Periodic solutions of nonlinear hyperbolic problems
This paper sketches a rather brief survey of methods employed in establishing existence of periodic solutions of nonlinear partial differential equations of the form \(u_{tt}+Au+f(t,x,u,u_ t)=p(t,x)\) where A is a linear operator with domain and range in \(L_ 2(0,\pi)\), the unknown function \(u:[0,\infty)\times [0,\pi]\to {\mathbb{R}},\) f is the prescribed nonlinearity and p(t,x) is periodic in t. Homogeneous boundary conditions are prescribed for A. Assuming solutions as finite combinations of eigenfunctions of A with functions of t as coefficients, there results a finite system of ordinary differential equations for these coefficients as in the classical elementary method of separation of variables. One method then is to obtain estimates for these approximate solutions and their derivatives that will help passage to the limit along a strongly convergent subsequence of these solutions to obtain a weak solution of the original problem. Another method is to constrct a Liapunov function for each of the ordinary differential equations with the aid of which it could be shown that the ordinary differential system is dissipative thereby establishing the existence of periodic solutions. A different approach to the treatment of the partial differential equation problem is to rewrite it as an operator equation \(Eu=Nu\) where E is a linear operator and N is a nonlinear operator from one Banach space to another. With appropriate decomposition properties for these spaces, the available theory of nonlinear operator equations in abstract spaces could be invoked to establish existence of a weak solution of the original problem. The problem of establishing existence of periodic solutions of the ordinary differential systems is described in some detail but the paper contains no theorems or proofs specifically indicated as such. Some of the details not provided here are promised to appear elsewhere and it is remarked that the methods indicated could be modified to deal with nonlinear A as well.
finite system of ordinary differential equations
scientific article; zbMATH DE number 3875864
scientific article