具时滞的单峰双稳态反应扩散方程单调波的整体延拓
Global continuation of monotone waves for a unimodal bistable reaction-diffusion equation with delay
摘要：
研究了一类具有时滞反应项$g$的双稳态反应扩散方程的单调波前的存在性.从以前的作品，我们不假设单调的$g(u,v)$相对于延迟变量$v$，不允许应用比较技术。因此，我们的证明是基于一个变种的Hale-Lin功能分析方法异宿解的泛函微分方程的Lyapunov-Schmidt减少是在适当的加权空间的$C^2$ -光滑函数。这种方法需要详细分析相关的线性微分弗雷德霍姆算子及其形式伴随。对于两种不同类型的$v-$单峰函数$g(u,v)$，我们证明了极大连续的双稳态单调波前族的存在性.取决于单峰的类型（等效地，取决于波速的符号），对于双稳态波可以观察到两种不同的情形：1）独立于延迟的大小，每个双稳态波前是单调的; 2）波前对于中等延迟值是单调的，并且可以对于大延迟振荡。
英文摘要：
We study the existence of monotone wavefronts for a general family of bistable reaction-diffusion equations with delayed reaction term $g$. Differently from previous works, we do not assume the monotonicity of $g(u,v)$ with respect to the delayed variable $v$ that does not allow to apply the comparison techniques. Thus our proof is based on a variant of the Hale-Lin functional-analytic approach to heteroclinic solutions of functional differential equations where Lyapunov-Schmidt reduction is done in appropriate weighted spaces of $C^2$-smooth functions. This method requires a detailed analysis of associated linear differential Fredholm operators and their formal adjoints. For two different types of $v-$unimodal functions $g(u,v)$, we prove the existence of a maximal continuous family of bistable monotone wavefronts.. Depending on the type of unimodality (equivalently, on the sign of the wave speed), two different scenarios can be observed for the bistable waves: 1) independently on the size of delay, each bistable wavefront is monotone; 2) wavefronts are monotone for moderate values of delays and can oscillate for large delays.
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