Tokyo Tech News
Published: November 30, 2011
The Fujita equation is a type of nonlinear diffusion equation. It is known to exhibit various interesting phenomena, such as ‘blow-up’ where the maximum of a solution can tend to infinity in a finite time. Now Eiji Yanagida from the Department of Mathematics at Tokyo Institute of Technology and Shota Sato from the Mathematical Institute at Tohoku University have found a new solution that is particularly unusual for nonlinear diffusion equations.
No solutions extend to times beyond the blow-up point. On the other hand, the Fujita equation does have a stationary solution with a singularity that is radially symmetric with respect to the singularity point. Eiji Yanagida and Shota Sato have now proved the existence of a solution whose singularity moves along a prescribed curve in time. Mathematically, such a solution is called a weak solution. This is the first example of a time-dependent weak solution of a nonlinear diffusion equation.
The researchers also carefully studied the behavior of singular solutions and found one whose singularity suddenly becomes anomalous. The study of singular solutions will open up a new field of analysis on partial differential equations.
The profile of a solution u(x, t) of the Fujita equation with a moving singularity at ξ(t), where t is time, x is the spatial variable in RN.
Graduate School of Science and Engineering Mathematics