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

Abstract

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 = i (t), and x = i (t), where 0 < 1 (t) < 2 (t) < … < N−1 (t) < 1, 0 < 1 (t) < 2 (t) < … < 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.

Corrigendum to “Existence, blow‐up, and exponential decay estimates for a system of nonlinear wave equations with nonlinear boundary conditions”

“…By giving corrections to the energy functionals E(t) and I(t) as above, with adding the functional (g * u)(t), we also have suitable corrections to our papers. [2][3][4][5][6][7][8][9][10][11][12][13][14][15] Note more that, in case of the problem considered containing the term ∫ t 0 g(t − s)Δu(x, s)ds, we onlygive corrections to the functional I(t) with adding the term ∫ t 0 g(t − s) ‖∇u(t) − ∇u(s)‖ 2 ds in the definition of I(t), where, for example, u ∈ C 0 ( R + ; H 1 0…”