Asymptotic Analysis - Volume 140, issue 1-2

Authors: Temam, Roger Meyer

Article Type: Editorial

DOI: 10.3233/ASY-241934

Citation: Asymptotic Analysis, vol. 140, no. 1-2, pp. 1-1, 2024

Authors: Berestycki, Henri | Coron, Jean-Michel

Article Type: Obituary

DOI: 10.3233/ASY-241935

Citation: Asymptotic Analysis, vol. 140, no. 1-2, pp. 3-4, 2024

Authors: Deng, Ting | Squassina, Marco | Zhang, Jianjun | Zhong, Xuexiu

Article Type: Research Article

Abstract: We are concerned with solutions of the following quasilinear Schrödinger equations − div ( φ 2 ( u ) ∇ u ) + φ ( u ) φ ′ ( u ) | ∇ u | 2 + λ u = f ( u ) , x ∈ R N with prescribed mass ∫ R N u 2 d x = c , …where N ⩾ 3 , c > 0 , λ ∈ R appears as the Lagrange multiplier and φ ∈ C 1 ( R , R + ) . The nonlinearity f ∈ C ( R , R ) is allowed to be mass-subcritical, mass-critical and mass-supercritical at origin and infinity. Via a dual approach, the fixed point index and a global branch approach, we establish the existence of normalized solutions to the problem above. The results extend previous results by L. Jeanjean, J. J. Zhang and X.X. Zhong to the quasilinear case. Show more

Keywords: Quasilinear Schrödinger equations, normalized solutions, mass critical exponent

DOI: 10.3233/ASY-241908

Citation: Asymptotic Analysis, vol. 140, no. 1-2, pp. 5-24, 2024

Authors: Selim, Salem | Yan, Lili

Article Type: Research Article

Abstract: We study inverse boundary problems for the magnetic Schrödinger operator with Hölder continuous magnetic potentials and continuous electric potentials on a conformally transversally anisotropic Riemannian manifold of dimension n ⩾ 3 with connected boundary. A global uniqueness result is established for magnetic fields and electric potentials from the partial Cauchy data on the boundary of the manifold provided that the geodesic X-ray transform on the transversal manifold is injective.

Keywords: Inverse problems, magnetic Schrödinger operator, partial data, CTA manifold

DOI: 10.3233/ASY-241909

Citation: Asymptotic Analysis, vol. 140, no. 1-2, pp. 25-36, 2024

Authors: Takahashi, Akihiko | Yamada, Toshihiro

Article Type: Research Article

Abstract: This paper presents a novel generic asymptotic expansion formula of expectations of multidimensional Wiener functionals through a Malliavin calculus technique. The uniform estimate of the asymptotic expansion is shown under a weaker condition on the Malliavin covariance matrix of the target Wiener functional. In particular, the method provides a tractable expansion for the expectation of an irregular functional of the solution to a multidimensional rough differential equation driven by fractional Brownian motion with Hurst index H < 1 / 2 , without using complicated fractional integral calculus for the singular kernel. In a numerical experiment, our expansion shows …a much better approximation for a probability distribution function than its normal approximation, which demonstrates the validity of the proposed method. Show more

Keywords: Asymptotic expansion, Wiener functional, Malliavin calculus, Rough differential equation, fractional Brownian motion

DOI: 10.3233/ASY-241910

Citation: Asymptotic Analysis, vol. 140, no. 1-2, pp. 37-58, 2024

Authors: Da Silva, João Pablo P.

Article Type: Research Article

Abstract: In this work, we consider a functional I : W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ) → R of the form I ( u , v ) = 1 p ∫ Ω ( | ∇ u | p + | ∇ v | p ) d x − ∫ Ω H …( x , u ( x ) , v ( x ) ) d x where Ω ⊂ R N is a smooth bounded domain, max { | ∂ s H ( x , s , t ) | , | ∂ t H ( x , s , t ) | } ⩽ C ( 1 + | s | σ 1 − 1 + | t | σ 2 − 1 ) a.e. x ∈ Ω , for some C > 0 , ∀ t , s ∈ R , p < σ i ⩽ p ∗ : = N p / ( N − p ) , i = 1 , 2 , and 1 < p < N . We prove that a local minimum in the topology of C 0 1 ( Ω ) × C 0 1 ( Ω ) is a local minimum in the topology of W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ) . An important application of this result is related to the question of multiplicity of solutions for a class of systems with concave-convex type nonlinearities. Show more

Keywords: Local minimization, p-Laplacian system, Sub-super solution method

DOI: 10.3233/ASY-241911

Citation: Asymptotic Analysis, vol. 140, no. 1-2, pp. 59-76, 2024

Authors: Zhao, Yongqing | Liu, Wenjun | Lv, Guangying | Wang, Yuepeng

Article Type: Research Article

Abstract: In this paper, the problem of continuous data assimilation of three dimensional primitive equations with magnetic field in thin domain is studied. We establish the well-posedness of the assimilation system and prove that the H 2 -strong solution of the assimilation system converges exponentially to the reference solution in the sense of L 2 as t → ∞ . We also study the sensitivity analysis of the assimilation system and prove that a sequence of solutions of the difference quotient equation converge to the unique solution of …the formal sensitivity equation. Show more

Keywords: Well-posedness, data assimilation, sensitivity analysis

DOI: 10.3233/ASY-241912

Citation: Asymptotic Analysis, vol. 140, no. 1-2, pp. 77-108, 2024

Authors: Fang, Xiang-Dong | Han, Zhi-Qing

Article Type: Research Article

Abstract: In this paper we consider the generalized quasilinear Schrödinger equations − div ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = h ( x , u ) , x ∈ R N , where V and h are periodic in x i , 1 ⩽ i ⩽ N …. By using variational methods, we prove the existence of ground state solutions, i.e., nontrivial solutions with least possible energy. Show more

Keywords: Quasilinear Schrödinger equation, ground state solution, asymptotically linear, Nehari manifold

DOI: 10.3233/ASY-241913

Citation: Asymptotic Analysis, vol. 140, no. 1-2, pp. 109-122, 2024

Authors: Nimi, Aymard Christbert | Langa, Franck Davhys Reval

Article Type: Research Article

Abstract: In this article, our objective is to explore a Cahn–Hilliard system with a proliferation term, particularly relevant in biological contexts, with Neumann boundary conditions. We commence our investigation by establishing the boundedness of the average values of the local cell density u and the temperature H . This observation suggests that the solution ( u , H ) either persists globally in time or experiences finite-time blow-up. Subsequently, we prove the convergence of u to 1 and H to 0 as time approaches infinity. Finally, we bolster our theoretical findings with numerical simulations.

Keywords: Cahn–Hilliard system, proliferation term, dissipativity, blow up, simulations

DOI: 10.3233/ASY-241915

Citation: Asymptotic Analysis, vol. 140, no. 1-2, pp. 123-145, 2024