Double-periodic boundary value problem for nonlinear dissipative hyperbolic equations (Q2638437)

The author considers the existence of the weak solution of the double- periodic problem for nonlinear dissipative hyperbolic equations of the form \[ \beta u_ t+u_{tt}-u_{xx}-g(t,x,u)=h(t,x)\text{ in } \Omega =[0,2\pi]\times [0,2\pi], \] with \(\beta\neq 0\), \(h\in L^ 2(\Omega)\), and g: \(\Omega\times {\mathbb{R}}\to {\mathbb{R}}\) is a Carathéodory function. In the last years several papers dealt with this problem for various types of the nonlinear term g. By use of an abstract continuation theorem of \textit{J. Mawhin} [Topological degree methods in nonlinear boundary value problems, Reg. Conf. Ser. Math. No.40, Am. Math. Soc. (1979; Zbl 0414.34025)] the author succeeds in weakening some conditions on g.