By Nakao Hayashi, Elena I. Kaikina, Pavel Naumkin, Ilya A. Shishmarev
Many of difficulties of the usual sciences bring about nonlinear partial differential equations. despite the fact that, just a couple of of them have succeeded in being solved explicitly. consequently various tools of qualitative research equivalent to the asymptotic tools play an important position. this is often the 1st publication on the planet literature giving a scientific improvement of a normal asymptotic thought for nonlinear partial differential equations with dissipation. Many normal famous equations are regarded as examples, equivalent to: nonlinear warmth equation, KdVB equation, nonlinear damped wave equation, Landau-Ginzburg equation, Sobolev sort equations, structures of equations of Boussinesq, Navier-Stokes and others.
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7. Let σ > n2 . Assume that the initial data u0 ∈ L1,a (Rn ) ∩ Lp (Rn ), a ∈ (0, 1] , and p > max 1, n2 σ . 8). 9) as t → ∞ uniformly with respect to x ∈ Rn , where 0 < γ < min and the constant ∞ u0 (x) dx − λ A= −1 , σ |u (τ, x)| u (τ, x) dx. 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.
We restrict our attention only to the case of the space dimensions n = 1, 2, 3. We assume the following asymptotic expansion for the symbols S (t, ξ) for n = 1, 2, 3 ∂tk S (t, ξ) = t −k −tL0 (ξ) e + ∂tk e−αt sin (tβ |ξ|) |ξ| + Rk (t, ξ) for ξ → ∞, where α > 0, β ∈ R, k = 0, 1. , n, k = 0, 1, with some N ≥ n + 2, γ > 0. is the symbol for the wave equation. g. 5 Estimates for linear semigroups φ (y) dy 1 2π W (t) φ = W (t) φ = 1 4πt , 2 |x−y|≤βt and for n = 3 35 β 2 t2 − |x − y| |x−y|=βt φ (y) dωy .
66) the estimates of the lemma follow. 33 is proved. 2 Estimates in the L2 - theory By the Sobolev Imbedding Theorem the Lp norms can be estimated by derivatives in L2 norms and weighted L1,a norms can be estimates by weighted L2,b norms with b > a + n2 . And the L2 - theory is convenient for passing to Fourier transform representations. 67) for all ξ ∈ Rn , where α > 0, δ > 0, µ ≥ 0. , n, with some N ≥ n + 2. We now prove L2 estimates for the Green operator G (t). 35. 68). Then the estimates are true ρ |∇| G (t) φ L2 n 1 1 − ρ−β δ − δ (q−2) ≤C t −αt |∇| + Ce ρ |∇| −µ i∇ φ β −µ i∇ φ Lq L2 for all t > 0, where 1 ≤ q ≤ 2, 0 ≤ β ≤ ρ, and ω |·| G (t) φ L2 ω n δ − 2δ ≤C t −α 2t ω · + Ce −µ i∇ i∇ for all t > 0, where 1 ≤ q ≤ 2, are ﬁnite.