The Cauchy problem and asymptotic decay for solutions of differential inequalities in Hilbert space
Gerald N. Hile, Murray H. Protter · Pacific Journal of Mathematics · 1982
From (2.4) we find + Ύp'V'"' Γ 1 and so \ ω(sy*~βdtds ^ γ\ \ 2/3r-^1ft)(s)e 2r " ί dtds 0 Js JO Js Hence " u{t) " 2 + &2Ti+1 " M( -t)uit) H 2 k 2 Thus for β sufficiently large, k 2 a sufficiently small constant, and C 1 = 2C X we find 2 ^τ-^21| u(t) fdt ^ cX° e^" β || L(ζu) \\ 2 dt .