1. Introduction We are going to investigate the equation ut(t, x ... - ICMS

26Therefore, by Lemma 2.6, which is obviously valid for R in place of (0,T), foranym 1 , ..., m n ,wehave∫n∏||u(t, e mi ·)ζ|| p H γ p dt ≤ NRi=1∫Rn∑i=1e 2m ip || ¯f mi (t, e mi ·)|| p H γ−2p∏||u(t, e mj ·)ζ|| p H dt.p γj≠iComing back to (6.2), we conclude||M −1 u|| npH γ,npp,θ∫≤ NRF (t)||u(t)|| (n−1)pH γ p,θ−pdt,whereF (t) :=∞∑m=−∞e m(θ+p) || ¯f m (t, e m ·)|| p .Hpγ−2Next we use (see [5]) that the operator M β is a bounded operator from H γ p,θ to Hγ p,θ+βpand that M∇ is a bounded operator from H γ p,θto Hγ−1p,θ. Then we find+N∞∑m=−∞F (t) ≤ N∞∑m=−∞e mθ ||u x (t, e m ·)ζ ′ || p H γ−2p≤ N(||Mf(t)|| p H γ−2p,θe m(θ+p) ||f(t, e m ·)ζ|| p H γ−2p+ N∞∑m=−∞+ ||Mu x (t)|| p H γ−2p,θ−pe m(θ−p) ||u(t, e m ·)ζ ′′ || p H γ−2p+ ||M −1 u(t)|| p )H γ−2p,θ≤ N(||Mf(t)|| p H γ−2p,θ+ ||u(t)|| p )H γ−1p,θ−p