A two-parameter shooting problem for a second-order differential equation
The author considers the nonlinear equation \(u''+ p| u|^{p-1} u'+ {t\over 2} u'+ {1\over 2(p-1)} u=0\), \(t\in \mathbb{R}\), under the conditions \(\lim_{t\to+\infty} u(t)=0\), \(\lim_{t\to -\infty} (- t)^{-1/(p-1)} u(t)= p^{-1/(p-1)}\), \(u\geq 0\), and show that when \(1< p<2\) its global solutions constitute a one-parameter ordered family \((u^ \lambda)\) with \(\lambda\geq 0\). They show also that the global solution which behaves like \(\exp\{-t^ 2/4\}\) is unique and \(u^ 0\). To arrive at the result they are based on 23 propositions, lemmas and corollaries. Notice that this particular equation and boundary conditions play an important role in the study of the long-time behavior of a nonlinear diffusion phenomenon.
scientific article; zbMATH DE number 665708