Asymptotic Analysis - Volume 118, issue 4

Authors: Challa, Durga Prasad | Mantile, Andrea | Sini, Mourad

Article Type: Research Article

Abstract: We deal with the time-harmonic acoustic waves scattered by a large number of small holes, with radius a , a ≪ 1 , arbitrarily distributed in a bounded part of the homogeneous background R 3 . We assume no periodicity in distributing these holes. Using the asymptotic expansions of the scattered field by this cluster, we show that as their number M grows following the law M ∼ a − s , a → 0 , the collection of these holes has one of the …following behaviors: (1) if s < 1 , then the scattered fields tend to vanish as a tends to zero, i.e., the cluster is a soft one. (2) if s = 1 , then the cluster behaves as an equivalent medium modeled by a refraction index, supported in a given bounded domain Ω, which is described by certain geometry properties of the holes and their local distribution. The cluster is a moderate (or intermediate) one. (3) if s > 1 , with additional conditions, then the cluster behaves as a totally reflecting extended body, modeled by a bounded domain Ω, i.e., the incident waves are totally reflected by the surface of this extended body. The cluster is a rigid one. Explicit errors estimates between the scattered fields due to the cluster of small holes and the ones due to equivalent media (i.e., the refraction index) or the extended body are provided. Show more

Keywords: Scattering by small particles, Foldy–Lax approximation, effective medium theory

DOI: 10.3233/ASY-191560

Citation: Asymptotic Analysis, vol. 118, no. 4, pp. 235-268, 2020

Authors: Belchior, P. | Bueno, H. | Miyagaki, O.H. | Pereira, G.A.

Article Type: Research Article

Abstract: With appropriate hypotheses on the nonlinearity f , we prove the existence of a ground state solution u for the problem − Δ + m 2 u + V u = ( W ∗ F ( u ) ) f ( u ) in  R N , where V is a bounded potential, not necessarily continuous, and F the primitive of f . We also show that any of this problem is a classical solution. Furthermore, we prove …that the ground state solution has exponential decay. Show more

Keywords: Variational methods, exponential decay, fractional Laplacian, Hartree equations

DOI: 10.3233/ASY-191561

Citation: Asymptotic Analysis, vol. 118, no. 4, pp. 269-295, 2020

Authors: Vodev, Georgi

Article Type: Research Article

Abstract: We prove semiclassical resolvent estimates for real-valued potentials V ∈ L ∞ ( R n ) , n ⩾ 3 , of the form V = V L + V S , where V L is a long-range potential which is C 1 with respect to the radial variable, while V S is a short-range potential satisfying V S ( x …) = O ( ⟨ x ⟩ − δ ) with δ > 1 . Show more

Keywords: Schrödinger operator, resolvent estimates, short-range potentials

DOI: 10.3233/ASY-191562

Citation: Asymptotic Analysis, vol. 118, no. 4, pp. 297-312, 2020

Authors: Xiang, Mingqi | Yang, Di | Zhang, Binlin

Article Type: Research Article

Abstract: In this paper, we consider the following Kirchhoff-type diffusion problem involving the fractional Laplacian and logarithmic nonlinearity at high initial energy level: u t + [ u ] s 2 ( θ − 1 ) ( − Δ ) s u = | u | q − 2 u ln | u | ( x , t ) ∈ Ω × R + , u ( x , t ) = 0 ( x , t …) ∈ ( R N ∖ Ω ) × R + , u ( x , 0 ) = u 0 ( x ) x ∈ Ω , where ( − Δ ) s is the fractional Laplacian with s ∈ ( 0 , 1 ) , N > 2 s , [ u ] s is the Gagliardo seminorm of u , Ω ⊂ R N is a bounded domain with Lipschitz boundary, 1 ⩽ θ < N / ( N − 2 s ) , 2 θ < q < 2 s ∗ . Based on the potential well theory, a sufficient condition is given for the existence of global solutions that vanish at infinity or solutions that blow up in finite time under some appropriate assumptions. In particular, the existence of ground state solutions for the above stationary problem is obtained by restricting the related discussion on Nehari manifold. Show more

Keywords: Fractional Kirchhoff problems, logarithmic nonlinearity, global existence, blow up

DOI: 10.3233/ASY-191564

Citation: Asymptotic Analysis, vol. 118, no. 4, pp. 313-329, 2020