Atsushi Yagi

Abstract parabolic evolution equations and their applications

Parabolic Evolution Equations and their Applications Atsushi Yagi Department of Applied Physics Graduate School of Engineering Osaka University Suita, Osaka 565-0871 Japan yagi@ap.eng.osaka-u.ac.jp ISSN 1439-7382 ISBN 978-3-642-04630-8 e-ISBN 978-3-642-04631-5 DOI 10.1007/978-3-642-04631-5 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009939692 Mathematics Subject Classification (2000): 35K90, 35K57, 37L30, 92D25, 92D40

Lower Estimate of the Attractor Dimension for a Chemotaxis Growth System

This paper estimates from below the attractor dimension of the dynamical system determined from a chemotaxis growth model which was presented by Mimura and Tsujikawa. It is already known that the dynamical system has exponential attractors and it is also known by numerical computations that the model contains various pattern solutions. This paper is then devoted to estimating the attractor dimension from below and in fact to showing that, as the parameter of chemotaxis increases and tends to infinity, so does the attractor dimension. Such a result is in a good correlation with the numerical results.

An L^{p}-approach to singular linear parabolic equations in bounded domains

Singular means here that the parabolic equation is not in normal form neither can it be reduced to such a form. For this class of problems, following the operator approach used in [1], we prove global in time existence and uniqueness theorems related to (spatial) -spaces. Various improvements to [2], [3] are given.

Applications of the purely imaginary powers of operators in Hilbert spaces

In C. R. Acad. Sci. Paris299 (1984), 173–176, we discussed purely imaginary powers Aiy(−∞ < y < + ∞) of linear operators A in Hilbert spaces. Here we utilize the results to consider the various problems: generation of cosine families in Hilbert spaces, coincidence of the definition domains of the fractional powers of operators, differentiability of the functions of the form A(·)0 (0 < θ < 1) where A(·) is an operator valued function defined on an interval [0, T], and so forth.

FLOCKING AND NON-FLOCKING BEHAVIOR IN A STOCHASTIC CUCKER–SMALE SYSTEM

We first present a new stochastic version of the Cucker-Smale model of the emergent behavior in flocks in which the mutual communication between individuals is affected by random factor. Then, the existence and uniqueness of global solution to this system are verified. We show a result which agrees with natural fact that under the effect of large noise, there is no flocking. In contrast, if noise is small, then flocking may occur. Paper ends with some numerical examples.

Dynamic of a stochastic predator–prey population

Abstract A stochastic predator–prey model is studied. Firstly, we prove the existence, uniqueness and positivity of the solution. Then, we show the upper bounds for moments and growth rate of population. In some cases, the growth rate is negative and the population dies out rapidly. The paper ends with some reviews for the paper [13] .

Optimal control for an adsorbate-induced phase transition model

This paper is concerned with the optimal control problem for an adsorbate-induced phase transition model. That is, we show the existence of the optimal control and derive the optimality conditions by showing the differentiability of the cost functional with respect to the control.

H∞ Functional Calculus and Characterization of Domains of Fractional Powers

Characterization of the domains of fractional powers of linear operators is very important in the study of (linear or nonlinear) abstract parabolic evolution equations. In this paper we present a new method of utilizing the H ∞ functional calculus of linear operators.

STATIONARY SOLUTIONS TO FOREST KINEMATIC MODEL

We continue the study of a mathematical model for a forest ecosystem which has been presented by Y. A. Kuznetsov, M. Y. Antonovsky, V. N. Biktashev and A. Aponina (A cross-diffusion model of forest boundary dynamics, J. Math. Biol. 32 (1994), 219–232). In the preceding two papers (L. H. Chuan and A. Yagi, Dynamical systemfor forest kinematic model, Adv. Math. Sci. Appl. 16 (2006), 393– 409; L. H. Chuan, T. Tsujikawa and A. Yagi, Aysmptotic behavior of solutions for forest kinematic model, Funkcial. Ekvac. 49 (2006), 427–449), the present authors already constructed a dynamical system and investigated asymptotic behaviour of trajectories of the dynamical system. This paper is then devoted to studying not only the structure (including stability and instability) of homogeneous stationary solutions but also the existence of inhomogeneous stationary solutions. Especially it shall be shown that in some cases, one can construct an infinite number of discontinuous stationary solutions. 2000 Mathematics Subject Classification. 35J60, 37L15, 37N25.

Exponential attractor for an adsorbate-induced phase transition model in non smooth domain

We improve our preceding result obtained in Tsujikawa and Ya gi [10]. We construct the similar exponential attractors to the same adsor bate-induced phase transition model as in [10] but in a convex domain by using the compac t smoothing property of corresponding nonlinear semigroup. In [10], the dom ain has been assumed to haveC3 regularity to ensure the squeezing property of semigroup.