Nonlinear singularly perturbed problem for time-delay evolution equation with two parameters is studied. Using the variables of the multiple scales method, homogeneous equilibrium method, and approximation expansion method from the singularly perturbed theories, the structure of the solution to the time-delay problem with two small parameters is discussed. Under suitable conditions, first, the outer solution to the time-delay initial boundary value problem is given. Second, the multiple scales variables are introduced to obtain the shock wave solution and boundary layer corrective terms for the solution. Then, the stretched variable is applied to get the initial layer correction terms. Finally, using the singularly perturbed theory and the fixed point theorem from functional analysis, the uniform validity of asymptotic expansion solution to the problem is proved. In addition, the proposed method possesses the advantages of being very convenient to use.

中文翻译：

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一类奇摄动非线性时滞发展方程的激波解

研究了带有两个参数的时滞演化方程的非线性奇摄动问题。利用奇异摄动理论中的多尺度方法，齐次均衡方法和逼近展开法的变量，讨论了具有两个小参数的时滞问题解的结构。在适当的条件下，首先，给出了时滞初始边界值问题的外部解。其次，引入多个尺度变量以获得冲击波解和该解的边界层校正项。然后，应用拉伸变量以获得初始层校正项。最后，利用奇摄动理论和泛函分析的不动点定理，证明了该问题渐近展开解的一致有效性。另外，所提出的方法具有使用非常方便的优点。