Alain Miranville | Université du Havre (original) (raw)

Papers by Alain Miranville

Mediterranean Journal of Mathematics

Mathematical Medicine and Biology: A Journal of the IMA

Our aim in this paper is to study a mathematical model for high grade gliomas, taking into accoun... more Our aim in this paper is to study a mathematical model for high grade gliomas, taking into account lactates kinetics, as well as chemotherapy and antiangiogenic treatment. In particular, we prove the existence and uniqueness of biologically relevant solutions. We also perform numerical simulations based on different therapeutical situations that can be found in the literature. These simulations are consistent with what is expected in these situations.

Mediterranean Journal of Mathematics, 2019

Our aim in this paper is to prove the existence of solutions to a Cahn-Hilliard type equation wit... more Our aim in this paper is to prove the existence of solutions to a Cahn-Hilliard type equation with a proliferation term and a logarithmic nonlinear term. Such an equation was proposed in view of biological applications. The main difficulty comes from the fact that we no longer have the conservation of the spatial average of the order parameter, contrary to the original Cahn-Hilliard equation. This makes the derivation of uniform (with respect to the regularization parameter) estimates on the solutions to approximated problems delicate, as blow up in finite time may occur.

International Mathematics Research Notices, 2021

We consider a nonlinear delay evolution equation with multivalued perturbation on a noncompact in... more We consider a nonlinear delay evolution equation with multivalued perturbation on a noncompact interval. The nonlinearity, having convex and closed values, is upper hemicontinuous with respect to the solution variable. A basic question on whether there exists a solution set carrying RdeltaR_{\delta }Rdelta-structure remains unsolved when the operator families generated by the principal part lack compactness. One of our main goals is to settle this question in the affirmative. Moreover, we prove that the solution map, having compact values, is an RdeltaR_{\delta }Rdelta-map, which maps any connected set into a connected set. It is then exploited to deal with the existence in the large for a nonlocal problem. Finally, several examples are worked out in detail, illustrating the applicability of our general results.

Applied Mathematics & Optimization, 2020

Our aim in this paper is to study a Cahn–Hilliard model with a symport term. This equation is pro... more Our aim in this paper is to study a Cahn–Hilliard model with a symport term. This equation is proposed to model some energy mechanisms (e.g., lactate) in glial cells. The main difficulty is to prove the existence of a biologically relevant solution. This is achieved by considering a modified equation and taking a logarithmic nonlinear term. A second difficulty is to prove additional regularity on the solutions which is essential to prove a strict separation from the pure states 0 and 1 in one and two space dimensions. We also consider a second model, based on the Cahn–Hilliard–Oono equation.

Our aim is to prove the existence and uniqueness of solutions for one-dimensional Cahn-Hilliard a... more Our aim is to prove the existence and uniqueness of solutions for one-dimensional Cahn-Hilliard and Allen-Cahn type equations based on a modification of the Ginzburg-Landau free energy proposed in [8]. In particular, the free energy contains an additional term called Willmore regularization and takes into account strong anisotropy effects.

Journal of Differential Equations, 2021

Abstract Our aim in this paper is to prove the existence of solutions to the Cahn–Hilliard equati... more Abstract Our aim in this paper is to prove the existence of solutions to the Cahn–Hilliard equation with a general nonlinear source term. An essential difficulty is to obtain a global in time solution. Indeed, due to the presence of the source term, one cannot exclude the possibility of blow up in finite time when considering regular nonlinear terms and when considering an approximated scheme. Considering instead logarithmic nonlinear terms, we give sufficient conditions on the source term which ensure the existence of a global in time weak solution. These conditions are satisfied by several important models and applications which can be found in the literature.

Communications on Pure & Applied Analysis, 2020

The main goal of this paper is to study the asymptotic behavior of a coupled Cahn-Hilliard/Allen-... more The main goal of this paper is to study the asymptotic behavior of a coupled Cahn-Hilliard/Allen-Cahn system with temperature. The work is divided into two parts: In the first part, the heat equation is based on the usual Fourier law. In the second one, it is based on the type III heat conduction law. In both parts, we prove the existence of exponential attractors and, therefore, of finite-dimensional global attractors.

Applied Mathematics Letters, 2019

Portal del coneixement obert de la UPC http://upcommons.upc.edu/e-prints Aquesta és una còpia de ... more Portal del coneixement obert de la UPC http://upcommons.upc.edu/e-prints Aquesta és una còpia de la versió author's final draft d'un article publicat a la revista Applied mathematics letters.

AIMS Mathematics, 2017

Our aim in this article is to review and discuss the Cahn–Hilliard equation, as well as someof it... more Our aim in this article is to review and discuss the Cahn–Hilliard equation, as well as someof its variants. Such variants have applications in, e.g., biology and image inpainting.

Cubo (Temuco), 2015

Our aim in this paper is to prove the existence and uniqueness of solutions for an Allen-Cahn typ... more Our aim in this paper is to prove the existence and uniqueness of solutions for an Allen-Cahn type system based on a modification of the Ginzburg-Landau free energy proposed in [11]. In particular, the free energy contains an additional term called Willmore regularization and takes into account anisotropy effects. RESUMEN Nuestro propósito en este trabajo es probar la existencia y unicidad de soluciones para un Sistema de tipo Allen-Cahn basados en una modificación de la energía libre Ginzburg-Landau propuesta en [11]. En particular, la energía libre contiene un término adicional llamado regularización de Willmore y considera efectos de anisotropía.

Communications on Pure and Applied Analysis

This preface gives a short scientific biography of Michel Pierre.

Discrete and Continuous Dynamical Systems, 2013

will celebrate his seventieth birthday on December 11, 2012, most likely in Heusy (Verviers), a l... more will celebrate his seventieth birthday on December 11, 2012, most likely in Heusy (Verviers), a lovely city in the foothills of the Ardennes, in Belgium, where he was born and resided all his life. He received his pre-university education there and then continued with his studies of mathematics at the Université de Liège, where he received the degree of Docteur en Sciences Mathématiques, avec la plus grande distinction on February 10, 1969. He continued as Maître de Conférences at Liège from 1969 until 1973 and simultaneously was appointed as Chargé de Cours at the (newly established) Université Catholique de Louvain at Louvain-la-Neuve, from 1970 to 1974, where he then served as Professeur from 1974 to 1977 and was promoted to Professeur Ordinaire in 1977. Jean's career as a mathematician has been unique and extraordinarily successful. His scientific contributions to the fields Nonlinear Differential Equations, Nonlinear Analysis, Variational Methods, History of Mathematics have been very substantial, deep and often trendsetting. His outstanding skills have been used in manifold ways: organizing relevant conferences on significant topics, giving superbly prepared lectures on pertinent research subjects, preparing many students for successful careers in mathematics and other academic fields. His achievements have been honored by invitations by many Universities of visiting professorships, by conferences to present plenary and distinguished lectures, by Academies with honorary memberships and Universities with honorary doctorates, besides several other awards and honorific distinctions. He has also been appointed President de l'Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique. Belgian law required his retirement at age sixty-five and he was then appointed ProfesseurÉmérite by his university. Yet his scientific interests and productivity were in no way diminished and they and the number of invitations to present lectures and the number of visiting professorships have increased since. While still being a student, just after having completed his (at that time compulsory) military service, Jean married Marguerite Bellefontaine, who tirelessly encouraged him throughout his career, and gave him three children. Jean's and Marguerite's marriage is a true confirmation of the dictum that "behind every great man there stands a great woman". All who know Jean Mawhin are thoroughly impressed by his warm personality, his keen sense of humor, his modesty, and his strict and high professional standards. His mathematical life and career, his clear interest in the work of others, have had striking impacts on the development of research in his home country and many other research centers around the world. His numerous students, as well as his great number of collaborators (we, the three editors of this special volume are among those), will always be grateful for his friendship, patient collaboration, and continuous encouragement. We extend our warmest and best wishes to Jean for many more years of good health and many more exciting discoveries.

Discrete and Continuous Dynamical Systems - Series S, 2012

Our aim in this paper is to define proper dynamic boundary conditions for a generalization of the... more Our aim in this paper is to define proper dynamic boundary conditions for a generalization of the Cahn-Hilliard system proposed by M. Gurtin. Such boundary conditions take into account the interactions with the walls in confined systems. We then study the existence and uniqueness of weak solutions. 2010 Mathematics Subject Classification. 35K55.

Applications of Mathematics, 2015

Our aim in this paper is to study the existence of solutions to a phase-field system based on the... more Our aim in this paper is to study the existence of solutions to a phase-field system based on the Maxwell-Cattaneo heat conduction law, with a logarithmic nonlinearity. In particular, we prove, in one and two space dimensions, the existence of a solution which is separated from the singularities of the nonlinear term.

Abstract. We consider a model of non-isothermal phase separation taking place in a confined conta... more Abstract. We consider a model of non-isothermal phase separation taking place in a confined container. The order parameter φ is governed by a viscous or non-viscous Cahn-Hilliard type equation which is coupled with a heat equation for the temperature θ. The former is subject to a nonlinear dynamic boundary condition recently proposed by physicists to account for interactions with the walls, while the latter is endowed with a standard (Dirichlet, Neumann or Robin) boundary condition. We indicate by α the viscosity coefficient, by ε a (small) relaxation parameter multiplying ∂tθ in the heat equation and by δ a small latent heat coefficient (satisfying δ ≤ λα, δ ≤ λε, λ, λ> 0) multiplying ∆θ in the Cahn-Hilliard equation and ∂tφ in the heat equation. Then, we construct a family of exponential attractorsMε,δ,α which is a robust perturbation of an exponential attractor M0,0,α of the (isothermal) viscous (α> 0) Cahn-Hilliard equation, namely, the symmetric Hausdorff distance between...