Asymptotic Dynamics of a New Mechanochemical Model in Biological Patterns

Liu, Aibo; Liu, Changchun

doi:10.3846/13926292.2017.1292324

Cited by 4 publications

(2 citation statements)

References 18 publications

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“…While there exist some numerical simulations of the new mechanochemical model (see previous studies 1,2 ), there are still only one results of the mathematical analysis of this model. Liu and Liu 3 proved the existence of attractor for the mechanochemical model.…”

Section: Introductionmentioning

confidence: 99%

Optimal control of a new mechanochemical model with state constraint

Liu

Zhang

2021

Math Methods in App Sciences

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In this paper, a state‐constrained optimal control problem is considered for the new mechanochemical model in biological patterns modelling the skin coating of some vertebrate marine animals. The model consists of the time‐dependent Ginzburg–Landau equation for the concentration difference of at least two pigments u coupled with the Swift–Hohenberg equation for the difference of dermal cellular densities of at least two types of cells. The objective is to force the state to be as close as possible to a desired state. Considering that the cost functional is discontinuous, which along with state constraint, we apply a new penalty functional to approximate the cost functional, in this case, so we first prove the existence of optimal control. Then we derive the necessary optimality conditions for the approximating optimal control problem. Finally, the necessary optimality conditions is obtained by considering the limits of the results for the approximating optimal control problem.

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Section: Introductionmentioning

confidence: 99%

Optimal control of a new mechanochemical model with state constraint

Liu

Zhang

2021

Math Methods in App Sciences

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“…They proved the existence and uniqueness of a strong solution and the first-order necessary optimality conditions for the optimal control problem. Liu and Liu [8] showed the existence of attractor for a new mechanochemical model with Neumann boundary conditions on a bounded domain with the space dimension n ≤ 3. Liu and Zhang [11] considered the optimal control problem for the new mechanochemical model in biological patterns,…”

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Time periodic solution to a mechanochemical model in biological patterns

Liu

2023

EECT

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<p style='text-indent:20px;'>In this paper, we consider a mechanochemical model in biological patterns in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ N\geq 5 $\end{document}</tex-math></inline-formula>. We first prove the existence of time periodic solution in <inline-formula><tex-math id="M3">\begin{document}$ BC(\mathbb{R}; L^{N,\infty}(\Omega)) $\end{document}</tex-math></inline-formula>. Then we obtain the existence, uniqueness and regularity of the mild solution of the problem. Finally, we prove that the mild solution can become strong solution in <inline-formula><tex-math id="M4">\begin{document}$ BC(\mathbb{R}; L^{N,\infty}(\Omega)) $\end{document}</tex-math></inline-formula>.</p>

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Global attractor for a smectic-A liquid crystal model in 2D

Liu

2018

Boll Unione Mat Ital

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Asymptotic Dynamics of a New Mechanochemical Model in Biological Patterns

Cited by 4 publications

References 18 publications

Optimal control of a new mechanochemical model with state constraint

Optimal control of a new mechanochemical model with state constraint

Time periodic solution to a mechanochemical model in biological patterns

Global attractor for a smectic-A liquid crystal model in 2D

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