Global analytic solutions of the radial nonlinear wave equation
The initial-valued problem \[ f''(z)+\alpha z^{-1}f'(z)=G(z,f(z)),\quad f(0)=\lambda,\quad f'(0)=0, \] under suitable assumptions on the function G(z,f(z)) is reduced by a well-known method to an operator equation of the form \(f=N_ 1(f)+\lambda e_ 1\) in an abstract Banach space \(H_ 1\). The operator \(N_ 1\) is a nonlinear compact and holomorphic operator, defined in an open sphere centered at the origin of \(H_ 1\). The bar in the number \(\lambda\) means complex conjugation of \(\lambda\) and \(e_ 1\) is a known element in \(H_ 1\). The existence of a solution to the above operator equation means that there exists an analytic function \(f(z)=\lambda +\sum^{\infty}_{n=1}b_ n(\lambda)z^ n,\) \(f'(0)=0\), which together with the first two derivatives converges absolutely for \(| z| \leq 1\). It is well known that such solutions exist under suitable assumptions on G(z,f(z)), for sufficiently small \(| \lambda |\). Here the existence of such analytic solutions is proved under more general assumptions on G(z,f(z)). Moreover for a given sufficiently small \(| \lambda |\) the theorems predict an upper bound of the solution f(z) for all \(| z| \leq 1\). The most important is that the existence theorems have a constructive character and can provide an answer to the question: How small is the initial point \(| f(0)| =| \lambda | ?\) As examples, some equations of particular interest are studied.
scientific article; zbMATH DE number 4071497
scientific article