In this paper, we consider the Robin-Dirichlet problem for a system of nonlinear pseudoparabolic equations with viscoelastic term. By the Faedo-Galerkin method, we first prove the existence and uniqueness of solution for the problem. Next, we give a sufficient condition to get the global existence and decay of the weak solution. Finally, by the concavity method, we prove the blow-up result of the solution when the initial energy is negative. Furthermore, we establish here the lifespan of the solution by finding the upper bound and the lower bound for the blow-up time.

In this paper, the Neumann-Dirichlet boundary problem for a system of nonlinear viscoelastic equations of Kirchhoff type with Balakrishnan-Taylor term is considered. At first, a local existence is established by the linear approximation together with the Faedo- Galerkin method. Then, by establishing several reasonable conditions and suitable energy inequalities, the solution of the problem admits a general decay in time.

This paper is devoted to the study of a Kirchhoff wave equation with a viscoelastic term in an annular associated with homogeneous Dirichlet conditions. At first, by applying the Faedo-Galerkin, we prove existence and uniqueness of the solution of the problem considered. Next, by constructing Lyapunov functional, we establish a sufficient condition such that any global weak solution is general decay as t → +∞.

In this paper, we consider the Dirichlet problem for a wave equation of Kirchhoff- Carrier type with a nonlinear viscoelastic term. It consists of two main parts. In Part 1, we establish existence and uniqueness of a weak solution by applying the Faedo- Galerkin method and the standard arguments of density corresponding to the regularity of initial conditions. In Part 2, we give a sufficient condition for the global existence and exponential decay of the weak solutions by defining a modified energy functional together with the technique of Lyapunov functional.

The report deals with the Robin problem for a nonlinear wave equation with viscoelastic term. Under some suitable conditions, we establish a high-order iterative scheme and then prove that the scheme converges to the weak solution of the original problem along with the error estimate. This result extends the result in [9].

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.

In this paper, we consider the Dirichlet boundary problem for a nonlinear wave equation of Kirchhoff-Carrier-Love type as follow   utt − B ? ∥u(t)∥2 , ∥ux(t)∥2
(uxx + uxxtt) = f(x, t, u, ux, ut, uxt) + Xp i=1 εifi(x, t, u, ux, ut, uxt) for 0 < x < 1, 0 < t < T, u(0, t) = u(1, t) = 0, u(x, 0) = ˜u0(x), ut(x, 0) = ˜u1(x), (1) where ˜u0, ˜u1, B, f, fi (i = 1, · · · , p) are given functions, ε1, · · · , εp are small parameters and ∥u(t)∥2 = Z 1 0 u2 (x, t) dx, ∥ux(t)∥2 = Z 1 0 u2 x (x, t) dx. First, a declaration of the existence and uniqueness of solutions provided by the linearly approximate technique and the Faedo-Galerkin method is presented. Then, by using Taylor’s expansion for the functions B, f, fi, i = 1, · · · , p, up to (N + 1)th order, we establish a high-order asymptotic expansion of solutions in the small parameters ε1, · · · , εp.

In this paper, a high-order iterative scheme is established in order to get a convergent sequence at a rate of order N, (N ≥ 2) to a local unique weak solution of a nonlinear Kirchhoff-type wave equation associated with Robin conditions.