On the long time behavior of a tumor growth model
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TLDR
In this article, the authors consider the problem of the long time dynamics for a diffuse interface model for tumor growth and prove that the corresponding initial-boundary value problem generates a dissipative dynamical system that admits the global attractor in a proper phase space.About:
This article is published in Journal of Differential Equations.The article was published on 2019-08-05 and is currently open access. It has received 53 citations till now. The article focuses on the topics: Attractor & Dissipative system.read more
Citations
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On a phase field model of Cahn–Hilliard type for tumour growth with mechanical effects
TL;DR: In this article, the authors studied a macroscopic mechanical model for phase field tumour growth in which cell-cell adhesion effects are taken into account with the help of a Ginzburg-Landau type energy.
Journal ArticleDOI
A Distributed Control Problem for a Fractional Tumor Growth Model
TL;DR: In this article, the distributed optimal control of a system of three evolutionary equations involving fractional powers of three self-adjoint, monotone, unbounded linear operators having compact resolvents was studied.
Journal ArticleDOI
Optimal control of a phase field system modelling tumor growth with chemotaxis and singular potentials
TL;DR: In this article, a distributed optimal control problem for an extended model of phase field type for tumor growth is addressed, where the chemotaxis effects are also taken into account, and the control is realized by two control variables that design the dispensation of some drugs to the patient.
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Mathematical analysis and simulation study of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects
Pierluigi Colli,Hector Gomez,Guillermo Lorenzo,Gabriela Marinoschi,Alessandro Reali,Elisabetta Rocca +5 more
TL;DR: The standard approach relies on cytotoxic drugs, which aim at inhibiting proliferation and promoting cell death as mentioned in this paper, which is a common treatment for advanced prostate cancer. Advanced prostatic...
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Weak and stationary solutions to a Cahn–Hilliard–Brinkman model with singular potentials and source terms
Matthias Ebenbeck,Kei Fong Lam +1 more
TL;DR: In this paper, a phase field model with singular potentials was proposed and the existence of weak and stationary solutions to the Cahn-Hilliard-brinkman inpainting model was shown.
References
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Journal ArticleDOI
Free Energy of a Nonuniform System. I. Interfacial Free Energy
John W. Cahn,John E. Hilliard +1 more
TL;DR: In this article, it was shown that the thickness of the interface increases with increasing temperature and becomes infinite at the critical temperature Tc, and that at a temperature T just below Tc the interfacial free energy σ is proportional to (T c −T) 3 2.
Book
Infinite-Dimensional Dynamical Systems in Mechanics and Physics
TL;DR: In this article, the authors give bounds on the number of degrees of freedom and the dimension of attractors of some physical systems, including inertial manifolds and slow manifolds.
Book
Attractors of Evolution Equations
A. Babin,Mark I. Vishik +1 more
TL;DR: In this article, the authors present an overview of Semigroups of linear operators and their relation to a parameter of Attractors of Differentiable Semigroup and Uniform Asymptotics of Trajectories.
Journal ArticleDOI
Three-dimensional multispecies nonlinear tumor growth--I Model and numerical method.
TL;DR: This is the first paper in a two-part series in which a diffuse interface continuum model of multispecies tumor growth and tumor-induced angiogenesis in two and three dimensions is developed, analyzed, and simulated.
Book ChapterDOI
Chapter 3 Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains
Alain Miranville,Sergey Zelik +1 more
TL;DR: In this paper, the authors considered the problem of dissipative dynamical systems in unbounded domains and showed that the dynamics generated by dissipative PDEs in such domains are purely infinite dimensional and do not possess any finite dimensional reduction principle.