2007 Fiscal Year Final Research Report Summary
Self-similar structure and singularity of solutions for nonlinear parabolic partial differential equations
Grant-in-Aid for Scientific Research (C)
|Allocation Type||Single-year Grants |
|Research Institution||Kobe University |
NAITO Yuki Kobe University, Graduate School of Engineering, Associate Professor (10231458)
ISHI Katsuyuki Kobe University, Graduate School of Maritime Sciences, Associate Professor (40232227)
KUWAMURA Masataka Kobe University, Graduate School of Human Development and Environment, Associate Professor (30270333)
SUZUKI Takashi Osaka University, Graduate Scool of Fundamental Engineering, Professor (40114516)
SATO Tokushi Tohoku University, Graduate School of Science, Assistant Professor (00261545)
TANAKA Satoshi Okayama University of Science, Department of Scienoe, Lecturer (90331959)
|Project Period (FY)
2005 – 2007
|Keywords||Nonlinear Problem / Self-similar solutions / Parabolic partial differential equaitons / Variationa method / Chemotaxis system|
Structure of self-similar solutions for nonlinear parabolic partial differential equations and the role of self-similar solutions for the asymptotic behavior of time-global solutions and the finite time blow up solutions are studied. In particular, we consider the structure of self-similar solutions for semilinear heat equations, the role of self-similar solutions in the blow-up phenomena for nonlinear heat equations, and the structure of self-similar solutions for chemotaxis system.
We show the multiple existence of positive self-similar solutions for semilinear heat equations with sub-critical and critical nonlinearity by employing variationa methods for semilinear elliptic partial differential equations. In the super-critical case, we show the existence and some properties of positive self-similar solutions by using of the ordinary differential methods.
We show some criterion for blow up rate of solutions of semilinear heat equations with critical Sobolev nonlinearity. In particular we verify that a solution must blow up in the self-similar rate, which is called type-I blow up rate, if the solution is positive on the backward parbolic space-time region. In addition, we show, in the case where space dimension is 3, there exist solutions which blow up in specific blow-up rate, called type II blow up rate, when the domain shrinks with suitable rate.
We show that time-global solutions for parabolic chemotaxis system asymptotic to forward self-similar solutions as time gtends to the infinity in the case where space dimensional is 2. We also show the non-existence of backward self-similar solutions for chemotaxis system in the 2-dimensional case.
We consider the representation of positive solutions for semilinear elliptic equations with singular forcing terms, and then, we show the existence of positive minimal solution and the second positive solutions.
Research Products (88 results)