scholarly journals Blow-Up Analysis for a Quasilinear Degenerate Parabolic Equation with Strongly Nonlinear Source

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Pan Zheng ◽  
Chunlai Mu ◽  
Dengming Liu ◽  
Xianzhong Yao ◽  
Shouming Zhou
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Degenerate Parabolic Equation ◽  
Blow Up ◽  
Single Point ◽  
Nonlinear Source ◽  
Strongly Nonlinear ◽  
Blow Up Analysis ◽  
The Cauchy Problem ◽  
Degenerate Parabolic

We investigate the blow-up properties of the positive solution of the Cauchy problem for a quasilinear degenerate parabolic equation with strongly nonlinear sourceut=div(|∇um|p−2∇ul)+uq,  (x,t)∈RN×(0,T), whereN≥1,p>2, andm,l,  q>1, and give a secondary critical exponent on the decay asymptotic behavior of an initial value at infinity for the existence and nonexistence of global solutions of the Cauchy problem. Moreover, under some suitable conditions we prove single-point blow-up for a large class of radial decreasing solutions.

LOCALIZATION OF SOLUTIONS TO A DOUBLY DEGENERATE PARABOLIC EQUATION WITH A STRONGLY NONLINEAR SOURCE

2012 ◽  
Vol 14 (03) ◽  
pp. 1250018 ◽  
Author(s):  
CHUNLAI MU ◽  
PAN ZHENG ◽  
DENGMING LIU
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Initial Data ◽  
Compact Support ◽  
Degenerate Parabolic Equation ◽  
Nonlinear Source ◽  
Strongly Nonlinear ◽  
The Cauchy Problem ◽  
Degenerate Parabolic

In this paper, we investigate the localization of solutions of the Cauchy problem to a doubly degenerate parabolic equation with a strongly nonlinear source [Formula: see text] where N ≥ 1, p > 2 and m, l, q > 1. When q > l + m(p - 2), we prove that the solution u(x, t) has strict localization if the initial data u0(x) has a compact support, and we also show that the solution u(x, t) has the property of effective localization if the initial data u0(x) satisfies radially symmetric decay. Moreover, when 1 < q < l + m(p - 2), we obtain that the solution of the Cauchy problem blows up at any point of RNto arbitrary initial data with compact support.

Single-point blow-up for a doubly degenerate parabolic equation with nonlinear source

2011 ◽  
Vol 141 (3) ◽  
pp. 641-654 ◽  
Author(s):  
Chunlai Mu ◽  
Rong Zeng
Keyword(s):  
Parabolic Equation ◽  
Positive Solution ◽  
Large Class ◽  
Degenerate Parabolic Equation ◽  
Blow Up ◽  
Single Point ◽  
Degenerate Equation ◽  
Nonlinear Source ◽  
Degenerate Parabolic ◽  
Image Position

This paper deals with the positive solution to the doubly degenerate equationwhere σ > 0, m > 1, β > m(1 + σ). We prove single-point blow-up for a large class of radial decreasing solutions. Furthermore, the upper and lower estimates of the blow-up solution near the single blow-up point are obtained.

scholarly journals On the Cauchy Problem of a Quasilinear Degenerate Parabolic Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-12
Author(s):  
Zongqi Liang ◽  
Huashui Zhan
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Classical Solution ◽  
Finite Time ◽  
Numerical Experiments ◽  
Degenerate Parabolic Equation ◽  
Blow Up ◽  
Strong Solutions ◽  
The Cauchy Problem ◽  
Degenerate Parabolic

By Oleinik's line method, we study the existence and the uniqueness of the classical solution of the Cauchy problem for the following equation in[0,T]×R2:∂xxu+u∂yu−∂tu=f(⋅,u), provided thatTis suitable small. Results of numerical experiments are reported to demonstrate that the strong solutions of the above equation may blow up in finite time.

scholarly journals On the Cauchy problem for a degenerate parabolic differential equation

1998 ◽  
Vol 21 (3) ◽  
pp. 555-558
Author(s):  
Ahmed El-Fiky
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Parabolic Equation ◽  
Degenerate Parabolic Equation ◽  
Parabolic Differential Equation ◽  
The Cauchy Problem ◽  
Uniqueness Of The Solution ◽  
Degenerate Parabolic

The aim of this work is to prove the existence and the uniqueness of the solution of a degenerate parabolic equation. This is done using H. Tanabe and P.E. Sobolevsldi theory.

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