Asymptotic Analysis - Volume 137, issue 1-2

Authors: Giacomoni, Jacques | Il’yasov, Yavdat | Kumar, Deepak

Article Type: Research Article

Abstract: We discuss the existence and non-existence of periodic in one variable and compactly supported in the other variables least energy solutions for equations with non-Lipschitz nonlinearity of the form: − Δ u = λ u p − u q in R N + 1 , where 0 < q < p < 1 and λ ∈ R . The approach is based on the Nehari manifold method supplemented by a one-sided constraint given through the functional of the suitable Pohozaev identity. …The limit value of the parameter λ , where the approach is applicable, corresponds to the existence of periodic in one variable and compactly supported in the other variables least energy solutions. This value is found through the extrem values of nonlinear generalized Rayleigh quotients and the so-called curve of the critical exponents of p , q . Important properties of the solutions are derived for suitable ranges of the parameters, such as that they are not trivial with respect to the periodic variable and do not coincide with compactly supported solutions on the entire space R N + 1 . Show more

Keywords: Semilinear elliptic equation, non-Lipschitz nonlinearity, compactly supported solutions, periodic solutions, generalized Rayleigh’s quotients, the Pohozaev identity

DOI: 10.3233/ASY-231878

Citation: Asymptotic Analysis, vol. 137, no. 1-2, pp. 1-25, 2024

Authors: Mel’nyk, Taras | Rohde, Christian

Article Type: Research Article

Abstract: This article completes the study of the influence of the intensity parameter α in the boundary condition ε ∂ ν ε u ε − u ε V ε → · ν ε = ε α φ ε given on the boundary of a thin three-dimensional graph-like network consisting of thin cylinders that are interconnected by small domains (nodes) with diameters of order O ( ε ) . Inside …of the thin network a time-dependent convection-diffusion equation with high Péclet number of order O ( ε − 1 ) is considered. The novelty of this article is the case of α < 1 , which indicates a strong intensity of physical processes on the boundary, described by the inhomogeneity φ ε (the cases α = 1 and α > 1 were previously studied by the same authors). A complete Puiseux asymptotic expansion is constructed for the solution u ε as ε → 0 , i.e., when the diffusion coefficients are eliminated and the thin network shrinks into a graph. Furthermore, the corresponding uniform pointwise and energy estimates are proved, which provide an approximation of the solution with a given accuracy in terms of the parameter ε . Show more

Keywords: Asymptotic expansion, convection-diffusion problem, boundary interactions, thin graph-like junction, hyperbolic limit model

DOI: 10.3233/ASY-231876

Citation: Asymptotic Analysis, vol. 137, no. 1-2, pp. 27-52, 2024

Authors: Angulo-Castillo, V. | Ferreira, L.C.F. | Kosloff, L.

Article Type: Research Article

Abstract: We are concerned with the long-time solvability for 2D inviscid Boussinesq equations for a larger class of initial data which covers the case of borderline regularity. First we show the local solvability in Besov spaces uniformly with respect to a parameter κ associated with the stratification of the fluid. Afterwards, employing a blow-up criterion and Strichartz-type estimates, the long-time solvability is obtained for large κ regardless of the size of initial data.

Keywords: Boussinesq equations, Convection problem, Long-time solvability, Dispersive effects, Besov spaces, Borderline regularity

DOI: 10.3233/ASY-231879

Citation: Asymptotic Analysis, vol. 137, no. 1-2, pp. 53-84, 2024

Authors: Qin, Yuming | Wang, Hongli | Yang, Bin

Article Type: Research Article

Abstract: This paper is concerned with the dimension of the global attractors for a time-dependent strongly damped subcritical Kirchhoff wave equation with a memory term. A careful analysis is required in the proof of a stabilizability inequality. The main result establishes the finite dimensionality of the global attractor.

Keywords: Fractal dimension, subcritical case, Kirchhoff wave equation, global attractor, stabilizability inequality

DOI: 10.3233/ASY-231881

Citation: Asymptotic Analysis, vol. 137, no. 1-2, pp. 85-95, 2024

Authors: Avila, Jake | Monsurrò, Sara | Raimondi, Federica

Article Type: Research Article

Abstract: In a bounded cylinder with a rough interface we study the asymptotic behaviour of the spectrum and its associated eigenspaces for a stationary heat propagation problem. The main novelty concerns the proof of the uniform a priori estimates for the eigenvalues. In fact, due to the peculiar geometry of the domain, standard techniques do not apply and a suitable new approach is developed.

Keywords: Eigenvalue problem, homogenization, rough surfaces, interface conditions

DOI: 10.3233/ASY-231882

Citation: Asymptotic Analysis, vol. 137, no. 1-2, pp. 97-121, 2024

Authors: Cabanillas Zannini, Victor | Quispe Méndez, Teófanes | Ramos, A.J.A.

Article Type: Research Article

Abstract: This article deals with the asymptotic behavior of a mathematical model for laminated beams with Kelvin–Voigt dissipation acting on the equations of transverse displacement and dimensionless slip. We prove that the evolution semigroup is exponentially stable if the damping is effective in the two equations of the model. Otherwise, we prove that the semigroup is polynomially stable and find the optimal decay rate when damping is effective only in the slip equation. Our stability approach is based on the Gearhart–Prüss–Huang Theorem, which characterizes exponential stability, while the polynomial decay rate is obtained using the Borichev and Tomilov Theorem.

Keywords: Laminated beam, Kelvin–Voigt damping, exponential stability, polynomial stability

DOI: 10.3233/ASY-231883

Citation: Asymptotic Analysis, vol. 137, no. 1-2, pp. 123-151, 2024