Stability of the steady state for multi-dimensional thermoelastic systems of shape memory alloys
Takashi Suzuki Shuji Yoshikawa
Discrete & Continuous Dynamical Systems - S 2012, 5(1): 209-217 doi: 10.3934/dcdss.2012.5.209
This paper studies a dynamical stability of the steady state for some thermoelastic and thermoviscoelastic systems in multi-dimensional space domain. More general nonlinear term can be taken here than the one in [6] which studied the stability for the one-dimensional system called the Falk model system. We also give applications to thermoviscoelastic systems treated in [8] and [9].
keywords: stability phase transition steady state. Shape memory alloys thermoelastic
Brownian point vortices and dd-model
Takashi Suzuki
Discrete & Continuous Dynamical Systems - S 2014, 7(1): 161-176 doi: 10.3934/dcdss.2014.7.161
We study the kinetic mean field equation on two-dimensional Brownian vortices; derivation, similarity to the DD-model, and existence and non-existence of global-in-time solution.
keywords: chemotaxis. Brownian point vortex DD model
Asymptotic behaviour of the solutions to a virus dynamics model with diffusion
Toru Sasaki Takashi Suzuki
Discrete & Continuous Dynamical Systems - B 2018, 23(2): 525-541 doi: 10.3934/dcdsb.2017206

Asymptotic behaviour of the solutions to a basic virus dynamics model is discussed. We consider the population of uninfected cells, infected cells, and virus particles. Diffusion effect is incorporated there. First, the Lyapunov function effective to the spatially homogeneous part (ODE model without diffusion) admits the $L^1$ boundedness of the orbit. Then the pre-compactness of this orbit in the space of continuous functions is derived by the semigroup estimates. Consequently, from the invariant principle, if the basic reproductive number $R_0$ is less than or equal to 1, each orbit converges to the disease free spatially homogeneous equilibrium, and if $R_0>1$, each orbit converges to the infected spatially homogeneous equilibrium, which means that the simple diffusion does not affect the asymptotic behaviour of the solutions.

keywords: Asymptotic behaviour Lyapunov functions reaction-diffution equations virus dynamics model
Blowup threshold and collapse mass separation for a drift-diffusion system in space-dimension two
Elio E. Espejo Masaki Kurokiba Takashi Suzuki
Communications on Pure & Applied Analysis 2013, 12(6): 2627-2644 doi: 10.3934/cpaa.2013.12.2627
We study a drift-diffusion system on bounded domain in two-space dimension. This model is provided with a hetero-separative and homo-aggregative feature subject to a gradient of physical or chemical potential which is proportional to their densities. We extend a criterion of global-in-time existence of the solution, especially for non-radially symmetric case. Then we perform the blowup analysis such as the formation of collapses and collapse mass separations. A slightly different model describing cross chemotaxis is also discussed.
keywords: formation of collapse blowup threshold mass separation. Drift-diffusion model chemotaxis
Reaction diffusion equation with non-local term arises as a mean field limit of the master equation
Kazuhisa Ichikawa Mahemauti Rouzimaimaiti Takashi Suzuki
Discrete & Continuous Dynamical Systems - S 2012, 5(1): 115-126 doi: 10.3934/dcdss.2012.5.115
We formulate a reaction diffusion equation with non-local term as a mean field equation of the master equation where the particle density is defined continuously in space and time. In the case of the constant mean waiting time, this limit equation is associated with the diffusion coefficient of A. Einstein, the reaction rate in phenomenology, and the Debye term under the presence of potential.
keywords: Reaction probability. Stochastic reaction Reaction radius
Global-in-time behavior of the solution to a Gierer-Meinhardt system
Georgia Karali Takashi Suzuki Yoshio Yamada
Discrete & Continuous Dynamical Systems - A 2013, 33(7): 2885-2900 doi: 10.3934/dcds.2013.33.2885
Gierer-Meinhardt system is a mathematical model to describe biological pattern formation due to activator and inhibitor. Turing pattern is expected in the presence of local self-enhancement and long-range inhibition. The long-time behavior of the solution, however, has not yet been clarified mathematically. In this paper, we study the case when its ODE part takes periodic-in-time solutions, that is, $\tau=\frac{s+1}{p-1}$. Under some additional assumptions on parameters, we show that the solution exists global-in-time and absorbed into one of these ODE orbits. Thus spatial patterns eventually disappear if those parameters are in a region without local self-enhancement or long-range inhibition.
keywords: Hamilton structure Turing pattern asymptotic behavior of the solution. Reaction-diffusion equation Gierer-Meinhardt system

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