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$$ u_{tt}- mbig(displaystyle int_{omega}|nabla u|^{2}, dxbig)triangle u + delta u_{t}= |u|^{rho-2}uhspace{1.0cm} text{in} omega times ]0,infty[,  $$wherebegin{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.