Finite time blow-up and global solutions for a class of semilinear parabolic equations at high energy level

2012 ◽  
Vol 13 (1) ◽  
pp. 197-202 ◽  
Author(s):  
Runzhang Xu ◽  
Xiuying Cao ◽  
Tao Yu
Keyword(s):  
Energy Level ◽  
Parabolic Equations ◽  
Finite Time ◽  
Blow Up ◽  
Global Solutions ◽  
High Energy ◽  
Semilinear Parabolic Equations ◽  
High Energy Level ◽  
Semilinear Parabolic

Global blow-up of a nonlinear heat equation

1986 ◽  
Vol 104 (1-2) ◽  
pp. 161-167 ◽  
Author(s):  
A. A. Lacey
Keyword(s):  
Heat Equation ◽  
Parabolic Equations ◽  
Finite Time ◽  
Positive Measure ◽  
Blow Up ◽  
Nonlinear Heat Equation ◽  
Semilinear Parabolic Equations ◽  
Semilinear Parabolic

SynopsisSolutions to semilinear parabolic equations of the form ut = Δu + f(u), x in Ω, which blow up at some finite time t* are investigated for “slowly growing” functions f. For nonlinearities such as f(s) = (2 +s)(ln(2 +s))1+b with 0 < b < l,u becomes infinite throughout Ω as t→t* −. It is alsofound that for marginally more quickly growing functions, e.g. f(s) = (2 + s)(ln(2 +s))2, u is unbounded on some subset of Ω which has positive measure, and is unbounded throughout Ω if Ω is a small enough region.

FINITE TIME BLOW UP OF FOURTH-ORDER WAVE EQUATIONS WITH NONLINEAR STRAIN AND SOURCE TERMS AT HIGH ENERGY LEVEL

2013 ◽  
Vol 24 (06) ◽  
pp. 1350043 ◽  
Author(s):  
JIHONG SHEN ◽  
YANBING YANG ◽  
SHAOHUA CHEN ◽  
RUNZHANG XU
Keyword(s):  
Energy Level ◽  
Fourth Order ◽  
Finite Time ◽  
Wave Equations ◽  
Blow Up ◽  
Initial Boundary ◽  
High Energy ◽  
Source Terms ◽  
High Energy Level ◽  
Nonlinear Strain

In this paper, we study the initial boundary value problem for fourth-order wave equations with nonlinear strain and source terms at high energy level. We prove that, for certain initial data in the unstable set, the solution with arbitrarily positive initial energy blows up in finite time.

scholarly journals Parabolic Equations with a Singular Absorption: Extinction Versus Blow-up

2017 ◽  
Vol 1 (2) ◽  
pp. 134
Author(s):  
Nguyen Anh Dao
Keyword(s):  
Parabolic Equations ◽  
Finite Time ◽  
Source Term ◽  
Blow Up ◽  
Local Existence ◽  
Qualitative Behavior ◽  
Semilinear Parabolic Equations ◽  
Creative Commons ◽  
Semilinear Parabolic ◽  
Open Access Article

We prove a local existence of weak solutions of semilinear parabolic equations with a strong singular absorption and a  source. Moreover, we consider the qualitative behavior of solutions. We show that any solution exists globally and vanishes after a finite time if either the initial data or the source term is small enough. On the other hand, we point out some criteria such that solutions are explosive in a finite time.  This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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