Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions of 2N‐point type

This paper is devoted to the study of a nonlinear wave equation with initial conditions and nonlocal boundary conditions of 2N‐point type, which connect the values of an unknown function u(x,t) at x = 1, x = 0, x = η_i(t) , and x = θ_i(t), where $0<{\eta }_1(t)<{\eta }_2(t)<\dots <{\eta }_{{}_{N?1}}(t)<1,$ $0<{\theta }_1(t)<{\theta }_2(t)<\dots <{\theta }_{{}_{N?1}}(t)<1,$ for all t ≥ 0. First, we prove local existence of a unique weak solution by using density arguments and applying the Banach's contraction principle. Next, under the suitable conditions, we show that the problem considered has a unique global solution u(t) with energy decaying exponentially as t → +∞ . Finally, we present numerical results.