scholarly journals Extinction Phenomenon and Decay Estimate for a Quasilinear Parabolic Equation with a Nonlinear Source

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Dengming Liu ◽  
Luo Yang
Keyword(s):  
Parabolic Equation ◽  
Decay Estimate ◽  
Energy Estimate ◽  
Quasilinear Parabolic Equation ◽  
Upper And Lower Solutions ◽  
Decay Estimates ◽  
Nonlinear Source ◽  
Quasilinear Parabolic

By energy estimate approach and the method of upper and lower solutions, we give the conditions on the occurrence of the extinction and nonextinction behaviors of the solutions for a quasilinear parabolic equation with nonlinear source. Moreover, the decay estimates of the solutions are studied.

COMPLETE BLOW-UP AND ESTIMATE ON THE LOCALIZATION FOR A QUASILINEAR PARABOLIC EQUATION WITH NONLINEAR SOURCE

2014 ◽  
Vol 12 (02) ◽  
pp. 209-225
Author(s):  
PAN ZHENG ◽  
CHUNLAI MU
Keyword(s):  
Parabolic Equation ◽  
Blow Up ◽  
Quasilinear Parabolic Equation ◽  
Nonlinear Source ◽  
The Cauchy Problem ◽  
Initial Support ◽  
Upper Bound Estimate ◽  
The Stability ◽  
Special Case ◽  
Quasilinear Parabolic

This paper deals with the Cauchy problem for a quasilinear parabolic equation with nonlinear source [Formula: see text] where N ≥ 1, p > 2, m, l, q > 1 and T < ∞ is blow-up time. When q > l + m(p - 2) + p, we first give an upper bound estimate on the localization in terms of the initial support supp u0(x) and the blow-up time T < ∞ provided that the initial function u0(x) is compactly supported. Moreover, for the special case m = l, when [Formula: see text], we obtain the result of complete blow-up and the stability of complete blow-up time provided that the solution u(x, t) blows up in finite time.

scholarly journals Blow-Up Analysis for a Quasilinear Parabolic Equation with Inner Absorption and Nonlinear Neumann Boundary Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhong Bo Fang ◽  
Yan Chai
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Neumann Boundary Condition ◽  
Blow Up ◽  
Quasilinear Parabolic Equation ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Blow Up Analysis ◽  
Inner Absorption ◽  
Quasilinear Parabolic

We investigate an initial-boundary value problem for a quasilinear parabolic equation with inner absorption and nonlinear Neumann boundary condition. We establish, respectively, the conditions on nonlinearity to guarantee thatu(x,t)exists globally or blows up at some finite timet*. Moreover, an upper bound fort*is derived. Under somewhat more restrictive conditions, a lower bound fort*is also obtained.

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