On solutions of Cauchy problem for equation $u_{xx}+Q(x)u-P(u)=0$ without singularities in a given interval

Abstract

The paper is devoted to Cauchy problem for equation $u_{xx}+Q(x)u-P(u)=0$, where $Q(x)$ is a $\pi$-periodic function. It is known that for a wide class of the nonlinearities $P(u)$ the “most part” of solutions of Cauchy problem for this equation are singular, i.e., they tend to infinity at some finite point of real axis. Earlier in the case $P(u)=u^3$ this fact allowed us to propose an approach for a complete description of solutions to this equations bounded on the entire line. One of the ingredients in this approach is the studying of the set $\mathcal U^+_L$ introduced as the set of the points $(u_*,u_*')$ in the initial data plane, for which the solutions to the Cauchy problem $u(0)=u_*$, $u_x(0)=u_*'$ is not singular in the segment $[0;L]$. In the present work we prove a series of statements on the set $\mathcal U^+_L$ and on their base, we classify all possible type of the geometry of such sets. The presented results of the numerical calculations are in a good agreement with theoretical statements.