久保 明逹

We consider a competitive system of differential equations describing the behavior of two biological species "u" and "v". The system is weakly coupled and one of the species has the capacity to diffuse and moves toward the higher concentration of the second species following its gradient, the density function satisfies a second order parabolic equation with chemotactic terms. The second species does not have motility capacity and satisfies an ordinary differential equation. We prove that the solutions are uniformly bounded and exist globally in time. The asymptotic behavior of solutions is also studied for a range of parameters and initial data. If the chemotaxis coefficient x is small enough the quadratic terms drive the solutions to the constant steady state.

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.

We investigate the global existence in time and the asymptotic profile of solutions of nonlinear evolution equations with strong dissipation. Applying our result to some models of mathematical biology and medicine, we discuss mathematical properties of them.

The main purpose in this paper is to investigate existence and non-existence of global solutions of the initial Dirichlet boundary value problem for evolution equations with strong dissipation. Many authors studied classes consisting of such types of equations for which initial boundary value problems possess global solutions. For this purpose we consider a related problem and seek global solutions and blow-up solutions of it depending on whether it belongs to such classes or not. (C) 2009 Elsevier Ltd. All rights reserved.

We first study a parabolic-ODE system modelling tumour growth proposed by Othmer and Stevens [Aggregation, blowup, and collapse: the ABC's of taxis in reinforced random walks, SIAM J. Appl. Math. 57 (4) (1997) 1044-1081]. According to Levine and Sleeman [A system of reaction and diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math. 57 (3) (1997) 683-730], we reduced it to a hyperbolic equation and showed the existence of collapse in [A. Kubo, T. Suzuki, Asymptotic behavior of the solution to a parabolic ODE system modeling tumour growth, Differential Integral Equations 17 (2004) 721-736]. We also deal with the system in case the reduced equation is elliptic and show the existence of collapse analogously. Next we apply the above result to another model proposed by Anderson and Chaplain arising from tumour angiogenesis and show the existence of collapse. Further we investigate a contact point between these two models and a common property to them. (C) 2006 Elsevier B.V. All rights reserved.

We study a parabolic ODE system modeling tumour growth proposed by Othmer and Stevens [8]. In use of the transformation of Levine and Sleeman [6], we reduce it to a hyperbolic equation with strong dissipation. Then, we show the existence of collapse in arbitrary space dimension by the method of energy.