星野 弘喜

We investigate the global existence in time and asymptotic profile of the solution of some nonlinear evolution equations with strong dissipation and proliferation arising in mathematical biology. We apply our result to mathematical models of tumour angiogenesis and invasion with proliferation of tumour cells.

This paper is concerned with traveling wave solutions to a malignant tumor invasion model. In 1999, Perumpanani et al. presented a mathematical model of malignant tumor cell invasion into connective tissue without cell diffusion. In their and subsequent researches, traveling waves have been studied mainly by numerical methods. In this paper, the existence of smooth traveling wave solutions to the model is proved rigorously. In order to show the existence of traveling waves, one considers heteroclinic orbits on a phase plane and constructs an invariant region for the orbits. Moreover, their asymptotic behavior is investigated. Finally, a remark on discontinuous traveling waves is given. © 2011, by Oldenbourg Wissenschaftsverlag, Toyoake, Aichi 470-1192, Germany. All rights reserved.

A reaction-diffusion system which has a balance law is studied. Homogeneous Neumann boundary conditions are imposed. When a nonnegative global solution uniformly converges to a steady state with a polynomial rate as time goes to infinity, it is proved that the spatial average of the solution to the system describes a large-time approximation of the solution itself with an exponential rate being sharper than those obtained before. The proof is based on properties of a solution for the corresponding system of ordinary differential equations and an L estimate of an analytic semigroup. © 2014, Oldenbourg Wissenschaftsverlag GmbH, Rosenheimer Str. 145, 81671 München. All rights reserved.

A reaction-diffusion system which is related to a simple irreversible chemical reaction between two chemical substances is considered. When a non-negative global solution for the system converges uniformly to zero with polynomial rate as time goes to infinity, large-time approximation of the solution is studied. It is shown that the difference of the solution and its spatial average tends to zero with exponential rate via a global solution for the corresponding system of ordinary differential equations.

Large time behavior of the solution to some simple reaction-diffusion system is studied. It is proved that the solution behaves like the solution to the corresponding system of ordinary differential equations as time goes to infinity. The proof is based on an energy method combined with the L(p)-L(q) estimate for the associated semigroup.