AN EFFECT OF DECAY RATES: SUPERCRITICAL DAMPING AND A VISCOELASTIC TERM

. We consider the following viscoelastic equation involving variable exponent nonlinearities: Z t utt - & UDelta;u + g(t - s)& UDelta;u(s)ds + a|ut |m(x)-2ut = |u|q(x)-2u. 0 Due to the failure of the embedding inequality for the supercritical case, the well-known technique is unsuccessful in our problem. To do this, our strategy is to give a priori estimate for the weighted integral f & omega; (2 + t)1-m(x)|u|m(x)dx and then apply modified multiplier method to prove that the energy functional decays logarithmically to zero when the relaxation function g decays exponentially to zero. Meanwhile, for more general cases, we also give the explicit dependence of decay rate on both the exponent m(x) and the relaxation function g. This differs from some results of [21] where the energy functional decays exponentially to zero when the relaxation function g decays exponentially to zero.