Attractors of resonance wave type equations: Discontinuous oscillations
In a previous paper [Russ. Acad. Sci., Sb. Math. 78, No.2, 367-378 (1994; Zbl 0817.35038); translation from Mat. Sb. 184, No. 3, 121-136 (1993)], the author defined the concepts of orbital exponential stability and of quasinormal form of a boundary problem. In this paper, he investigates the boundary value problem: \[ {\partial^2 u\over \partial t^2}- \varepsilon {\partial\over \partial t} \Biggl(u+ a^2 {\partial^2 u\over \partial x^2}\Biggr)+ u^2 {\partial u\over \partial t}= {\partial^2 u\over \partial x^2};\;u|_{x= 0}= u|_{x= 1}= 0\tag{1} \] with \(0< a\pi< 1\). He states that, for \(\varepsilon\) small enough, problem (1) has an orbitally exponential stable cycle \(U\) whose asymptotic representation is of the form \[ U= \varepsilon^{1/2}[v(t- x)- v(t+ x)]+ O(\varepsilon^{3/2}), \] where the 2-periodic function \(v\) is the equilibrium state of the quasinormal form of problem (1). The result is proved by modifying the constructions of loc. cit.
scientific article; zbMATH DE number 830687