Semnan University International Journal of Nonlinear Analysis and Applications 2008-6822 14 10 2023 10 01 Global solutions for a nonlinear degenerate nonlocal problem 9 17 8092 10.22075/ijnaa.2023.26510.3338 EN Eugenio Cabanillas Lapa Instituto de Investigacion, FCM-UNMSM, Av. Venezuela S/N, Lima, Peru Journal Article 2022 03 22 In this paper, we consider the existence and asymptotic behavior of solutions to the following new nonlocal problem<br />$$ u_{tt}- M\Big(\displaystyle \int_{\Omega}|\nabla u|^{2}\, dx\Big)\triangle u + \delta u_{t}= |u|^{\rho-2}u\hspace{1.0cm} \text{in}\ \Omega \times ]0,\infty[,  $$<br />where<br />\begin{equation*}<br />M(s)=\begin{cases}<br />a-bs &\text{for } \ \, s \in [0,\frac{a}{b}[,\\<br />0, &\text{for }  s \in [\frac{a}{b}, +\infty[.<br />\end{cases}<br />\end{equation*}<br />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. https://ijnaa.semnan.ac.ir/article_8092_630eff1b380505a67570dff952ce4ad7.pdf