International Journal of Nonlinear Analysis and Applications

Global solutions for a nonlinear degenerate nonlocal problem

Document Type : Research Paper

Author

Instituto de Investigacion, FCM-UNMSM, Av. Venezuela S/N, Lima, Peru

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.

Keywords

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International Journal of Nonlinear Analysis and Applications
Volume 14, Issue 10
October 2023
Pages 9-17
  • Receive Date: 22 March 2022
  • Revise Date: 12 August 2023
  • Accept Date: 15 August 2023