The integral manifold of a nonlinear elliptic equation in a cylinder (Q1274128)

In the cylinder \(\Omega=\mathbb{R}\times\omega\), where \(\omega= [\alpha_1, \alpha_2] \Subset\mathbb{R}\), we consider the elliptic equation \[ \partial^2_t u+2b\partial_tu- Au-F(t,u)= g(t,x),\quad u=0\text{ on }\partial w.\tag{1} \] Here \(A=-\partial^2_x\), \(b>0\) is some constant, \(u=u(t,x)= (u_1, \dots, u_m)\), \(F(t,u)\), and \(g(t,x)\) are vector functions, and \((t,x)\in \mathbb{R} \times \omega\). We prove the existence of an integral manifold for (1).