id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-629	Liu, Qiang; Zhu, Wanyu; Ye, Hailong	Mild solutions to fourth-order parabolic equations modeling thin film growth with time fractional derivative	2024	14	.pdf	application/pdf	5339	270	77	However, to the best of our knowledge, the well-posedness for the solutions of the time fractional thin film growth equation is not clear, which is the main motivation of the present work. Ct αγ 4β2 − α 2β R(t)1+β j , and similarly ∥∇2uj+1∥ βN 2−β ≤ ∥∇2Eα(−tαA)φ∥ βN 2−β + ∫ t 0 (t− s)α−1∥∇2Eα,α(−(t− s)αA)∇ · f(∇uj)∥ βN 2−β ds ≤ ∥∇2u0∥ βN 2−β + C ∫ t 0 (t− s) α 2 −1−αγ 4β ∥∇ · f(∇uj)∥ βN 2−β+γ ds ≤ ∥∇2u0∥ βN 2−β + CR(t)1+β j ∫ t 0 (t− s) α 2 −1−αγ 4β s−α+αγ 4β ds (3.6) ≤ ∥∇2u0∥ βN 2−β + Ct− α 2 R(t)1+β j . Combining (3.5) and (3.6), for any fixed T > 0, we have R(T )j+1 ≤ R(T )0 + CR(T )1+β j , where C > 0 is independent of T .	cache/ejde-629.pdf	txt/ejde-629.txt