Asymptotic Analysis - Volume 128, issue 4

Authors: Vetro, Calogero | Wardowski, Dariusz

Article Type: Research Article

Abstract: We discuss a third-order differential equation, involving a general form of nonlinearity. We obtain results describing how suitable coefficient functions determine the asymptotic and (non-)oscillatory behavior of solutions. We use comparison technique with first-order differential equations together with the Kusano–Naito’s and Philos’ approaches.

Keywords: Nonlinear differential equation, oscillation and non-oscillation, asymptotic behavior, comparison technique, third-order differential equation

DOI: 10.3233/ASY-211710

Citation: Asymptotic Analysis, vol. 128, no. 4, pp. 445-463, 2022

Authors: Suzuki, Takashi

Article Type: Research Article

Abstract: We study the family of blowup solutions to semilinear elliptic equations in two-space dimensions with exponentially-dominated nonnegative nonlinearities. Such a family admits an exclusion of the boundary blowup, finiteness of blowup points, and pattern formation. Then, Hamiltonian control of the location of blowup points, residual vanishing, and mass quantization arise under the estimate from below of the nonlinearity. Finally, if the principal growth rate of nonlinearity is exactly exponential and the residual part has a gap relative to this term, there is a locally uniform estimate of the solution which ensures its asymptotic non-degeneracy.

Keywords: Semilinear elliptic equation, blowup analysis, point vortex Hamiltonian, recursive hierarchy, Onsager’s theory

DOI: 10.3233/ASY-211713

Citation: Asymptotic Analysis, vol. 128, no. 4, pp. 465-494, 2022

Authors: Ben Hassen, Moahmed Fahmi | Hamouda, Makram | Hamza, Mohamed Ali | Teka, Hanen Khaled

Article Type: Research Article

Abstract: In this article, we consider the damped wave equation in the scale-invariant case with time-dependent speed of propagation, mass term and time derivative nonlinearity. More precisely, we study the blow-up of the solutions to the following equation: ( E ) u t t − t 2 m Δ u + μ t u t + ν 2 t 2 u = | u t | p , …in  R N × [ 1 , ∞ ) , that we associate with small initial data. Assuming some assumptions on the mass and damping coefficients, ν and μ > 0 , respectively, we prove that blow-up region and the lifespan bound of the solution of ( E ) remain the same as the ones obtained for the case without mass, i.e. ( E ) with ν = 0 which constitutes itself a shift of the dimension N by μ 1 + m compared to the problem without damping and mass. Finally, we think that the new bound for p is a serious candidate to the critical exponent which characterizes the threshold between the blow-up and the global existence regions. Show more

Keywords: Blow-up, generalized Tricomi equation, Glassey exponent, lifespan, nonlinear wave equations, scale-invariant damping, time-derivative nonlinearity

DOI: 10.3233/ASY-211714

Citation: Asymptotic Analysis, vol. 128, no. 4, pp. 495-515, 2022

Authors: Costa, Augusto | Pinheiro, Andréia

Article Type: Research Article

Abstract: This article concerns the multiplicity of solutions for a class of nonlocal and nonhomogeneous Neumann boundary value problems involving the p ( x ) -Laplacian, in which both nonlinear terms assume critical growth. We use variational method, exploring an important truncation argument and properties of the genus.

Keywords: Nonlocal problem, Neumann boundary conditions, Sobolev spaces with variable exponent, critical exponent, truncation argument

DOI: 10.3233/ASY-211715

Citation: Asymptotic Analysis, vol. 128, no. 4, pp. 517-538, 2022

Authors: Zeng, Shengda | Papageorgiou, Nikolaos S.

Article Type: Research Article

Abstract: In the present paper, we consider a nonlinear Robin problem driven by a nonhomogeneous differential operator and with a reaction which is only locally defined. Using cut-off techniques and variational tools, we show that the problem has a sequence of nodal solutions converging to zero in C 1 ( Ω ‾ ) .

Keywords: Nonhomogeneous differential operator, locally defined reaction, cut-off function, extremal constant sign solution, nodal solution

DOI: 10.3233/ASY-211716

Citation: Asymptotic Analysis, vol. 128, no. 4, pp. 539-553, 2022

Authors: Iosifescu, Oana | Licht, Christian

Article Type: Research Article

Abstract: By using a nonlinear version of Trotter’s theory of approximation of semi-groups acting on variable Hilbert spaces, we propose an asymptotic modeling for the behavior of a linearly elastic plate in bilateral contact with a rigid body along part of its lateral boundary with Norton or Tresca friction.

Keywords: Linearly elastic thin plates, Norton or Tresca friction, dynamics, maximal-monotone operators, Trotter’s theory of approximation of semi-groups

DOI: 10.3233/ASY-211717

Citation: Asymptotic Analysis, vol. 128, no. 4, pp. 555-570, 2022

Authors: Dipierro, Serena | Proietti Lippi, Edoardo | Valdinoci, Enrico

Article Type: Research Article

Abstract: We consider here a new type of mixed local and nonlocal equation under suitable Neumann conditions. We discuss the spectral properties associated to a weighted eigenvalue problem and present a global bound for subsolutions. The Neumann condition that we take into account comprises, as a particular case, the one that has been recently introduced in (Rev. Mat. Iberoam. 33 (2) (2017), 377–416). Also, the results that we present here find a natural application to a logistic equation motivated by biological problems that has been recently considered in (Dipierro, Proietti Lippi and Valdinoci (2021)).

Keywords: Long-range interactions, zero-flux condition, spectral theory, boundedness of subsolutions

DOI: 10.3233/ASY-211718

Citation: Asymptotic Analysis, vol. 128, no. 4, pp. 571-594, 2022