具非線性阻尼項(xiàng)和源函數(shù)項(xiàng)雙曲方程解爆破時(shí)間的下界估計(jì)

孫愛慧, 曹春玲

(1.吉林師范大學(xué) 數(shù)學(xué)學(xué)院, 吉林 四平 136000； 2.吉林大學(xué) 數(shù)學(xué)學(xué)院, 長春 130012)

研究簡報(bào)

具非線性阻尼項(xiàng)和源函數(shù)項(xiàng)雙曲方程解爆破時(shí)間的下界估計(jì)

孫愛慧1, 曹春玲2

(1.吉林師范大學(xué) 數(shù)學(xué)學(xué)院, 吉林 四平 136000； 2.吉林大學(xué) 數(shù)學(xué)學(xué)院, 長春 130012)

考慮雙曲方程初邊值問題解的性質(zhì).利用能量估計(jì)方法和Sobolev嵌入不等式, 給出一個(gè)具非線性阻尼項(xiàng)和源函數(shù)項(xiàng)雙曲方程解爆破時(shí)間的下界估計(jì).

阻尼項(xiàng)； 非線性源； 雙曲方程； 爆破時(shí)間； 下界估計(jì)

考慮如下半線性雙曲方程的初邊值問題:

其中:Ω?N(N≥3)是有界區(qū)域, 且邊界?Ω滑;p>2;m>2;g≥0.許多實(shí)際問題都可以用模型(1)刻畫, 例如通過滲流介質(zhì)的流體過程、與溫度相關(guān)的黏彈性流問題等, 文獻(xiàn)[1-3]給出了這類問題的研究結(jié)果.當(dāng)黏彈性項(xiàng)g=0時(shí), 問題(1)為具有非線性阻尼項(xiàng)波的方程

先引進(jìn)如下能量泛函：

其中(g°u)(t)=g(t-τ)‖v(t)τ.根據(jù)參考文獻(xiàn)[10]中引理2.1, 有:

引理1如果下列條件成立, 則E′(t)≤0,t≥0且E(t)≤E1:

(H3)α=(C/k1/2)-p/(p-2),E1=(1/2-1/p)α2, 其中C為最優(yōu)嵌入常數(shù).

下面分兩種情形討論.

1) 當(dāng)2

由于p>N(p-2)=μ, 應(yīng)用H?lder’s不等式, 有

結(jié)合式(7),(8)有

由2(p-μ)/(Np)=2(p-Np+2N)/(Np)及不等式aαbβ≤(a+b)α+β(a>0,b>0,α>0,β>0), 有

其中:C為最優(yōu)嵌入常數(shù);q1=3-4/p>1.

2) 當(dāng)2N/(N-1)

對(duì)式(10)應(yīng)用不等式aαbβ≤(a+b)α+β(a>0,b>0,α>0,β>0), 有

其中:C為最優(yōu)嵌入常數(shù);q2=[2(N-2)-(N-4)p]/[2N-p(N-2)]>1.

k‖‖

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LowerBoundEstimationfortheBlow-upTimeofSolutionstoaClassofNonlinearDampedHyperbolicEquationswithSources

SUN Aihui1, CAO Chunling2
(1.CollegeofMathematics,JilinNormalUniversity,Siping136000,JilinProvince,China;
2.CollegeofMathematics,JilinUniversity,Changchun130012,China)

The authors studied the properties of solutions for the initial boundary value problem to a hyperbolic equation and obtained a lower bound estimation of blow-up time for a class of nonlinear damped hyperbolic equations with sources by using the method of energy estimate and Sobolev embedding inequalities.

damped term; nonlinear sources; hyerbolic equations; blow-up time; lower bound estimation

2014-07-14.

孫愛慧(1978—), 女, 漢族, 碩士, 講師, 從事偏微分方程的研究, E-mail: sunaihui2002@126.com.

吉林省自然科學(xué)基金(批準(zhǔn)號(hào)： 20115222)、吉林省科技發(fā)展計(jì)劃項(xiàng)目(批準(zhǔn)號(hào)： 201201082； 201201081)和吉林省教育廳“十二五”科學(xué)技術(shù)研究項(xiàng)目(批準(zhǔn)號(hào)： 吉教科合字[2013]第445號(hào)).

O175.8

A

1671-5489(2014)06-1227-03

10.13413/j.cnki.jdxblxb.2014.06.24

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