Long-time dynamics for a Cahn–Hilliard tumor growth model with chemotaxis

Garcke, Harald; Yayla, Sema

doi:10.1007/s00033-020-01351-3

Cited by 7 publications

(6 citation statements)

References 18 publications

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“…It is easy to see that the critical points of the functional

scriptG false(u false) : = \frac{1}{2} {(‖‖, {false(- normalΔ false)}^{1 false/ 2} u)}_{L_{2} false(normalΩ false)}^{2} + \int_{normalΩ} false(F ((), u false(x false)) - f false(0 false) u false(x false) false) d x + \int_{normalΩ} {false(- normalΔ false)}^{- 1} h false(x false) u false(x false) d x

are the strong solutions of problem (). Moreover, by using the standard techniques, it is easy to prove the following lemma (see, e.g., Garcke and Yayla 35 ).…”

Section: Global Attractorsmentioning

confidence: 99%

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Long‐time dynamics of the 2D nonisothermal hyperbolic Cahn–Hilliard equations

Yayla

2022

Math Methods in App Sciences

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In this paper, we consider the hyperbolic relaxation of the nonisothermal Cahn-Hilliard equation based on either Fourier law or Maxwell-Cattaneo law for heat conduction. In the Maxwell-Cattaneo case, we reformulate the problem by using enthalpy instead of relative temperature. In both cases, we prove the existence of the global attractor for the weak solutions of the problem. Moreover, for both laws, we establish that every full trajectory in the global attractor converges to a single stationary point as t → ∞ and another single stationary point as t → −∞. Consequently, we infer that the global attractor is equal to a union of the unstable manifolds emanating from the stationary points.

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“…It is easy to see that the critical points of the functional

scriptG false(u false) : = \frac{1}{2} {(‖‖, {false(- normalΔ false)}^{1 false/ 2} u)}_{L_{2} false(normalΩ false)}^{2} + \int_{normalΩ} false(F ((), u false(x false)) - f false(0 false) u false(x false) false) d x + \int_{normalΩ} {false(- normalΔ false)}^{- 1} h false(x false) u false(x false) d x

are the strong solutions of problem (). Moreover, by using the standard techniques, it is easy to prove the following lemma (see, e.g., Garcke and Yayla 35 ).…”

Section: Global Attractorsmentioning

confidence: 99%

“…are the strong solutions of problem (3.2). Moreover, by using the standard techniques, it is easy to prove the following lemma (see, e.g., Garcke and Yayla 35 ).…”

Section: Set Of the Stationary Pointsmentioning

confidence: 99%

Long‐time dynamics of the 2D nonisothermal hyperbolic Cahn–Hilliard equations

Yayla

2022

Math Methods in App Sciences

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“…In the past several decades, there have been many recent contributions regarding the wellposedness, asymptotic behavior, and optimal control for phase field tumor growth models; see, for example, [3][4][5][6][7][8][9]15,[18][19][20][21]23,[32][33][34][36][37][38][39] where the velocity contributions were neglected, and [12,13,16,17,[27][28][29][30][31]43] considering the velocity contributions. In this paper, we focus on the case that velocity contributions are neglected.…”

Section: Introductionmentioning

confidence: 99%

Optimal control of a phase field tumor growth model with chemotaxis and active transport

2023

J. Nonlinear Var. Anal.

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This paper is concerned with a distributed optimal control problem for a phase field model describing tumor growth with chemotaxis and active transport. First, comparing with the results in [H. Garcke, K.F. Lam, Well-posedness of a Cahn-Hilliard system modelling tumour growth with chemotaxis and active transport, European J. Appl. Math. 28 (2017), 284-316], we prove the existence of solutions for such a system with more general potential, the regularity of solutions and the continuous dependence of initial data as well as control variable with respect to a strong topology. It is worth pointing out that the potentials cover the case of classical quartic double-well potential, which is the standard approximation for the physical relevant logarithmic potential. Furthermore, the existence of an optimal control is proved by monotonicity arguments and compactness theorems. Beyond that, by overcoming some difficulties in mathematical analysis and calculation, especially in the proof of the Fréchet differentiability of the control-to-state operator, we derive the corresponding first-order necessary conditions of optimality in terms of the adjoint variables and the usual variational inequality.

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“…Multi-timescale dynamics exist in various research fields [1][2][3][4][5]. The common characteristics of multi-timescale dynamic systems are that state variables interact on different time scales, leading to complex bursting oscillation patterns.…”

Section: Introductionmentioning

confidence: 99%

Bursting dynamics and the bifurcation mechanism of a modified Rayleigh-van der Pol-Duffing oscillator

Zhao¹,

Ma²,

Zhang³

et al. 2022

Phys. Scr.

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In this paper, complex bursting patterns in a modified Rayleigh–van der Pol-Duffing oscillator(abbreviated as MRPDO) under the simultaneous effect of parametric and external excitations are given theoretically by fast-slow analysis method. Four different kinds of bursting oscillations, i.e. bursting of compound asymmetric “Homoclinic/Homoclinic” type, bursting of compound asymmetric “delayed Homoclinic/Homoclinic” type, bursting of compound “fold/delayed Homoclinic connection/supHopf-supHopf/supHopf” type and bursting of “fold/delayed Homoclinic” type, are studied. For bursting of compound “fold/delayed Homoclinic connection/supHopf-supHopf/supHopf” type, we explain the delay generation mechanism. Based on the characteristics of the Homoclinic-limit cycles, we propose the concepts of the approximate quiescent state and peculiar limit cycle attractor.

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Long-time dynamics for a Cahn–Hilliard tumor growth model with chemotaxis

Cited by 7 publications

References 18 publications

Long‐time dynamics of the 2D nonisothermal hyperbolic Cahn–Hilliard equations

Long‐time dynamics of the 2D nonisothermal hyperbolic Cahn–Hilliard equations

Optimal control of a phase field tumor growth model with chemotaxis and active transport

Bursting dynamics and the bifurcation mechanism of a modified Rayleigh-van der Pol-Duffing oscillator

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