Small time periodic solutions of fully nonlinear telegraph equations in more spatial dimensions
This paper is devoted to the study of the existence of a classical solution to the problem \((1)\quad Au+\partial_ tu+\kappa \partial^ 2_ tu+\sum^{n}_{r=1}\kappa_ r\partial_{x_ r}\partial_ tu+G(Ju)+f=0\) in \(\Omega\) \(\times {\mathbb{R}}\), (2) \(u(x,t)=0\) for \(x\in \partial \Omega\), \(t\in {\mathbb{R}}\), \((3)\quad u(x,t)=u(x,t+2\pi)\) for \(x\in \Omega\), \(t\in {\mathbb{R}}\), where \(\kappa\) and \(\kappa_ r\) are constants, A is an elliptic operator, J is the jet of all derivatives up to order 2 and f is a smooth and small function. The solution u to (1)-(3) is written in the form \(u(x,t)=\int^{t}_{0}v(x,\tau)d\tau +b(x),\) where, obviously, \(v=\partial_ tu\) and, by virtue of \(2\pi\)-periodicity of u in t, v satisfies \(\int^{2\pi}_{0}v(x,t)dt=0\). This device makes avoiding the occurence of the ``loss of derivatives'' possible. Hence, by using Schauder's theorem, the author is able to extend earlier results of Rabinowitz to more spatial variables and to suppress the explicit appearance of the small parameter, with, in addition, an improvement of the assumptions on the regularity of f and G.
scientific article; zbMATH DE number 4113055