Asymptotic Analysis - Volume 115, issue 1-2

Authors: Kon’kov, Andrej | Shishkov, Andrey

Article Type: Research Article

Abstract: We obtain sharp conditions guaranteeing that every non-negative weak solution of the inequality ∑ | α | = m ∂ α a α ( x , t , u ) − u t ⩾ f ( x , t ) g ( u ) in  R + n + 1 = R n × ( 0 , ∞ ) , m , n ⩾ 1 , stabilizes to zero …as t → ∞ . These conditions generalize the well-known Keller–Osserman condition on the growth of the function g at infinity. Show more

Keywords: Higher order evolution inequalities, nonlinearity, stabilization

DOI: 10.3233/ASY-191522

Citation: Asymptotic Analysis, vol. 115, no. 1-2, pp. 1-17, 2019

Authors: Aiyappan, S. | Jose, Editha C. | Lomerio, Ivy Carol B. | Nandakumaran, A.K.

Article Type: Research Article

Abstract: This paper is concerned with the asymptotic analysis of optimal control problems posed on a rough circular domain. The domain has two parts, namely a fixed outer part and an oscillating inner part. The period of the oscillation is of order ε > 0 , a small parameter which approaches zero and the amplitude of the oscillation is fixed. We pose a periodic control on the oscillating part of the domain and study the homogenization of this problem using an unfolding operator suitably defined for this domain. One of the novelties of this paper is that we use …the unfolding operator to characterize the optimal control in the non-homogenized level. Show more

Keywords: Homogenization, optimal control, oscillating boundary, unfolding operator, rough circular domain

DOI: 10.3233/ASY-191526

Citation: Asymptotic Analysis, vol. 115, no. 1-2, pp. 19-46, 2019

Authors: Liang, Sihua | Zhang, Binlin

Article Type: Research Article

Abstract: In this paper, we study the following Kirchhoff type problems involving fractional p -Laplacian and critical exponents: M ( [ u ] s , p p ) ( − Δ ) p s u = λ | u | p s ∗ − 2 u + b ( x ) | u | p − 2 u + f ( x , u ) , in  Ω , u = 0 in  R …N ∖ Ω , where Ω is a bounded domain in R N (N ⩾ 3 ) with Lipshcitz boundary, 1 < p < N / s with s ∈ ( 0 , 1 ) , p s ∗ = N p / ( N − p s ) is the fractional critical Sobolev exponent, λ is a positive parameter, M : [ 0 , + ∞ ) → R + and f : Ω ‾ × R → R are continuous functions, b : Ω → R is a sign-changing function. By using the fractional version of concentration compactness principle together with mountain pass theorem, we obtained the multiplicity of solutions for the above problem. Show more

Keywords: Kirchhoff type problem, fractional p-Laplacian, mountain pass theorem, critical exponent, concentration–compactness principle

DOI: 10.3233/ASY-191527

Citation: Asymptotic Analysis, vol. 115, no. 1-2, pp. 47-61, 2019

Authors: Calvez, Vincent | Gabriel, Pierre | Mateos González, Álvaro

Article Type: Research Article

Abstract: Subdiffusive motion takes place at a much slower timescale than diffusive motion. As a preliminary step to studying reaction-subdiffusion pulled fronts, we consider here the hyperbolic limit ( t , x ) → ( t / ε , x / ε ) of an age-structured equation describing the subdiffusive motion of, e.g. , some protein inside a biological cell. Solutions of the rescaled equations are known to satisfy a Hamilton–Jacobi equation in the formal limit ε → 0 . In this work we derive uniform Lipschitz estimates, and establish the convergence towards the viscosity solution …of the limiting Hamilton–Jacobi equation. The two main obstacles overcome in this work are the non-existence of an integrable stationary measure, and the importance of memory terms in subdiffusion. Show more

Keywords: Age-structured PDE, renewal equation, anomalous diffusion, WKB approximation, Hamilton–Jacobi equation

DOI: 10.3233/ASY-191528

Citation: Asymptotic Analysis, vol. 115, no. 1-2, pp. 63-94, 2019

Authors: Li, Zhiyuan | Kian, Yavar | Soccorsi, Éric

Article Type: Research Article

Abstract: We examine initial-boundary value problems for diffusion equations with distributed order time-fractional derivatives. We prove existence and uniqueness results for the weak solution to these systems, together with its continuous dependency on initial value and source term. Moreover, we provide energy estimates of the solution for small and large times. Finally, under suitable assumption on the source term, we establish that the solution is analytic in time.

Keywords: Diffusion equation, distributed order time-fractional derivative, initial boundary value problem, small-time and large-time energy estimates

DOI: 10.3233/ASY-191532

Citation: Asymptotic Analysis, vol. 115, no. 1-2, pp. 95-126, 2019