擬線性拋物方程解的爆破時(shí)間下界

盧 靜

(1. 天津大學(xué) 理學(xué)院 數(shù)學(xué)系，天津 300072；2. 天津大學(xué) 應(yīng)用數(shù)學(xué)中心，天津 300072)

擬線性拋物方程解的爆破時(shí)間下界

盧 靜1，2

(1. 天津大學(xué) 理學(xué)院 數(shù)學(xué)系，天津 300072；2. 天津大學(xué) 應(yīng)用數(shù)學(xué)中心，天津 300072)

研究了下面的方程

ut=Δum+up-uqin Ω×(0,t*),

u(x,t)=0 on ? Ω×(0,t*),

u(x,0)=u0(x) in Ω,這里Ω?RN是一個(gè)光滑有界的開(kāi)區(qū)域且N≥3. 可以得到方程解的爆破時(shí)間下界.

擬線性拋物方程；有限時(shí)間爆破；爆破時(shí)間下界

ut=Δum+up-uqin Ω×(0,t*),

u(x,t)=0 on ?Ω×(0,t*),

u(x,0)=u0(x) in Ω,

Where Ω?RNwas a smooth bounded open domain andN≥3. Obtained the lower bound for the blow-up time of the solution.

本文主要研究的是如下方程解的爆破時(shí)間下界，

ut=Δum+up-uqin Ω×(0,t*),

u(x,t)=0 on ?Ω×(0,t*),

u(x,0)=u0(x) in Ω,

(1)

這里Ω?RN(N≥3)是一個(gè)光滑有界開(kāi)區(qū)域 ，p,q≥0其中u代表熱傳導(dǎo)模型中物質(zhì)的溫度.

本文受到了文獻(xiàn)[1]的啟發(fā)而加以延伸.但是我們所研究的是非線性項(xiàng)被吸收的情況. 關(guān)于拋物方程解的爆破時(shí)間下界有很多相關(guān)的結(jié)果，可以查閱文獻(xiàn)[2-7].

定義

(2)

其中

利用格林公式，得到

(3)

利用H?lder不等式和Yong不等式，我們有

(4)

且0<α<1.利用施瓦茲不等式，得到

(5)

利用Sobolev不等式(見(jiàn)文獻(xiàn)[8])，有

再次利用Yong不等式，得到

(6)

最后得到

其中，

(7)

(8)

(9)

(10)

將式(10)從0到t*進(jìn)行積分，有

(11)

其中

[1] BAO A G, SONG X F. Bounds for the blowup time of the solutions to quasi-linear parabolic problems [J]. Z. Angew. Math. Phys., 2014, 65: 115-123.

[2] LV X ,SONG X . Bounds of the blowup time in parabolic equations with weighted source under nonhomogeneous Neumann boundary condition [J]. Math. Meth. Appl. Sci., 2014, 37: 1019-1028.

[3] PAYNE L E, PHILIPPIN G A, SCHAEFER P W. Blow-up phenomena for some nonlinear parabolic problems [J]. Nonlinear Anal., 2008, 69: 3495-3502.

[4] PAYNE L E, PHILIPPIN G A, SCHAEFER P W.Bounds for blow-up time in nonlinear parabolic problems [J]. J. Math. Anal. Appl., 2008, 338: 438-447.

[5] PAYNE L E, SCHAEFER P W. Lower bounds for blow-up time in parabolic problems under Dirichlet conditions [J]. J. Math. Anal. Appl., 2007, 328: 1196-1205.

[6] L E PAYNE, J C SONG. Lower bounds for the blow-up time in a nonliner parabolic problem [J]. J. Math. Anal. Appl., 2009, 354, 394-396.

[7] SONG X F, LV X S. Bounds for the blowup time and blowup rate estimates for a type of parabolic equations with weighted source [J]. Appl. Math.Comput., 2014, 236: 78-92.

[8] TALENTI G. Best Constant in Sobolev Inequality [J]. Ann. Mat. Pura Appl., 1976, 110: 353-372.

Lower bound for blowup time of solution to quasilinearparabolic equation

LU Jing1, 2

(1. Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, China;2. Center for Applied Mathematics, Tianjin University, Tianjin 300072, China)

In this paper,the following problem were diseussed

quasilinear parabolic equation; blow up in finite time; lower bound for blow-up time

2014-09-17.

盧 靜(1988-)，女，碩士，研究方向：應(yīng)用數(shù)學(xué).

O175

A

1672-0946(2015)06-0751-02