By Yuming Qin

This booklet provides 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 constructing (uniform) bounds, worldwide life, large-time habit, decay charges and blow-up of options to varied sessions of evolutionary differential equations. Summarizing effects from an enormous variety of literature assets equivalent to released papers, preprints and books, it categorizes inequalities by way of their assorted properties.

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**Additional info for Analytic Inequalities and Their Applications in PDEs**

**Example text**

J=1 j (1+β)/(2(1−β)) (ηβ)jβ/(1−β) 1−β , where η > 0 is arbitrary. 33) with η := (1 + ε)/β. Now it is easy to prove the following existence and uniqueness theorem for abstract linear Volterra equations. 5 ([40]). Assume that α, β ∈ [0, 1) and k ∈ K(E, α). 32). 28 Chapter 1. Integral Inequalities Proof. 35). The second one can be treated in a similar manner. 29), respectively. 35). Let T ∈ J˙ be ﬁxed. 31) that ∗k ∈ L(K∞ (E, F, β)) and that the spectral radius of this operator equals zero. 35) has at most one solution ‘on ˙ This proves the lemma.

1. 3) still holds. 1. 2 (The Kawashima–Nakao–Ono Inequality [423]). 7) 0 with some constants k0 , k1 > 0, α, β, γ ≥ 0 and 0 ≤ μ < 1.

Now we shall prove assertion (ii). 135) 0 where v(t) = (e−t u(t))q , φ(t) = 2q−1 aq (t). 126). 5. 123) involving multiple integrals. We do not give the details here. 4. 123) for r = 2, β = 1 was studied by Pachpatte in [723], where a result obtained by Ou-Yang [715] is generalized. Applying ˇ [606], the following theorem can be proved in the method developed in Medved ˇ [606]. Medved ˇ Inequality [606]). 16 (The Medved 1 decreasing C -function on the interval [0, T ] (0 < T < +∞), let F (t) be a nonnegative, continuous function on [0, T ], 0 < β < 1, r ≥ 1, and let ω : R+ → R+ be a continuous, non-decreasing, positive function.