Asymptotic Analysis - Volume 133, issue 4

Authors: Cantin, Guillaume

Article Type: Research Article

Abstract: In this paper, we study the asymptotic behaviour of the solutions to a degenerate reaction–diffusion system. This system admits a continuum of discontinuous stationary solutions due to the effect of a hysteresis process, but only one discontinuous stationary solution is compatible with a principle of preservation of locally invariant regions. Using a macroscopic mass effect which guarantees that fast particles help slow particles to displace, we establish a novel result of convergence of a non trivial set of trajectories towards a discontinuous pattern.

Keywords: Reaction-diffusion, degenerate, discontinuous pattern, asymptotic behavior, hysteresis

DOI: 10.3233/ASY-221818

Citation: Asymptotic Analysis, vol. 133, no. 4, pp. 447-462, 2023

Authors: Pan, Guofu | Ji, Chao

Article Type: Research Article

Abstract: In this paper, we study the least energy sign-changing solutions to the following nonlinear Kirchhoff equation − ( a + b ∫ V | ∇ u | 2 d μ ) Δ u + c ( x ) u = f ( u ) on a locally finite graph G = ( V , E ) , where a , b are positive constants. We use the constrained variational method to prove the existence of a least energy sign-changing …solution u b of the above equation if c ( x ) and f satisfy certain assumptions, and to show the energy of u b is strictly larger than twice that of the least energy solutions. Moreover, if we regard b as a parameter, as b → 0 + , the solution u b converges to a least energy sign-changing solution of a local equation − a Δ u + c ( x ) u = f ( u ) . Show more

Keywords: Kirchhoff equations, locally finite graphs, sign-changing solutions, variational methods

DOI: 10.3233/ASY-221819

Citation: Asymptotic Analysis, vol. 133, no. 4, pp. 463-482, 2023

Authors: Jabin, Pierre-Emmanuel | Perthame, Benoît

Article Type: Research Article

Abstract: A classical problem describing the collective motion of cells, is the movement driven by consumption/depletion of a nutrient. Here we analyze one of the simplest such model written as a coupled Partial Differential Equation/Ordinary Differential Equation system which we scale so as to get a limit describing the usually observed pattern. In this limit the cell density is concentrated as a moving Dirac mass and the nutrient undergoes a discontinuity. We first carry out the analysis without diffusion, getting a complete description of the unique limit. When diffusion is included, we prove several specific a priori estimates and interpret …the system as a heterogeneous monostable equation. This allow us to obtain a limiting problem which shows the concentration effect of the limiting dynamics. Show more

Keywords: Asymptotic analysis, pattern formation, reaction–diffusion equations

DOI: 10.3233/ASY-221820

Citation: Asymptotic Analysis, vol. 133, no. 4, pp. 483-497, 2023

Authors: Mheich, Rim

Article Type: Research Article

Abstract: We will study in this article the nonlinear Cahn–Hilliard equation with proliferation and regularization terms with regular and logarithmic potentials. First, we consider the regular potential case, we show that the solutions blow up in finite time or exist globally in time. Furthermore, we prove that the model possess a global attractor. In addition, we construct a robust family of exponential attractors, i.e. attractors which are continuous with respect to the perturbation parameter. In the second part, we consider the logarithmic potential case and show the existence of a global solution.

Keywords: Cahn–Hilliard Equation, global attractor, exponential attractor, well-posedness, logarithmic nonlinear term, blow up, existence, regularization term, regular nonlinear term

DOI: 10.3233/ASY-221821

Citation: Asymptotic Analysis, vol. 133, no. 4, pp. 499-533, 2023

Authors: Tao, Huo | Li, Lin | Yang, Xiao-Qiong

Article Type: Research Article

Abstract: In this paper, we study the existence results of a quasilinear elliptic problem involving the 1-biharmonic operator in R N , whose nonlinearity satisfies appropriate conditions. The existence theorem is proved through a new version of the Mountain Pass Theorem to locally Lipschitz functionals, where it is considered the Cerami compactness condition rather than the Palais–Smale one.

Keywords: 1-biharmonic operator, BL space, Variational methods

DOI: 10.3233/ASY-221822

Citation: Asymptotic Analysis, vol. 133, no. 4, pp. 535-553, 2023

Authors: Raad, Hussein | Cherfils, Laurence | Allery, Cyrille | Guillevin, Rémy

Article Type: Research Article

Abstract: The aim of this paper is to first study a Cahn-Hilliard model for brain lactate kinetics with a control function. This control allows for optimal treatment administered to ill patients suffering from glioma, in order to reduce their brain lactate concentrations, and thereby to slow down the tumor growth. We establish the well-posedness of the problem and the continuity of the control-to-state mapping, the existence of a minimizer of the objective functional, and its Fréchet differentiability in suitable Banach spaces with respect to the control and with respect to time. Moreover, we derive the first-order necessary conditions that an optimal …control has to satisfy. In the second part of the paper, we illustrate our theoretical results with numerical simulations using MRI data from the University Hospital of Poitiers. Show more

Keywords: Cahn-Hilliard equation, well-posedness, optimal control, lactate kinetics, adjoint equations, Fréchet differentiability with respect to the control, numerical simulations

DOI: 10.3233/ASY-221823

Citation: Asymptotic Analysis, vol. 133, no. 4, pp. 555-586, 2023