... ends the proof of Theorem 2.4.Acknowledgments The authors express their sincere gratitude to the anonymous referees for a number of valuable comments and suggestions. The work was supported by the ... t−p/p−2.4.15This is 2.9 and we complete the proof of Theorem 2.3.Journal of Inequalities and Applications 15 The applications of Lemma 4.1 and 4.19 yields thatutt22∇ut22utα1α1≤ ... Lp and W1,p0 for LpΩ and W1,p0Ω and relevant normsby ·p and ·1,p, respectively. In general, ·Xdenotes the norm of Banach space X.Wealso denote by ·, · and ·, · the...