Abstract :
In this paper, we consider the existence and asymptotic behavior of solutions to the following new nonlocal problem$$ u_{tt}- M\Big(\displaystyle \int_{\Omega}|\nabla u|^{2}\, dx\Big)\triangle u + \delta u_{t}= |u|^{\rho-2}u\hspace{1.0cm} \text{in}\ \Omega \times ]0,\infty[, $$where\begin{equation*}M(s)=\begin{cases}a-bs \text{for } \ \, s \in [0,\frac{a}{b}[,\\0, \text{for } s \in [\frac{a}{b}, +\infty[.\end{cases}\end{equation*}We first state a local existence theorem. Next, if the initial energy is appropriately small, by using Tartar’s method and the decay rate of the energy, we derive the global existence theorem. As a biproduct, we also obtain the exponential decay property of the global solution.
