By Yuming Qin

This publication provides a few analytic inequalities and their purposes in partial differential equations. those contain quintessential inequalities, differential inequalities and distinction inequalities, which play a very important function in constructing (uniform) bounds, international lifestyles, large-time habit, decay premiums and blow-up of recommendations to numerous periods of evolutionary differential equations. Summarizing effects from an unlimited variety of literature assets similar to released papers, preprints and books, it categorizes inequalities by way of their diversified properties.

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**Example text**

Proof. 149) 0 t t1 1/2 + · · · + et ηm 0 tm−1 ··· 0 1/2 2 Fm (s)e−2s ω 2 (u(s))ds · · · dt1 , 0 where ηi (i = 1, 2, . . 148). Here we have used the following estimate t 0 t1 ti−1 ··· 0 (ti−1 − s)2βi −1 e2s ds · · · dt1 0 t t1 = 0 0 2t ti−2 ··· ti−1 e2ti−1 0 σ 2βi −1 e−2σ dσ · · · dt1 0 t t1 ti−2 e ≤ 2βi Γ(2βi − 1) ··· e2ti−2 dti−1 · · · dt1 2 0 0 0 e2t Γ(2βi − 1) ≤ , i = 1, 2, . . , m. 151) 0 t t1 + · · · + e2t ηm 0 0 tm−1 ··· 0 2 Fm (s)e−2s ω 2 (u(s))ds · · · dt1 50 Chapter 1. 152) 0 t t1 + · · · + ηm 0 where tm−1 ··· 0 Fm (s)R(s)ω(u(s))ds · · · dt1 , 0 v(t) = (e−t u(t))2 .

159) Proof. 156), we obtain t 0 H(t) H (s) ds = + ω(V (s)) ω(V (t)) t H(s) 0 ω (V (s)) H(t) . V (s)ds ≥ [ω(V (s))]2 ω(V (t)) Now let us continue the proof of the theorem. 156), we have t t Vm−1 (s) Vm−1 (t) ≤ ds ≤ hm (s)ds. 160), it follows Vm−2 (t) ≤ ω(V (t)) t 0 Vm−2 (s) ds ≤ ω(V (t)) t ≤ t t hm−1 (s)ds + 0 t 0 t1 hm−1 (s)ds + 0 hm (s)dsdt1 . 5. 162) 0 t t1 +··· + 0 tm−1 ··· 0 hm (s)dsdtm−1 · · · dt1 . 14. 159). 6. 6. We omit the details 1 for a real number z ≥ 1 is more complicated and we also here.

160), it follows Vm−2 (t) ≤ ω(V (t)) t 0 Vm−2 (s) ds ≤ ω(V (t)) t ≤ t t hm−1 (s)ds + 0 t 0 t1 hm−1 (s)ds + 0 hm (s)dsdt1 . 5. 162) 0 t t1 +··· + 0 tm−1 ··· 0 hm (s)dsdtm−1 · · · dt1 . 14. 159). 6. 6. We omit the details 1 for a real number z ≥ 1 is more complicated and we also here. The case βi > z+1 omit it here. 5 Integral inequalities leading to upper bounds and decay rates In this section, we shall introduce some integral inequalities leading to upper bounds and decay rates. Bae and Jin [57] proved the following theorem.