By Yuming Qin

This e-book offers a few analytic inequalities and their purposes in partial differential equations. those contain essential inequalities, differential inequalities and distinction inequalities, which play a very important position in setting up (uniform) bounds, worldwide life, large-time habit, decay premiums and blow-up of recommendations to varied 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 varied properties.

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**Example text**

27). With this new deﬁnition of K(E, F, α), and by using obvious notation, observe that K(F, G, β) ∗ K(E, F, α) → K(E, G, α + β − 1) ∩ C(JΔ , L(E, G)) if α + β < 1. 2, we prove the following generalized Bellman–Gronwall inequality (see [40]). 6. e. e. t ∈ J, u(t) ≤ At−β 1 + cBt1−α e(1+ε)μ(α,B)t , where μ(α, B) := (Γ(1 − α)B)1/(1−α) . 4. The inequalities of Henry’s type 29 ∗ Proof. Let E := F := R and k(t, s) := B(t − s)−α for (t, s) ∈ JΔ . Then that −β ˙ Let a(t) := At and observe that k ∈ K(E, α) and ||k||(α),T = B for T ∈ J.

111) 0 If t ≥ 2, then t/2 t/2 (s + 1)−b s−d ds = 0 1 (s + 1)−b s−d ds + 1 (s + 1)−b s−d ds 0 t/2 1 ≤C (s + 1)−b−d ds + C s−d ds 1 0 ⎧ if b + d < 1, ⎨ C(t + 1)1−b−d , C ln(t + 1), if b + d = 1, ≤ ⎩ C, if b + d > 1. 112) If t ≤ 2, then t/2 0 1 (s + 1)−b s−d ds ≤ C (s + 1)−b s−d ds ≤ C. 112) ⎧ if b + d < 1, ⎨ Ct−a (t + 1)1−b−d , Ct−a ln(t + 1), if b + d = 1, II ≤ ⎩ if b + d > 1. 4. 114) that t (t − s)−a (s + 1)−b s−d ds 0 ⎧ ⎨ Ct1−a−d (t + 1)−b + Ct−a (t + 1)1−b−d , Ct1−a−d (t + 1)−b + Ct−a ln(t + 1), ≤ ⎩ Ct1−a−d (t + 1)−b + Ct−a , if b + d < 1, if b + d = 1, if b + d > 1.

30) where B is Euler’s beta function. It follows from Fubini’s theorem that the operation ∗ is associative. In the sequel, we shall prove the following generalized Bellman–Gronwall inequality. First, we give the following two lemmas. Assume that k ∈ K(E, α) for some α ∈ [0, 1). By an easy induction argument, we see that for all n ∈ N and all 0 ≤ s < t ≤ T , || k ∗ k ∗ · · · ∗ k(t, s) ||L(E) ≤ [Γ(1 − α)||k||(α),T ]n (t − s)n(1−α)−1 . 31) n Set +∞ k ∗ ··· ∗ k. 32) j Then we have the following lemma.