Global solutions for a hyperbolic–parabolic system of chemotaxis

Granero-Belinchón, Rafael

doi:10.1016/j.jmaa.2016.12.050

Cited by 21 publications

(9 citation statements)

References 27 publications

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“…Indeed, some components in tumor development, such as immune cells, exhibit an anomalous diffusion dynamics (as it observed in experiments [28]), but other components, like chemical potential and nutrient concentration are possibly governed by different fractional or non-fractional flows. However, taking all this into account, it is the case of pointing out that fractional operators are becoming more and more implemented in the field of biological applications: to this concern, a selection of notable and meaningful references is given by [1,7,28,29,45,48,50,51,54,62,64,70].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic analysis of a tumor growth model with fractional operators

Colli

Gilardi

Sprekels

2020

ASY

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In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalized and relaxed version of a phase field system of Cahn-Hilliard type modelling tumor growth that has originally been proposed in Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3-24). The original phase field system and certain relaxed versions thereof have been studied in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn-Hilliard equation for the tumor cell fraction ϕ, coupled to a reaction-diffusion equation for a function S representing the nutrient-rich extracellular water volume fraction. Effects due to fluid motion are neglected. Motivated by the possibility that the diffusional regimes governing the evolution of the different constituents of the model may be of different (e.g., fractional) type, the present authors studied in a recent note a generalization of the systems investigated in the abovementioned works. Under rather general assumptions, well-posedness and regularity results have been shown. In particular, by writing the equation governing the evolution of the chemical potential in the form of a general variational inequality, also singular or nonsmooth contributions of logarithmic or of double obstacle type to the energy density could be admitted. In this note, we perform an asymptotic Colli -Gilardi -Sprekels analysis of the governing system as two (small) relaxation parameters approach zero separately and simultaneously. Corresponding well-posedness and regularity results are established for the respective cases; in particular, we give a detailed discussion which assumptions on the admissible nonlinearities have to be postulated in each of the occurring cases.

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Section: Introductionmentioning

confidence: 99%

Asymptotic analysis of a tumor growth model with fractional operators

Colli

Gilardi

Sprekels

2020

ASY

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“…To the best of our knowledge, the question of global existence vs. finite time blow up of the fully parabolic Keller-Segel system (τ, κ > 0) with fractional diffusion and arbitrary initial data remains open (compare with Wu & Zheng [79] and [18]). Similarly, as far as we know, the finite time blow up for the parabolic-hyperbolic case (the extreme case κ = 0), remains an open problem even for low values of α, compare [33,34].…”

Section: 31mentioning

confidence: 95%

Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations

Burczak¹,

Granero-Belinchón²

2020

Discrete &Amp; Continuous Dynamical Systems - S

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In this paper we consider a d-dimensional (d = 1, 2) parabolic-elliptic Keller-Segel equation with a logistic forcing and a fractional diffusion of order α ∈ (0, 2). We prove uniform in time boundedness of its solution in the supercritical range α > d (1 − c), where c is an explicit constant depending on parameters of our problem. Furthermore, we establish sufficient conditions for u(t) − u∞ L ∞ → 0, where u∞ ≡ 1 is the only nontrivial homogeneous solution. Finally, we provide a uniqueness result.1 1 Observe that the Patlak-Keller-Segel equation is often written for unknowns (u, −v)

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“…, global existence and exponential decay rates of solutions under Neumann boundary conditions were obtained in [24] for small data, and local existence of solutions in two dimensions with Dirichlet boundary conditions was given in [16]. Finally we mention that when the Laplacian (diffusion) in (1.1) was modified to a fractional Laplacian, the global existence of solutions of (1.1) in a torus with periodic boundary conditions in some dissipation regimes was established in [5,6].…”

Section: Introductionmentioning

confidence: 91%

On a parabolic-hyperbolic chemotaxis system with discontinuous data: Well-posedness, stability and regularity

Peng

Wang

2020

Journal of Differential Equations

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The global dynamics and regularity of parabolic-hyperbolic systems is an interesting topic in PDEs due to the coupling of competing dissipation and hyperbolic effects. This paper is concerned with the Cauchy problem of a parabolic-hyperbolic system derived from a chemotaxis model describing the dynamics of the initiation of tumor angiogenesis. It is shown that, as time tends to infinity, the Cauchy problem with large-amplitude discontinuous data admit global weak solutions which converge to a constant state (resp. a viscous shock wave) if the asymptotic states of initial values at far field are equal (resp. unequal). Our results improve the previous results where initial value was required to be continuous and have small amplitude. Numerical simulations are performed to verify our analytical results, illustrate the possible regularity of solutions and speculate the minimal regularity of initial data required to obtain the smooth (classical) solutions of the concerned parabolic-hyperbolic system. MSC 2000: 35A01, 35B40, 35B44, 35K57, 35Q92, 92C17 KEYWORDS: Parabolic-hyperbolic system, discontinuous initial data, weak solutions, effective viscous flux, regularity with the initial valueand the far-field behavior (i.e., asymptotic state at ±∞):where u ± ≥ 0, χ and D are positive constants. The system (1.1) is transformed from the following PDE-ODE singular chemotaxis model proposed in [19] (see [18,28] for mathematical derivation) to describe

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Global solutions for a hyperbolic–parabolic system of chemotaxis

Abstract: We study a hyperbolic-parabolic model of chemotaxis in dimensions one and two. In particular, we prove the global existence of classical solutions in certain dissipation regimes.

Cited by 21 publications

References 27 publications

Asymptotic analysis of a tumor growth model with fractional operators

Asymptotic analysis of a tumor growth model with fractional operators

Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations

On a parabolic-hyperbolic chemotaxis system with discontinuous data: Well-posedness, stability and regularity

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