具对称性的非线性非自治波动方程的周期解

Large Amplitude time Periodic Solutions of a Semilinear Wave Equation

本文讨论了偏微分方程周期解问题u_(it)-u_(xx)+g(t,x,ξ)=0(t,x)∈Q=(0,2π)×(0,π)(0.1)u(t,0)=0=u(t,π)t∈[0,2π](0.2)u(t,x)=u(t+2π,x)(t,x)∈其中函数g∈C(×R^1,R^1),且满足条件(g_1)g(t,x;ξ)关于变元ξ是严格单增的(g_2)存在数μ>2,r>0.使当|ξ|>r时有0<μG(t,x;ξ)≡μg(t,x;s)ds≤ξg(t,x;ξ)利用Z_2-指标理论和极大极小论证方法,当函数g(t,x;ξ)关于变元为奇函数时得到了问题的无穷多解存在性定理。

This paper is concerned with the existence and multiplicity results for a nonlinear wave equation of the type u_(tt)-u_(xx)+g(t,x;u)=0(t,x)∈Q=(0,2π)×(0,π) together with the boundary and periodicity conditions u(t_s 0)=0=u(t,π)t∈[0,2π] u(t,x)=u(t+2π,X)(t,x)∈Q where 9∈C(Q×R^1,R^1) Moreover we assume that g satisfies the following conditions (g1)g(t,x;ξ)is strictly monotone increasing in ξ; (g2)there exists μ>2 and r>0,such that for|ξ|≥r Provided that g is odd in ξ,using minimax arguments and Z_2—index theory,we have proved that the proceding eqution possesses an imbounded sequence of weak solutions. Next we replace 2π—period by T—period which is rational multiple of 2π. Under the same assumption we obtain the same conclusion.