global solutions for a nonlocal problem with logarithmic source term

The current paper discusses the global existence and asymptotic behavior of solutions of the following new nonlocal problem utt −m (∫ ω |∇u|2 dx ) △u + δut = |u|ρ−2u log |u|, in ω×]0,∞[,where m(s) = { a − bs, for s ∈ [0, a b [, 0, for s ∈ [ a b ,+∞[.if the initial data are appropriately small, we derive existence of global strong solutions and the exponential decay of the energy.