10/02/03 MATHEMATICS DEPARTMENT, FERRIS STATE UNIVERSITY
MATH COLLOQUIUM, THURSDAY, OCTOBER 09, 3:00pm, STARR 136
INVITED Dr. Zhengfang Zhou, Professor
SPEAKER: Department of Mathematics, Michigan State University
TITLE: Some recent results and applications of nonlinear heat equations
Even though tremendous progress is made in the study of nonlinear parabolic equations during last thirty years, there are many fundamental questions to be answered mathematically and physically. We will first discuss the uniqueness of nonnegative solutions. The uniqueness of positive solutions for the usual heat equation is well known, but it is not trivial to prove. For nonlinear heat equations, the problem is much harder and needs some heavy machinery in analysis and elliptic partial differential equations. Then we will turn our attention to two extreme cases for existence: nonlocal existence in L^p spaces, and extinction in finite time. It turns out that some simple equations have no local nonnegative solutions, while the solutions to other equations with "quick" diffusion will be identically zero when time is sufficiently large. Some critical exponents will be introduced and explained. Finally, some applications in imaging process will be discussed, and see how to smooth the image using the regularity of the solution with bigger diffusion, and how to keep the sharp edges with vanishing diffusion.
REFRESHMENTS: 3:00 pm, STARR 136