This paper analyzes the structure of the set of positive solutions of a class of one-dimensional superlinear indefinite bvp's. It is a paradigm of how mathematical analysis aids the numerical study of a problem, whereas simultaneously its numerical study confirms and illuminates the analysis. On the analytical side, we establish the fast decay of the positive solutions as $\lambda\downarrow -\infty$ in the region where $a(x)<0$ (see (1.1)), as well as the decay of the solutions of the parabolic counterpart of the model (see (1.2)) as $\lambda\downarrow-\infty$ on any subinterval of $[0,1]$ where $u_0=0$, provided $u_0$ is a subsolution of (1.1). This result provides us with a proof of a conjecture of [28] under an additional condition of a dynamical nature. On the numerical side, this paper ascertains the global structure of the set of positive solutions on some paradigmatic prototypes whose intricate behavior is far from predictable from existing analytical results.
翻译：本文分析了一组单维超线性无限期 bvp 的正面解决方案的结构。 它是一个数学分析如何帮助对问题进行数字研究的范例, 同时其数字研究确认并照亮了分析。 在分析方面, 我们将积极解决方案的快速衰减确定为 $\ lambda\ downrow -\ infty$( $) <0$) (参见 1.1) 的区域( $a( x) < 0$) (参见 1.1)), 以及模型的对立方方( 参见 ( 1.2) 的对应方( $( $) lambda\ downrow- infty) 的解决方案衰减为 $ $ $ $[0, 1美元=0美元, 提供 $_ 0$( $) 是 1. 1 的子溶解法。 这个结果为我们提供了在动态性质附加条件下[ 28] 的预测力的证明。 在数字方面, 本文确定对于某些复杂行为与现有分析结果完全无法预测的范式原型的全套积极解决方案的全球结构。
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