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

Abstract

This note presents corrections to our paper.

High-Order Iterative Scheme for a Viscoelastic Wave Equation and Numerical Results

“…If a � 0, it is well known that the source term b|u| p− 2 u causes a finite-time blow-up phenomenon of the solution with negative initial energy; see [20][21][22][23]. If b � 0, the damped term a|u t | q− 2 u t assures the global existence for suitably initial data; see [24][25][26][27][28][29]. e interactions between the damping and the source terms give the results that the solutions with any initial data continue to exist globally in time if q ≥ p and blow up in finite time if q < p and the initial energy is sufficiently negative.…”

“…. ., N. Normally, this system will be converted into a system of 2N first-order integrodifferential equations by using the auxiliary functions _ v (m) i (t) � u (m) i (t); see the same transformations in [14,17,23,26,29,31]; however, these transformations will increase computations. To reduce the computations, the above second-order integrodifferential system will be transformed into a system of N first-order integrodifferential equations by integrating in time on interval (0, t).…”

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

“…In this section, we present corrections to Proof of Theorem 4.1 related to Lemma 4.3 in Triet et al 1 (pp. 685), by the fact that Lemma 4.3 is wrong with the functional I ( t ) defined by (74) of Triet et al 1 …”

On the Robin-Dirchlet problem for a nonlinear wave equation with the term 1 n Xn i=1 u2(i−1 n , t)

“…On the other hand, the idea for considering Prob. (1.1) also comes from the study of problems containing the partitions of domain, for example, see [5], [11], [17] and the references therein. In [5], Il'in established the uniqueness theorem for generalized solution to the mixed problem for a wave equation with nonlocal boundary conditions (also called multi-point type conditions)…”