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 uxx Qpxqu P puq 0, where Qpxq is a π-periodic function. It is known that for a wide class of the nonlinearities P puq the “most part” of solutions of Cauchy problem for this equation are singular, i.e., they tend to infinity at some finite point of the real axis. Earlier in the case P puq u3 this fact allowed us to propose an approach for a complete description of solutions to this equation bounded on R. One of the ingredients in this approach is the studying of the set U L introduced as the set of the points pu , u1 q in the initial data plane, for which the solutions to the Cauchy problem up0q u , uxp0q u 1 are not singular in the segment r0;Ls. In the present work we prove a series of statements on the set 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.