Abstract

We describe contrast structures (CSs) arising from the simulation of reaction–diffusion (RD) processes in inhomogeneous media with the power dependence of the source density on the concentration in the vicinity of the roots. The results obtained earlier for homogeneous media are generalized to the case of inhomogeneous media. Sufficient conditions for the existence of a solution of the type of CS are strictly substantiated. Unlike previously known results, the power of the root function of the right-hand side is assumed to be noninteger (irrational powers are admitted as well). It is shown that the front (with respect to the direction of the movement) part of the front is an exponential function and the rear part of the front is a power function, which is a fundamentally new (previously unknown) result. The family of exact solutions of the evolution equation is found. The formal asymptotics of the solution of the initial-boundary value problem for the RD equation is constructed. The correctness of the partial sum of an asymptotic series is justified by the method of differential inequalities.

中文翻译：

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非均质介质中具有非整数倍根的反应扩散问题的非平稳对比度结构

摘要

我们描述了对比结构（CSs），该结构是在不均匀介质中模拟反应扩散（RD）过程而产生的，其源密度与根部附近浓度的功率相关。较早获得的均质介质结果可推广到非均质介质的情况。严格证明存在CS类型解决方案的充分条件。与先前已知的结果不同，右侧根函数的幂被假定为非整数（也可以接受非理性的幂）。结果表明，前部的前部（相对于运动方向）是指数函数，前部的后部是幂函数，这从根本上是新的（以前未知）结果。找到了演化方程的精确解族。构造了RD方程初边值问题解的形式渐近线。通过微分不等式的方法证明了渐近级数的部分和的正确性。