Analytic Inequalities and Their Applications in PDEs by Yuming Qin

By Yuming Qin

This e-book provides a few analytic inequalities and their functions in partial differential equations. those comprise essential inequalities, differential inequalities and distinction inequalities, which play a vital position in developing (uniform) bounds, worldwide life, large-time habit, decay charges and blow-up of strategies to numerous periods of evolutionary differential equations. Summarizing effects from an enormous variety of literature assets resembling released papers, preprints and books, it categorizes inequalities by way of their various properties.

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91) 0 for some a ∈ (0, 1), b > 0, positive constants C1 and C2 . 92) for a constant C > 0 independent of t. 92) holds true in the limit case a + b = 1 under the weaker assumption t h(t) ≤ C1 (1 + t) + C2 0 provided that C2 > 0 is sufficiently small. 4. The inequalities of Henry’s type 39 Proof. 93) the inequality h(t) ≤ C1 (1 + t) + C2 K(a, b) sup h(τ ), 0≤τ ≤t where t K(a, b) = 1 (t − τ )−a τ −b dτ = 0 (1 − s)−a s−b ds. 0 Consequently, sup h(τ ) ≤ 0≤τ ≤t C1 (1 + t), 1 − C2 K(a, b) provided that C2 < 1/K(a, b).

As s → +∞, we have Eβ,γ (s) = O s1/2(ν/β−γ) exp β ν/β s γ . 8) Cm [bΓ(β)]m tmν + B n+1 u(t). 9) Proof. 10) 0 where for γ ≥ 1, Kn (t, s) ≤ Qn t(n−1)(γ−1) (t − s)nβ−1 , Q1 = b, Qn+1 /Qn = bΓ(β)/Γ(nβ)/Γ(nβ + β). If γ ∈ (0, 1), we have Kn (t, s) ≤ Qn (t − s)nν−γ , Q1 = b, Qn+1 /Qn = bΓ(β)/Γ(nν)/Γ(nν + β). In either case, Qn+1 /Qn = O(n−β ) as n → +∞, so B n u(t) → 0 as n → +∞, and u(t) ≤ aEβ,γ [bΓ(β)]1/ν t . 11) Now Γ(z + p)/Γ(z + q) = z p−q {1 + (p − q)(p + q − 1)/2z + O(z −2 )} as z → +∞, so if δ = (βγ + ν)/2ν, then Γ((n + 1)β + δ)Cn+1 = (β/ν)β [1 + O(n−2 )].

Set φ(t) s≤t φ(s), ψ(t) = sups≤t ψ(s). 7) hold for φ ˜ and ψ. Indeed, if τ < t, then τ τ α1 1 ˜ (t − s)−α1 s−β1 ψ(s)ds = τ 1−β1 0 ˜ z)dz (1 − z)−α1 z −β1 ψ(τ 0 1 ˜ (1 − z)−α1 z −β1 ψ(tz)dz ≤ t1−β 0 t ˜ (t − s)−α1 s−β1 ψ(s)ds, = tα1 0 so that, τ φ(τ ) ≤ A + kτ α1 t ˜ (τ − s)−α1 s−β1 ψ(s)ds ≤ A + ktα1 0 ˜ (t − s)−α1 s−β1 ψ(s)ds, 0 whence t ˜ ≤ A + ktα1 φ(t) ˜ (t − s)−α1 s−β1 ψ(s)ds. 8) for φ, ˜ ψ. ˜ This A similar estimate holds for ψ. is why we assume that φ and ψ are non-decreasing functions. 8) for t small.

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