Asymptotic Analysis - Volume 136, issue 3-4

Authors: Shapiro, Jacob

Article Type: Research Article

Abstract: We consider, for h , E > 0 , resolvent estimates for the semiclassical Schrödinger operator − h 2 Δ + V − E . Near infinity, the potential takes the form V = V L + V S , where V L is a long range potential which is Lipschitz with respect to the radial variable, while V S = O ( | x | − 1 ( log | x | …) − ρ ) for some ρ > 1 . Near the origin, | V | may behave like | x | − β , provided 0 ⩽ β < 2 ( 3 − 1 ) . We find that, for any ρ ˜ > 1 , there are C , h 0 > 0 such that we have a resolvent bound of the form exp ( C h − 2 ( log ( h − 1 ) ) 1 + ρ ˜ ) for all h ∈ ( 0 , h 0 ] . The h -dependence of the bound improves if V S decays at a faster rate toward infinity. Show more

Keywords: Resolvent estimate, Schrödinger operator, short range potential

DOI: 10.3233/ASY-231872

Citation: Asymptotic Analysis, vol. 136, no. 3-4, pp. 157-180, 2024

Authors: Jleli, Mohamed | Samet, Bessem | Vetro, Calogero

Article Type: Research Article

Abstract: We consider a higher order (in time) evolution inequality posed in the half ball, under Dirichlet type boundary conditions. The involved elliptic operator is the sum of a Laplace differential operator and a Leray–Hardy potential with a singularity located at the boundary. Using a unified approach, we establish a sharp nonexistence result for the evolution inequalities and hence for the corresponding elliptic inequalities. We also investigate the influence of a nonlinear memory term on the existence of solutions to the Dirichlet problem, without imposing any restrictions on the sign of solutions.

Keywords: Higher order evolution inequalities, Leray–Hardy potential, half ball, nonexistence result

DOI: 10.3233/ASY-231873

Citation: Asymptotic Analysis, vol. 136, no. 3-4, pp. 181-202, 2024

Authors: Ayouch, C. | Meskine, D. | Tilioua, M.

Article Type: Research Article

Abstract: In this paper, the Landau–Lifshitz–Baryakhtar (LLBar) equation for magnetization dynamics in ferrimagnets is considered. We prove global existence of a periodic solutions as well as local existence and uniqueness of regular solutions. We also study the relationships between the Landau–Lifshitz–Baryakhtar equation and both Landau–Lifshitz–Bloch and harmonic map equations.

Keywords: Landau–Lifshitz–Baryakhtar equation, harmonic map equation, local well-posedness, asymptotic behaviour

DOI: 10.3233/ASY-231874

Citation: Asymptotic Analysis, vol. 136, no. 3-4, pp. 203-229, 2024

Authors: Charif, Mirna | Fino, Ahmad | Zielinski, Lech

Article Type: Research Article

Abstract: We prove that the spectrum of the asymmetric quantum Rabi model consists of two eigenvalue sequences ( E m + ) m = 0 ∞ , ( E m − ) m = 0 ∞ , satisfying a two-term asymptotic formula with error estimate of the form O ( m − 1 / 4 ) , when m tends to infinity.

Keywords: Quantum Rabi model, unbounded self-adjoint operators, discrete spectrum, asymtotic distribution of eigenvalues

DOI: 10.3233/ASY-231875

Citation: Asymptotic Analysis, vol. 136, no. 3-4, pp. 231-256, 2024

Authors: Dai, Guowei | Zhang, Zhitao

Article Type: Research Article

Abstract: By bifurcation and topological methods, we study the existence/nonexistence and multiplicity of one-sign or nodal solutions of the following k -th mean curvature problem in Minkowski spacetime r N − k v ′ 1 − v ′ 2 k ′ = λ N C N k r N − 1 H k ( r , v ) …in  ( 0 , R ) , | v ′ | < 1 in  ( 0 , R ) , v ′ ( 0 ) = v ( R ) = 0 . As a previous step, we investigate the spectral structure of its linearized problem at zero. Moreover, we also obtain a priori bounds and the asymptotic behaviors of solutions with respect to λ . Show more

Keywords: Spectrum, Bifurcation, k-th mean curvature, Nodal solutions, A priori bounds, Asymptotic behaviors, One-sign solution

DOI: 10.3233/ASY-231877

Citation: Asymptotic Analysis, vol. 136, no. 3-4, pp. 257-289, 2024

Authors: Dhanya, R. | Pramanik, Sarbani | Harish, R.

Article Type: Research Article

Abstract: We analyze a non-linear elliptic boundary value problem that involves ( p , q ) Laplace operator, for the existence of its positive solution in an arbitrary smooth bounded domain. The non-linearity here is driven by a singular, monotonically increasing continuous function in ( 0 , ∞ ) which is eventually positive. The novelty in proving the existence of a positive solution lies in the construction of a suitable subsolution. Our contribution marks an advancement in the theory of existence of positive solutions for infinite semipositone problems in arbitrary bounded domains, whereas the prevailing …theory is limited to addressing similar problems only in symmetric domains. Additionally, using the ideas pertaining to the construction of subsolution, we establish the exact behavior of the solutions of “q-sublinear” problem involving ( p , q ) Laplace operator when the parameter λ is very large. The parameter estimate that we derive is non-trivial due to the non-homogeneous nature of the operator and is of independent interest. Show more

Keywords: p−q Laplacian, infinite semipositone problem, maximal solution, asymptotic estimate, singular problem

DOI: 10.3233/ASY-231880

Citation: Asymptotic Analysis, vol. 136, no. 3-4, pp. 291-307, 2024