In this work, the behaviour of solutions for the Dirichlet problem of the non-local equation ut = ∆(κ(u)) + λf(u) ( Ω f(u) dx)p , Ω ⊂ RN , N = 1, 2, is studied, mainly for the case where f(s)=eκ(s). More precisely, the interplay of exponent p of the non-local term and spatial dimension N is investigated with regard to the existence and non-existence of solutions of the associated steady-state problem as well as the global existence and finite-time blow-up of the time-dependent solutions u(x, t). The asymptotic stability of the steady-state solutions is also studied.