Asymptotic Analysis - Volume 122, issue 1-2

Authors: Castro, Carlos | Münch, Arnaud

Article Type: Research Article

Abstract: The Petrowsky type equation y t t ε + ε y x x x x ε − y x x ε = 0 , ε > 0 encountered in linear beams theory is null controllable through Neumann boundary controls. Due to the boundary layer of size of order ε occurring at the extremities, these boundary controls get singular as ε goes to 0. Using the matched asymptotic method, we describe the boundary layer of the solution y …ε and derive a rigorous second order asymptotic expansion of the control of minimal weighted L 2 -norm, with respect to the parameter ε . The weight in the norm is chosen to guarantee the smoothness of the control. In particular, we recover and enrich earlier results due to J.-L. Lions in the eighties showing that the leading term of the expansion is a null Dirichlet control for the limit hyperbolic wave equation. The asymptotic analysis also provides a robust discrete approximation of the control for any ε small enough. Numerical experiments support our study. Show more

Keywords: Singular controllability, boundary layers, asymptotic analysis, numerical experiments

DOI: 10.3233/ASY-201607

Citation: Asymptotic Analysis, vol. 122, no. 1-2, pp. 1-33, 2021

Authors: Zhang, Yu | Zhang, Yanyan | Wang, Jinhuan

Article Type: Research Article

Abstract: By introducing an isentropic Euler system with a new version of extended Chaplygin gas equation of state, we study two kinds of occurrence mechanism on the phenomenon of concentration and the formation of delta shock waves in the zero-exponent limit of solutions to the extended Chaplygin gas equations as the two exponents tend to zero wholly or partly. The Riemann problem is first solved. Then, we show that, as both the two exponents tend to zero, that is, the extended Chaplygin gas pressure tends to a constant, any two-shock-wave Riemann solution of the extended Chaplygin gas equations converges to a …delta-shock solution to the zero-pressure flow system, and the intermediate density between the two shocks tends to a weighted δ -measure which forms a delta shock wave; any two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution to the zero-pressure flow system, and the nonvacuum intermediate state in between tends to a vacuum. It is also shown that, as one of the exponents goes to zero, namely, the extended Chaplygin gas pressure approaches to some special generalized Chaplygin gas pressure, any two-shock-wave Riemann solution tends to a delta-shock solution to the generalized Chaplygin gas equations. Show more

Keywords: Isentropic Euler equations, extended Chaplygin gas, generalized Chaplygin gas, delta shock wave, Riemann problem, zero-exponent limit

DOI: 10.3233/ASY-201609

Citation: Asymptotic Analysis, vol. 122, no. 1-2, pp. 35-67, 2021

Authors: de Albuquerque, J.C. | do Ó, J.M. | dos Santos, E.O. | Severo, U.B.

Article Type: Research Article

Abstract: In this work we study the existence of solutions for the following class of elliptic systems involving Kirchhoff equations in the plane: m ( ‖ u ‖ 2 ) [ − Δ u + u ] = λ f ( u , v ) , x ∈ R 2 , ℓ ( ‖ v ‖ 2 ) [ − Δ v + v ] = λ g ( u , v ) , x ∈ R 2 , …where λ > 0 is a parameter, m , ℓ : [ 0 , + ∞ ) → [ 0 , + ∞ ) are Kirchhoff-type functions, ‖ · ‖ denotes the usual norm of the Sobolev space H 1 ( R 2 ) and the nonlinear terms f and g have exponential critical growth of Trudinger–Moser type. Moreover, when f and g are odd functions, we prove that the number of solutions increases when the parameter λ becomes large. Show more

Keywords: Kirchhoff systems, exponential critical growth, Trudinger–Moser inequality

DOI: 10.3233/ASY-201610

Citation: Asymptotic Analysis, vol. 122, no. 1-2, pp. 69-85, 2021

Authors: Dinh, Thu | Xin, Jack

Article Type: Research Article

Abstract: We consider diffusivity of random walks with transition probabilities depending on the number of consecutive traversals of the last traversed edge, the so called senile reinforced random walk (SeRW). In one dimension, the walk is known to be sub-diffusive with identity reinforcement function. We perturb the model by introducing a small probability δ of escaping the last traversed edge at each step. The perturbed SeRW model is diffusive for any δ > 0 , with enhanced diffusivity (≫ O ( δ 2 ) ) in the small δ regime. We further …study stochastically perturbed SeRW models by having the last edge escape probability of the form δ ξ n with ξ n ’s being independent random variables. Enhanced diffusivity in such models are logarithmically close to the so called residual diffusivity (positive in the zero δ limit), with diffusivity between O ( 1 | log δ | ) and O ( 1 log | log δ | ) . Finally, we generalize our results to higher dimensions where the unperturbed model is already diffusive. The enhanced diffusivity can be as much as O ( log − 2 δ ) . Show more

Keywords: Reinforced random walk, symmetric perturbation, enhanced diffusivity, asymptotic analysis

DOI: 10.3233/ASY-201611

Citation: Asymptotic Analysis, vol. 122, no. 1-2, pp. 87-104, 2021

Authors: Vetro, Calogero

Article Type: Research Article

Abstract: We give sufficient conditions for the existence of weak solutions to quasilinear elliptic Dirichlet problem driven by the A -Laplace operator in a bounded domain Ω. The techniques, based on a variant of the symmetric mountain pass theorem, exploit variational methods. We also provide information about the asymptotic behavior of the solutions as a suitable parameter goes to 0 + . In this case, we point out the existence of a blow-up phenomenon. The analysis developed in this paper extends and complements various qualitative and asymptotic properties for some cases described by homogeneous differential operators.

Keywords: Dirichlet boundary value problem, A-Laplace operator, asymptotic analysis, Orlicz–Sobolev space

DOI: 10.3233/ASY-201612

Citation: Asymptotic Analysis, vol. 122, no. 1-2, pp. 105-118, 2021

Authors: Aghajani, A. | Mottaghi, S.F.

Article Type: Research Article

Abstract: In this paper we consider the p -Laplace equation − Δ p u = λ f ( u ) in a smooth bounded domain Ω ⊂ R N with zero Dirichlet boundary condition, where p > 1 , λ > 0 and f : [ 0 , ∞ ) → R is a C 1 function with f ( 0 ) > 0 , f ′ ⩾ 0 …and lim t → ∞ f ( t ) t p − 1 = ∞ . For the sequence ( u λ ) 0 < λ < λ ∗ of minimal semi-stable solutions, by applying the semi-stability inequality we find a class of functions E that asymptotically behave like a power of f at infinity and show that ‖ E ( u λ ) ‖ L 1 ( Ω ) is uniformly bounded for λ < λ ∗ . Then using elliptic regularity theory we provide some new L ∞ estimates for the extremal solution u ∗ , under some suitable conditions on the nonlinearity f , where the obtained results require neither the convexity of f nor the strictly convexity of the domain. In particular, under some mild assumptions on f we show that u ∗ ∈ L ∞ ( Ω ) for N < p + 4 p / ( p − 1 ) , which is conjectured to be the optimal regularity dimension for u ∗ . Show more

Keywords: p-Laplacian, semi-stability, extremal solution, regularity, asymptotic behavior

DOI: 10.3233/ASY-201613

Citation: Asymptotic Analysis, vol. 122, no. 1-2, pp. 119-130, 2021

Authors: Bostan, Mihaï

Article Type: Research Article

Abstract: The subject matter of this work concerns the asymptotic behavior of the wave problems, when the propagation speed in one direction is much larger than in the other directions. We establish weak and strong convergence results. We appeal to homogenization arguments, based on average operators with respect to unitary groups.

Keywords: Average operators, ergodic means, unitary groups, homogenization

DOI: 10.3233/ASY-201614

Citation: Asymptotic Analysis, vol. 122, no. 1-2, pp. 131-164, 2021

Authors: Kielty, Derek

Article Type: Research Article

Abstract: Consider the eigenvalue problem generated by a fixed differential operator with a sign-changing weight on the eigenvalue term. We prove that as part of the weight is rescaled towards negative infinity on some subregion, the spectrum converges to that of the original problem restricted to the complementary region. On the interface between the regions the limiting problem acquires Dirichlet-type boundary conditions. Our main theorem concerns eigenvalue problems for sign-changing bilinear forms on Hilbert spaces. We apply our results to a wide range of PDEs: second and fourth order equations with both Dirichlet and Neumann-type boundary conditions, and a problem where …the eigenvalue appears in both the equation and the boundary condition. Show more

Keywords: Spectral theory, indefinite, singular limits, mixed boundary conditions, Dirichlet, Neumann, Laplacian, bi-Laplacian, dynamical boundary conditions

DOI: 10.3233/ASY-201615

Citation: Asymptotic Analysis, vol. 122, no. 1-2, pp. 165-200, 2021