Anti-periodic solution for \(u_{tt}-(\sigma (u_ x))_ x - u_{xxt} = f(x,t)\)
The authors establish the existence of time anti-periodic solutions for the quasilinear wave equation with viscosity \[ u_{tt}-(\sigma(u_x))_x-u_{xxt}=f(x,t),\quad (x,t)\in[0,\pi]\times\mathbb{R},\tag{E} \] \[ u(0,t)=u(\pi,t)=0,\quad t\in\mathbb{R}, \] where \(\sigma:\mathbb{R}\to\mathbb{R}\) is an odd smooth function satisfying \(\sigma'>0\) and \(|\sigma''/\sigma'|\leq L\) for some \(L>0\), and \(f\) is a given \(t\)-anti-periodic function. A typical example is \(\sigma(v)=v/(1+v^2){1\over 2}\), \(v\in\mathbb{R}\). This case is not covered by earlier periodicity (rather than anti-periodicity) results for (E), due to \textit{T. Narazaki} [Proc. Fac. Sci. Tokai Univ. 18, 87-101 (1983; Zbl 0555.35095)], and \textit{C. O. A. Sowunmi} [Rend. Ist. Mat. Univ. Trieste 8, 58-68 (1976; Zbl 0341.35004)]. Two theorems are proved. A remark on the existence of periodic solutions to the equation obtained from (E) by adding the term \(\lambda u\) \((\lambda>0)\) on the left-hand side is also included.
quasilinear wave equation with viscosity
scientific article; zbMATH DE number 902474