scholarly journals Critical Exponents of Fujita Type for Inhomogeneous Parabolic Equations and Systems

2000 ◽  
Vol 251 (2) ◽  
pp. 624-648 ◽  
Author(s):  
C Bandle ◽  
H.A Levine ◽  
Qi S Zhang
Keyword(s):  
Critical Exponents ◽  
Parabolic Equations

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.

On the continuation of solutions of non-autonomous semilinear parabolic problems

2015 ◽  
Vol 59 (1) ◽  
pp. 17-55 ◽  
Author(s):  
Alexandre N. Carvalho ◽  
Jan W. Cholewa ◽  
Marcelo J. D. Nascimento
Keyword(s):  
Banach Spaces ◽  
Critical Exponents ◽  
Parabolic Equations ◽  
Parabolic Problems ◽  
Scale Of Banach Spaces ◽  
Semilinear Parabolic ◽  
Semilinear Parabolic Problems

AbstractWe study non-autonomous parabolic equations with critical exponents in a scale of Banach spaces Eσ, σ ∈ [0,1 + μ). We consider a suitable E1+ε-solution and describe continuation properties of the solution. This concerns both a situation when the solution can be continued as an E1+ε-solution and a situation when the E1+ε-norm of the solution blows up, in which case a piecewise E1+ε-solution is constructed.

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