By Yuming Qin
This ebook provides a couple of analytic inequalities and their purposes in partial differential equations. those comprise vital inequalities, differential inequalities and distinction inequalities, which play a very important function in developing (uniform) bounds, international 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 resources comparable to released papers, preprints and books, it categorizes inequalities when it comes to their diverse properties.
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Extra info for Analytic Inequalities and Their Applications in PDEs
Proof. Let J(t) = t −1/2 s x(s)ds. 0 Then J(t) satisﬁes J (t) ≤ Ct−α t−1/2 + εt−1 J(t) which implies d −ε t J(t) dt ≤ Ct−α−1/2−ε . 4) over (0, t) gives us t t−ε J(t) ≤ C s−α−1/2−ε ds. 5) 0 Here we used the assumption that lim t−ε J(t) = 0. t→0+ Assume that α + ε < 1/2. 3 yields t s−α−1/2−ε ds ≤ Ct−α+1/2−ε . 3). 1. 3) still holds. 1. 2 (The Kawashima–Nakao–Ono Inequality ). 7) 0 with some constants k0 , k1 > 0, α, β, γ ≥ 0 and 0 ≤ μ < 1.
0 Consequently, sup h(τ ) ≤ 0≤τ ≤t C1 (1 + t), 1 − C2 K(a, b) provided that C2 < 1/K(a, b). 92). 91), we write b = b1 + η for a + b1 = 1 and η > 0, and we ﬁx t1 > 0 such that C2 (1 + t1 )−η < 1 . 92) at t1 yields h(t) ≤ C(1 + t) + C2 K(a, b1 )(1 + t1 )−η sup h(τ ) 0≤τ ≤t for some constant C > 0 independent of t. Hence the conclusion follows. 2 in Henry ˇ . 3) and is due to Medved ˇ Inequality ). 12 (The Medved 1 decreasing C function on [0, T ) for 0 < T ≤ +∞, and F (t) a continuous, nonnegative function on [0, T ).
108) 0 If b + d > 1, then for all t > 0, t (t − s)−a (s + 1)−b s−d ds ≤ Ct−a . 109) 0 Proof. Set t t/2 (t − s)−a (s + 1)−b s−d ds, I := II := t/2 (t − s)−a (s + 1)−b s−d ds. 110) (s + 1)−b s−d ds. 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.