Periodic solutions for a class of nonautonomous wave equations
The authors investigate the existence of weak nontrivial, time \(2\pi\)-periodic solutions to the wave equation \(u_{tt}-u_{xx}=f(t,x,u)\), \(0<x<\pi\), with the Dirichlet boundary conditions and the forcing term of the form \(f(t,x,u)=bu+g(t,x,u)\). Here, \(b\) is a constant and \(g\) is a \(2\pi\)-periodic function in time, continuous on \(\mathbb{R}\times[0,\pi]\times\mathbb{R}\). The forcing term is supposed to satisfy: (\(f_0\)) There exist \(s>2\), \(a_1,a_2>0\) such that \(|f(t,x,\xi)|\leq a_1+a_2|\xi|^{s-1}\) for all \(t,x,\xi \); (\(f_1\)) There exists \(\varepsilon >0\) such that \((f(t,x,\xi)-f(t,x,\eta))(\xi-\eta)\geq\varepsilon|\xi-\eta|^s\) for all \(t,x,\xi,\eta\); (\(f_2\)) \(g(t,x,\xi)=o(|\xi|)\) uniformly in \(t,x\) as \(\xi\to 0\); (\(f_3\)) There exists \(\overline r>0\) such that the function \(F(t,x,\xi)=\int_0^\xi f(t,x,s)ds\) satisfies \(0<sF(t,x,\xi)\leq\xi f(t,x,\xi)\) for all \(t,x,\xi\) with \(|\xi|\geq\overline r\). When \(\sigma(\square)\) denotes the spectrum of the corresponding wave operator and \(G(t,x,\xi)=\int_0^\xi g(t,x,s)ds\), then the main result of the paper can be stated as follows. Under the above conditions there is at least one nontrivial solution to the problem in question in each of the following cases: (1) \(b\not\in\sigma(\square)\) or \(b=0\); (2) \(b\in\sigma(\square)\) and there exists a \(\delta>0\) such that \(G(t,x,\xi)\geq 0\) for all \(|\xi|\leq\delta\), \(t\in\mathbb{R}\) and \(x\in(0,\pi)\); (3) \(b\in\sigma(\square)\) and there exists a \(\delta>0\) such that \(G(t,x,\xi)\leq 0\) for all \(|\xi|\leq\delta\), \(t\in\mathbb{R}\) and \(x\in(0,\pi)\). The solution is obtained as a critical point of a suitable functional which admits the application of local linking. There are two corollaries of the main result. Firstly, it is proved that the above results (1)--(3) remain valid even when hypothesis \((f_0)\) is omitted. Secondly, if \(b=0\), then condition \((f_1)\) may be replaced by the assumption that the mapping \(\xi \to f(t,x,\xi)\) is nondecreasing for all \(t,x,\xi\).
scientific article; zbMATH DE number 1095807