveling front,by using spectral method,the uniform negative bound of the essential spectrum and point spectrum with non-zero but finite algebraic multiplicities of the linear operator is proved in some exponentially weighted spaces,so the exponential stability of the traveling front can be obtained from the classical analytic semi-group theory.[Findings]The front is asymptotic exponential stable with shift in the exponentially weighted spaces.[Conclusions]The asymptotic exponential stability of the traveling wave for this reaction-diffusion equation is obtained.