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.