Blow-up and critical exponents for parabolic equations with non-divergent operators: dual porous medium and thin film operators

2006 ◽  
Vol 6 (1) ◽  
pp. 45-69 ◽  
Author(s):  
V. A. Galaktionov ◽  
S. I. Pohozaev
Keyword(s):  
Thin Film ◽  
Porous Medium ◽  
Critical Exponents ◽  
Parabolic Equations ◽  
Blow Up

The critical exponents of parabolic equations and blow-up in Rn

1998 ◽  
Vol 128 (1) ◽  
pp. 123-136 ◽  
Author(s):  
Yuan-wei Qi
Keyword(s):  
Cauchy Problem ◽  
Global Solution ◽  
Nontrivial Solution ◽  
Critical Exponents ◽  
Parabolic Equations ◽  
Finite Time ◽  
Blow Up ◽  
Initial Value ◽  
The Cauchy Problem

In this paper we study the Cauchy problem in Rn of general parabolic equations which take the form ut = Δum + ts|x|σup with non-negative initial value. Here s ≧ 0, m > (n − 2)+/n, p > max (1, m) and σ > − 1 if n = 1 or σ > − 2 if n ≧ 2. We prove, among other things, that for p ≦ pc, where pc ≡ m + s(m − 1) + (2 + 2s + σ)/n > 1, every nontrivial solution blows up in finite time. But for p > pc a positive global solution exists.

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