孫仁斌
(中南民族大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)學(xué)院,武漢 430074)
一類對數(shù)形式退縮拋物型方程組解的整體存在性與爆破
孫仁斌
(中南民族大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)學(xué)院,武漢 430074)
考慮了一類由對數(shù)函數(shù)產(chǎn)生的退縮拋物型方程組的初邊值問題,利用逼近的方法得到了解的局部存在性.通過特征函數(shù)構(gòu)造的上解得到了解整體存在的條件,利用特征函數(shù)得到了解在有限時(shí)刻爆破的條件及爆破時(shí)間的上限.
退縮拋物型方程組;局部解;整體解;爆破
本文考慮如下由對數(shù)函數(shù)產(chǎn)生的退縮拋物型方程組解的初邊值問題:
(1)
其中Ω為RN(N≥1為正整數(shù))中具有光滑邊界?Ω的有界區(qū)域,u0(x),v0(x)為Ω上非負(fù)連續(xù)可微函數(shù),在?Ω上,u0(x)=0,v0(x)=0,f(s),g(s)在s≥0時(shí)為非負(fù)連續(xù)可微函數(shù),常數(shù)α,β>0.
對于拋物型方程與方程組在有界區(qū)域上的初邊值問題解的存在性與爆破性的討論已有很長時(shí)間了,早期的討論是針對半線性方程與方程組進(jìn)行,主要討論整體解存在的條件以及當(dāng)整體解不存在而在有限時(shí)刻發(fā)生爆破時(shí)對爆破時(shí)間上限的估計(jì)[1-4].隨著討論的深入,對爆破點(diǎn)集的分布和爆破速率的估計(jì)也開始進(jìn)行并逐步精確[5-7];另一方面討論的方程及方程組的復(fù)雜程度逐漸加大,從最初的半線性方程到后來的退縮方程[8-11],再到不同耦合形式的方程組[12-14],以及各種類型的退縮方程組[15,16].對于退縮性是由冪函數(shù)引起的方程及方程組的討論已有不少成果,文[15]中討論的方程組是ut=vα(uxx+au),vt=uα(vxx+bv),文[16]中討論的方程組是ut=up(Δu+av),vt=vq(Δv+bu),文[17]對文[16]中的方程組在無界區(qū)域上進(jìn)行了討論.
本文討論的初邊值問題(1)中,方程的退縮性是由對數(shù)函數(shù)引起,當(dāng)u=0或v=0時(shí),方程出現(xiàn)退縮.首先利用逼近的方法給出解的局部存在性,其次討論解整體存在的條件,最后討論解在有限時(shí)刻發(fā)生爆破的條件.
對于退縮拋物型方程組初邊值問題解的局部存在性與唯一性,討論的方法有幾種,但基本相同,在此只簡要介紹主要步驟,不作詳細(xì)的證明.首先,構(gòu)造一個(gè)近似的非退縮拋物方程組的初邊值問題.對常數(shù)ε>0,考慮如下問題:
(2)
初邊值問題(2)為非退縮的擬線性拋物型方程組,由經(jīng)典的拋物型方程理論可知,其解是局部存在的,即存在σ>0,使問題(2)在Ω×[0,σ)上至少存在一個(gè)正解,仍記為(uε,vε),再由解的延展知,存在最大值Tε,使問題(2)在Ω×[0,Tε)上至少存在一個(gè)正解,且由極大值原理知uε≥ε,vε≥ε,于是(uε,vε)是如下初邊值問題的解:
其次,由比較原理[18],當(dāng)ε1>ε2時(shí),對應(yīng)的解滿足uε1≥uε2,vε1≥vε2,即(uε,vε)隨ε的減少而減少,于是,存在T>0,當(dāng)ε→0+時(shí),Tε→T, (uε,vε)→(u,v),由此得到問題(1)解的存在性.進(jìn)一步通過經(jīng)典的方法還可以得到解的唯一性[18],即定理1.
由比較原理[18],容易得到下面的引理1.
設(shè)φ(x)是特征值問題:
(3)
定理2 設(shè)存在正常數(shù)a,b和k,l,且al≤λ1k,bk≤λ1l,使函數(shù)f(s),g(s)滿足:f(s)≤as,g(s)≤bs,s>0,函數(shù)u0(x),v0(x)滿足u0(x)≤kφ(x),v0(x)≤lφ(x),x∈Ω,則問題(1)存在整體解.
(1+kφ)[ln(1+kφ)]α(λ1kφ-f(lφ))≥
(1+kφ)[ln(1+kφ)]α(λ1kφ-alφ)≥0,
-(1+lφ)[ln(1+lφ)]β(lΔφ+g(kφ)),
利用(3)式,有:
(1+lφ)[ln(1+lφ)]β(λ1lφ-g(kφ))≥
(1+lφ)[ln(1+lφ)]β(λ1lφ-bkφ)≥0.
于是問題(1)存在上解(kφ(x),lφ(x)),由引理1得u(x,t)≤kφ(x),v(x,t)≤lφ(x)對任意t>0都成立,故u(x,t),v(x,t)對任意t>0都有定義,證畢.
引理2 設(shè)初值函數(shù)u0(x),v0(x)滿足:
u0(x)≥φ(x),v0(x)≥φ(x),
(4)
且存在常數(shù)a>λ1,b>λ1,使函數(shù)f(s),g(s)滿足:
f(s)≥as,g(s)≥bs,s>0,
(5)
則問題(1)的解滿足u(x,t)≥φ(x),v(x,t)≥φ(x),x∈Ω,t>0.
證明 令w(x,t)=u(x,t)-φ(x),z(x,t)=v(x,t)-φ(x),則t=0時(shí),w≥0,z≥0,x∈?Ω時(shí),w=z=0,而x∈Ω,t>0時(shí),wt=ut=(1+u)[ln(1+u)]α(Δu+f(v))=(1+u)[ln(1+u)]α(Δw+Δφ+f(v)),
利用(3)、(5)式,有:
wt=(1+u)[ln(1+u)]αΔw+(1+u)[ln(1+u)]α(f(v)-λ1φ)≥(1+u)[ln(1+u)]αΔw+(1+u)[ln(1+u)]α(av-λ1φ)=(1+u)[ln(1+u)]αΔw+(1+u)[ln(1+u)]α(az+aφ-λ1φ),
即有:wt≥(1+u)[ln(1+u)]αΔw+
(1+u)[ln(1+u)]αaz.
(6)
zt=vt=(1+v)[ln(1+v)]β(Δv+g(u))=(1+v)[ln(1+v)]β(Δz+Δφ+g(u)),
利用(3)、(5)式,有:
zt=(1+v)[ln(1+v)]βΔz+(1+v)[ln(1+v)]β(g(u)-λ1φ)≥(1+v)[ln(1+v)]βΔz+(1+v)[ln(1+v)]β(bu-λ1φ)=(1+v)[ln(1+v)]βΔz+(1+v)[ln(1+v)]β(bw+bφ-λ1φ),
即有:zt≥(1+v)[ln(1+v)]βΔz+
(1+v)[ln(1+v)]βbw.
(7)
由(6)、(7)式及拋物方程組的極大值原理知w≥0,z≥0,證畢.
v(x,t))-1[ln(1+v(x,t))]-βvt(x,t)φ(x)dx=
由(4)、(5)式得:
由引理2得:
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Global Existence and Blow-up for a Degenerate Parabolic System Come from Logarithmic Function
SunRenbin
(College of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China)
In this paper, a class of initial-boundary value problem of degenerate parabolic system come from logarithmic function was considered.The local existence of solution was proved by approximation method.The condition of global existence of the solution was obtained by the upper solution of the characteristic function, and the blow-up condition and the upper bound of the blow-up time were obtained by using the characteristic function.
degenerate parabolic system;local solution;global solution;blow-up
2017-01-04
孫仁斌(1964-),男,副教授,研究方向:拋物型偏微分方程, E-mail: sunrenbin@foxmail.com
國家自然科學(xué)基金資助項(xiàng)目(61374085)
O175.26
A
1672-4321(2017)01-0119-04