DCDS-S

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].

DCDS-S

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.

DCDS-B

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.

CPAA

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.

DCDS-S

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.

DCDS

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.

DCDS-B

This paper considers quadratic and super-quadratic reaction-diffusion systems, for which all species satisfy uniform-in-time $L^1$ a-priori estimates, for instance, as a consequence of suitable mass conservation laws. A new result on the global existence of classical solutions is proved in three and higher space dimensions by combining regularity and interpolation arguments in Bochner spaces, a bootstrap scheme and a weak comparison argument. Moreover, provided that the considered system allows for entropy entropy-dissipation estimates proving exponential convergence to equilibrium, we are also able to prove that solutions are bounded uniformly-in-time.