10/02/03 MATHEMATICS
DEPARTMENT, FERRIS STATE UNIVERSITY

MATH
COLLOQUIUM, THURSDAY, OCTOBER 09, 3:00pm, STARR 136

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**INVITED Dr.
Zhengfang Zhou, Professor**

**SPEAKER: Department
of Mathematics, Michigan State University**

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**TITLE: ****Some recent results and applications of nonlinear heat
equations**

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**Abstract**

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*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.*

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**REFRESHMENTS: **3:00
pm, STARR 136

http://www.ferris.edu/htmls/colleges/artsands/Math/MATH_COLLOQUIUM/ColloquiumWeb/index.html

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