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Asymptotic behaviour of the scattering phase for non-trapping obstacles

Veselin Petkov, Georgi Popov (1982)

Annales de l'institut Fourier

Let S ( λ ) be the scattering matrix related to the wave equation in the exterior of a non-trapping obstacle 𝒪 R n , n 3 with Dirichlet or Neumann boundary conditions on 𝒪 . The function s ( λ ) , called scattering phase, is determined from the equality e - 2 π i s ( λ ) = det S ( λ ) . We show that s ( λ ) has an asymptotic expansion s ( λ ) j = 0 c j λ n - j as λ + and we compute the first three coefficients. Our result proves the conjecture of Majda and Ralston for non-trapping obstacles.

Asymptotic stability for a nonlinear evolution equation

Zhang Hongwei, Chen Guowang (2004)

Commentationes Mathematicae Universitatis Carolinae

We establish the asymptotic stability of solutions of the mixed problem for the nonlinear evolution equation ( | u t | r - 2 u t ) t - Δ u t t - Δ u - δ Δ u t = f ( u ) .

Asymptotics for conservation laws involving Lévy diffusion generators

Piotr Biler, Grzegorz Karch, Wojbor A. Woyczyński (2001)

Studia Mathematica

Let -ℒ be the generator of a Lévy semigroup on L¹(ℝⁿ) and f: ℝ → ℝⁿ be a nonlinearity. We study the large time asymptotic behavior of solutions of the nonlocal and nonlinear equations uₜ + ℒu + ∇·f(u) = 0, analyzing their L p -decay and two terms of their asymptotics. These equations appear as models of physical phenomena that involve anomalous diffusions such as Lévy flights.

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