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

By Nguyen Vu Dzung, Tran Trinh Manh Dung
DOI: 10.37550/tdmu.EJS/2023.05.422

In this paper, we study the Robin-Dirchlet problem (Pn) for a wave equation with the
term
1
n
Xn
i=1
u2( i−1
n , t), n ∈ N. First, for each n ∈ N, under suitable conditions, we prove
the local existence and uniqueness of the weak solution un of (Pn). Next, we prove that
the sequence of solutions un of (Pn) converges strongly in appropriate spaces to the weak
solution u of the problem (P), where (P) is defined by (Pn) by replacing
1
n
Xn
i=1
u2( i−1
n , t)
by
Z 1
0
u2(y, t)dy. The main tools used here are the linearization method together with
the Faedo-Galerkin method and the weak compact method. We end the paper with a
remark related to a similar problem.