Asymptotic Analysis - Volume 117, issue 1-2

Authors: Hutridurga, Harsha | Mula, Olga | Salvarani, Francesco

Article Type: Research Article

Abstract: This article addresses the homogenization of linear Boltzmann equation when the optical parameters are highly heterogeneous in the energy variable. We employ the method of two-scale convergence to arrive at the homogenization result. In doing so, we show the induction of a memory effect in the homogenization limit. We also provide some numerical experiments to illustrate our results. In the Appendix, we treat a related case of the harmonic oscillator.

Keywords: Linear Boltzmann equation, self-shielding, two-scale convergence, memory effects

DOI: 10.3233/ASY-191544

Citation: Asymptotic Analysis, vol. 117, no. 1-2, pp. 1-25, 2020

Authors: Morisse, Baptiste

Article Type: Research Article

Abstract: In this paper we study the action of pseudo-differential operators on Gevrey spaces. We introduce classes of classical symbols with spatial Gevrey regularity. As the spatial Gevrey regularity of a symbol p ( · , ξ ) may depend on the frequency ξ , the action of the associated pseudo-differential operator op ( p ) may induce a loss of regularity. The proof is based on a para-product decomposition.

Keywords: Gevrey regularity, pseudo-differential operators, para-product

DOI: 10.3233/ASY-191545

Citation: Asymptotic Analysis, vol. 117, no. 1-2, pp. 27-42, 2020

Authors: Signori, Andrea

Article Type: Research Article

Abstract: A distributed optimal control problem for a phase field system which physical context is that of tumor growth is discussed. The system we are going to take into account consists of a Cahn–Hilliard equation for the phase variable (relative concentration of the tumor) coupled with a reaction-diffusion equation for the nutrient. The cost functional is of standard tracking-type and the control variable models the intensity at which it is possible to dispense medication. The model we deal with presents two small and positive parameters which are introduced in previous contributions as relaxation terms. Here, starting from the already investigated optimal …control problem for the relaxed model, we aim at confirming the existence of optimal control and characterizing the first-order necessary optimality condition, via asymptotic schemes, when one of the two occurring parameters goes to zero. Show more

Keywords: Asymptotic analysis, distributed optimal control, phase field model, tumor growth, cancer treatment, evolution equations, Cahn–Hilliard equation, optimal control, adjoint system, necessary optimality conditions

DOI: 10.3233/ASY-191546

Citation: Asymptotic Analysis, vol. 117, no. 1-2, pp. 43-66, 2020

Authors: Almeida, A.F. | Cavalcanti, M.M. | Gonzalez, R.B. | Gonzalez Martinez, V.H. | Zanchetta, J.P.

Article Type: Research Article

Abstract: We consider a coupled semilinear wave system posed in an inhomogeneous medium, with smooth boundary, subject to a nonlinear damping distributed around a neighborhood of the boundary according to the Geometric Control Condition. We show that the energy of the coupled system goes uniformly to zero, for all initial data of finite energy taken in bounded sets of finite energy phase-space. The approach involves refined techniques of microlocal analysis and follows ideas due to Burq and Gérard given in (Burq and Gérard (2001 )).

Keywords: Coupled system, observability, uniform decay rate

DOI: 10.3233/ASY-191547

Citation: Asymptotic Analysis, vol. 117, no. 1-2, pp. 67-111, 2020

Authors: Chen, Wenhui

Article Type: Research Article

Abstract: In this paper we consider elastic waves with Kelvin–Voigt damping in 2D. For the linear problem, applying pointwise estimates of the partial Fourier transform of solutions in the Fourier space and asymptotic expansions of eigenvalues and their eigenprojections, we obtain sharp energy decay estimates with additional L m regularity and L p − L q estimates on the conjugate line. Furthermore, we derive asymptotic profiles of solutions under different assumptions of initial data. For the semilinear problem, we use the derived L 2 …− L 2 estimates with additional L m regularity to prove global (in time) existence of small data solutions to the weakly coupled system. Finally, to deal with elastic waves with Kelvin–Voigt damping in 3D, we apply the Helmholtz decomposition. Show more

Keywords: Elastic waves, Kelvin–Voigt damping, decay property, asymptotic profile, weakly coupled system, global existence

DOI: 10.3233/ASY-191548

Citation: Asymptotic Analysis, vol. 117, no. 1-2, pp. 113-140, 2020