Global Solutions for a Nonlocal Problem with Logarithmic Source Term - Eugenio Lapa (Sahand Communications in Mathematical Analysis)
The current paper discusses the global existence and asymptotic behavior of solutions of the following new nonlocal problem$$ u_{tt}- M\left(\displaystyle \int_{\Omega}|\nabla u|^{2}\, dx\right)\triangle u + \delta u_{t}= |u|^{\rho-2}u \log|u|, \quad \text{in}\ \Omega \times ]0,\infty[,  $$where\begin{equation*}M(s)=\left\{\begin{array}{ll}{a-bs,}&{\text{for}\ s \in [0,\frac{a}{b}[,}\\{0,}&{\text{for}\ s \in [\frac{a}{b}, +\infty[.}\end{array}\right.\end{equation*}If the initial data are appropriately small, we derive existence of global strong solutions and the  exponential decay of the energy.