Global solutions for a nonlinear degenerate nonlocal problem

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|^{۲}\, dx\Big)\triangle u + \delta u_{t}= |u|^{\rho-۲}u\hspace{۱.۰cm} \text{in}\ \Omega \times ]۰,\infty[,  where\begin{equation*}M(s)=\begin{cases}a-bs &\text{for } \ \, s \in [۰,\frac{a}{b}[,\\۰, &\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.

کلیدواژه ها:

global solutions ، degenerate nonlocal problem ، asymptotic behavior