Asymptotics for Dissipative Nonlinear Equations by Nakao Hayashi, Elena I. Kaikina, Pavel Naumkin, Ilya A.

By Nakao Hayashi, Elena I. Kaikina, Pavel Naumkin, Ilya A. Shishmarev

Many of difficulties of the ordinary sciences result in nonlinear partial differential equations. notwithstanding, just a couple of of them have succeeded in being solved explicitly. consequently varied tools of qualitative research resembling the asymptotic tools play a crucial position. this is often the 1st publication on the planet literature giving a scientific improvement of a normal asymptotic conception for nonlinear partial differential equations with dissipation. Many common recognized equations are regarded as examples, reminiscent of: nonlinear warmth equation, KdVB equation, nonlinear damped wave equation, Landau-Ginzburg equation, Sobolev sort equations, platforms of equations of Boussinesq, Navier-Stokes and others.

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8. 25 for the case of large initial data and λ < 0. 7 we prepare the following lemma. 9. 1) −n 2 G0 (t, x) = (4π (t + 1)) |x|2 e− 4(t+1) in spaces X, Z. 5) is valid. Proof. 1 General approach 55 for all t ≥ 0. Hence we see that G0 ∈ X. 28 with δ = ν = 2 to obtain ≤ Ct− 2 ( r − q )− n b |·| ∂xβj G (t) φ Lq 1 1 > 0. 11) L1,a φ (x) dx, 1 ≤ r ≤ q ≤ ∞, β ≥ 0, 0 ≤ b ≤ a. 12) Z for all t ∈ (0, 1] . 13) Z for all t > 1. 3) with γ = a2 . 5). Also in view of the definition of the norm Y we have |f (N (u (τ )))| ≤ N (u (τ )) − nσ 2p ≤ C {τ } L1 ≤ C − nσ 2 τ − nσ 2p {τ } u σ X τ − nσ 2 N (u) Y .

Y)dy. 40 is then proved. Let us now compute the asymptotics of the Green function 1 G (ξ) = √ 2π for large values of ξ. 41. Let α > 0. 87) is true for |ξ| → ∞. Proof. Denote m = [a] + 1. We integrate by parts m times with respect to η to get α 1 −m G (ξ) = √ (iξ) eiξη ∂ηm e−|η| dη 2π R α (α − 1) · · · (α + 1 − m) α−m √ = eiξη |η| dη ξ m 2π R + Cξ −m − ∂ηm e−|η| α−m |η| eiξη α dη R = I1 + I2 . 87); it can be computed explicitly (see Erd´elyi et al. [1954]) α (α − 1) · · · (α − m + 1) α−m √ eiξη |η| dη ξ m 2π R √ 2 πα −1−α .

By the estimates of the Macdonald function we see that for any k ≥ 1 k− n+1 2 e−|x| , for |x| ≥ 1, C |x| |Bk (x)| ≤ 1 2k−n−1 C |x| y dy, for |x| < 1 . 51) for all 1 ≤ p ≤ ∞ and b ≥ 0. 30. 31. 50) takes place. Then the estimates are true ≤ Ct− δ ( r − p )+ δ φ n b |·| G (t) φ Lp 1 1 b + Ct− δ ( r − p ) φ n 1 Lr 1 + e− 2 t φ α Lr,b Lp,b and b |·| ≤ Ct G (t) φ − ϑt− δ G0 (·) t− δ n 1 b −n δ (1− p )+ δ t 1 −a δ Lp a − γδ + t |·| φ + e− 2 t |·| φ α L1 b Lp for all t > 0, where ϑ = Rn φ (x) dx, 1 ≤ r ≤ p ≤ ∞, 0 ≤ b ≤ a, provided that the right-hand sides are finite.

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