Analytic Inequalities and Their Applications in PDEs by Yuming Qin PDF

By Yuming Qin

ISBN-10: 3073123253

ISBN-13: 9783073123258

ISBN-10: 3319008307

ISBN-13: 9783319008301

ISBN-10: 3319008315

ISBN-13: 9783319008318

This publication offers a couple of analytic inequalities and their functions in partial differential equations. those comprise indispensable inequalities, differential inequalities and distinction inequalities, which play a vital position in developing (uniform) bounds, international life, large-time habit, decay charges and blow-up of ideas to numerous periods of evolutionary differential equations. Summarizing effects from an enormous variety of literature assets comparable to released papers, preprints and books, it categorizes inequalities by way of their assorted properties.

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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.

0 Consequently, v(t) ≤ V (t) ≤ Ω−1 [Ω(φ(t)) + g2 (t)]. 60). 6, we have the following corollary. 4. The inequalities of Henry’s type 37 ˇ Inequality [606]). 83) 0 for a constant β > 0. Then the following assertions hold: (1) If β > 12 , then for all t ∈ [0, T ), u(t) ≤ (2) If β = 1 z+1 √ 2Γ(2β − 1) 2a(t) exp 4β t F 2 (s)ds + t . 61), q = z + 2. 83), where a(t), F (t), and u(t) are integrable on [0, T ). ˇ Inequality [606]). e. on [0, T ), t (t − s)β−1 F (s)u(s)ds. e. 87) where t Φ(t) = 2a2 (t) + 2Kb2(t) t a2 (s)F 2 (s) exp K s 0 K= b2 (r)F 2 (r)dr ds, Γ(2β − 1) .

606]) If H(t) is a C 1 -function on [0, T ), H(t) ≥ 0 for all t ∈ [0, T ), and H(0) = 0, then for all t ∈ [0, T ), t 0 H(t) H (s) ds ≥ . 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 .

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Analytic Inequalities and Their Applications in PDEs by Yuming Qin


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