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 functionu(x,t) atx = 1,x = 0, x = eta(i)(t), andx = theta(i)(t),where0= 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 solutionu(t) with energy decaying exponentially ast -> +infinity. Finally, we present numerical results.

Automated summary

This paper focuses on a nonlinear wave equation with initial conditions and nonlocal boundary conditions, proving the existence of a unique weak solution and a unique global solution under certain conditions, with the solution exhibiting exponential energy decay as t approaches positive infinity.