SYMMETRY PROPERTIES OF POSITIVE SOLUTIONS OF PARABOLIC EQUATIONS: A SURVEY

2009 ◽  
Author(s):  
P. Poláčik
Keyword(s):  
Positive Solutions ◽  
Parabolic Equations ◽  
Symmetry Properties

Symmetry properties of positive solutions of elliptic equations in an infinite sectorial cone

1992 ◽  
Vol 122 (1-2) ◽  
pp. 137-160
Author(s):  
Chie-Ping Chu ◽  
Hwai-Chiuan Wang
Keyword(s):  
Boundary Conditions ◽  
Positive Solution ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Semilinear Elliptic Equations ◽  
Spherically Symmetric ◽  
Semilinear Elliptic ◽  
Symmetry Properties

SynopsisWe prove symmetry properties of positive solutions of semilinear elliptic equations Δu + f(u) = 0 with Neumann boundary conditions in an infinite sectorial cone. We establish that any positive solution u of such equations in an infinite sectorial cone ∑α in ℝ3 is spherically symmetric if the amplitude α of ∑α is not greater than π.

scholarly journals Life Span of Positive Solutions for the Cauchy Problem for the Parabolic Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16
Author(s):  
Yusuke Yamauchi
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Life Span ◽  
Recent Result ◽  
Positive Solutions ◽  
Parabolic Equations ◽  
Blow Up ◽  
Parabolic Problems ◽  
Survey Paper ◽  
The Cauchy Problem

Since 1960's, the blow-up phenomena for the Fujita type parabolic equation have been investigated by many researchers. In this survey paper, we discuss various results on the life span of positive solutions for several superlinear parabolic problems. In the last section, we introduce a recent result by the author.

scholarly journals The uniqueness of positive solutions of parabolic equations of divergence form on an unbounded domain

1993 ◽  
Vol 130 ◽  
pp. 111-121 ◽  
Author(s):  
Masaharu Nishio
Keyword(s):  
Euclidean Space ◽  
Positive Solutions ◽  
Unbounded Domain ◽  
Parabolic Equations ◽  
Divergence Form ◽  
Dimensional Euclidean Space

Let Rn+1 = Rn × R be the (n + 1)-dimensional Euclidean space (n ≥ 1). For X ∈ Rn+1, we write X = (x, t) with x ∈ Rn and t ∈ R. We consider parabolic operators of the following form:(1)

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