Autonomous Formation of Spatial Structures in Solutions of Parabolic Partial Differential Equations

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This objective of this project is to pursue the behavior of solutions of nonlinear partial differential equations of parabolic type.In collaboration with Wei-Ming Ni (University of Minnesota) and Kanako Suzuki (Tohoku University), Takagi studied the behavior of solutions of a reaction-diffusion system of activator-inhibitor type proposed by Gierer and Meinhardt and obtained the following results : (i)In the case where the initial data are constant functions, there exist solutions blowing up in finite time if the activator activates the its production stronger than that of the inhibitor. There are two types of blow-up solutions. Either (a) only the activator blows up or (b) both the activator and the inhibitor blow up. In the former case, we can choose the initial value so that the inhibitor converges to any specified positive number. (ii)In the case where the equation for the activator does not contain the source term, no solution blows up in finite time if the activator produces the inhibitor more than itself. Moreover, in this case the collapse of patterns can occur. Here, by the collapse of patterns we mean that the solution converges to the origin as the time variable tends to infinity.Nishiura studied scattering phenomena of pulse solutions and spot solutions. He found that various input/output relationships can be formed depending on the local dynamics in the neighborhood of unstable steady-states or periodic solutions and on the location of solution orbits.Yanagida considered a certain quasilinear parabolic equation and showed that the solution is either (a) globally increasing, (b) a traveling wave, or (c) extinct in finite time, depending on the initial data. The asymptotic behavior of the solution is also investigated.Kozono proved that in three dimensional exterior domains one can construct weak solutions of the Navier-Stokes equations which satisfy the strong energy inequality for all square-integrable initial data.