Asymptotic analysis for reaction-diffusion equations with absorption

Du Wanjuan, Li, Li Zhongping

Description

In this paper, we study the blow-up and nonextinction phenomenon of reaction-diffusion equations with absorption under the null Dirichlet boundary condition. We at first discuss the existence and nonexistence of global solutions to the problem, and then give the blow-up rate estimates for the nonglobal solutions. In addition, the nonextinction of solutions is also concerned. MSC: 35B33, 35K55, 35K60.

Sujets

• Système à réaction-diffusion
• Absorption
• Blow-Up

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 7
• Langue: English

Extrait

DuandLiBoundaryValueProblems2012,2012:84
http://www.boundaryvalueproblems.com/content/2012/1/84
RESEARCH OpenAccess
Asymptoticanalysisforreaction-diffusion
equationswithabsorption
*WanjuanDu andZhongpingLi
*Correspondence:
[e-mail] Abstract
CollegeofMathematicand
Inthispaper,westudytheblow-upandnonextinctionphenomenonofInformation,ChinaWestNormal
University,Nanchong,637009,P.R. reaction-diffusionequationswithabsorptionunderthenullDirichletboundary
China condition.Weatfirstdiscusstheexistenceandnonexistenceofglobalsolutionsto
theproblem,andthengivetheblow-uprateestimatesforthenonglobal.In
addition,thenonextinctionofsolutionsisalsoconcerned.
MSC: 35B33;35K55;35K60
Keywords: reaction-diffusion;absorption;blow-up;blow-uprate;non-extinction
1 Introduction
Inthispaper,weconsiderthereaction-diffusionequationswithabsorption
m p qu = u +u –u , x ∈ ,t>,t
u(x,t)=, x ∈ ∂ ,t>, (.)
u(x,)=u (x), x ∈ ,
Nwhere m>, p>, q ≥ , p = q, ⊂R is a bounded domain with smooth boundary
∂ ,and u (x) is a nontrivial, nonnegative, bounded, and appropriately smooth function.
Parabolic equations like (.) appear in population dynamics, chemical reactions, heat
transfer, and so on. We refer to [, , ] for details on physical models involving more
generalreaction-diffusionequations.
The semilinear case (m=)of(.) has been investigated by Bedjaoui and Souplet [].
Theyobtainedthatthesolutionsexistgloballyifeitherp<max{q,}orp=max{q,},and
thesolutionsmayblowupinfinitetimeforlargeinitialvalueifp>max{q,}.Recently,Xi-
angetal.[]consideredtheblow-uprateestimatesfornonglobalsolutionsof(.)(m=)
–p–with p > max{q,},andobtainedthat(i) max u(x,t) ≥ c(T – t) ; (ii) max u(x,t) ≤
– p–C(T – t) if p ≤+ ,where c,C >  are positive constants. Liu et al. [] studiedN+
the extinction phenomenon of solutions of (.)forthecase<m<with q=andob-
tained some sufficient conditions about the extinction in finite time and decay estimates
Nofsolutionsin ⊂R (N>).
Recently, Zhou et al. [] investigated positive solutions of the degenerate parabolic
equationnotindivergenceform
p q ru =u u+au –bu , x ∈ ,t>,t
u(x,t)=, x ∈ ∂ ,t>, (.)
u(x,)=u (x), x ∈ ,
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where p ≥, q,a,b>, r >. They at first gave some conditions about the existence and
nonexistenceofglobalsolutionsto(.),andthenstudiedthelargetimebehaviorforthe
globalsolutions.
Motivatedbytheabovementionedworks,theaimofthispaperisthreefold.First,wede-
termineoptimalconditionsfortheexistenceandnonexistenceofglobalsolutionsto(.).
Secondly, by using the scaling arguments we establish the exact blow-up rate estimates
forsolutionswhichblowupinafinitetime.Finally,weprovethateverysolutionto(.)is
nonextinction.
As it is well known that degenerate equations need not possess classical solutions, we
giveaprecisedefinitionofaweaksolutionto(.).
p qDefinition . Let T > and Q = × (,T), E = {u ∈ L (Q ) ∩ L (Q );u ,∇u ∈T T T t
L (Q )},E = {u ∈E;u=on ∂ },anonnegativefunction u(x,t) ∈E iscalledaweakup-T 
per(orlower)solutionto(.)inQ ifforanynonnegativefunction ϕ ∈E ,onehasT 

m p qu ϕdxdt+ ∇u ∇ϕdxdt ≥(≤) u ϕ–u ϕdxdt,t
Q Q QT T T
u(x,t) ≥(≤) on ∂ ×(,T)and u(x,) ≥(≤)u (x)a.e.in .
In particular, u(x,t) is called a weak solution of (.) if it is both a weak upper and a
weak lower solution. For every T < ∞,if u(x,t) is a weak solution of (.)in Q ,wesayT
thatu(x,t)isglobal.Thelocalintimeexistenceofnonnegativeweaksolutionshavebeen
established (see the survey []), and the weak comparison principle is stated and proved
intheAppendixinthispaper.
Thebehavioroftheweaksolutionsisdeterminedbytheinteractionsamongthemulti-
nonlinearmechanismsinthenonlineardiffusionequationsin(.).Wedividethe(m,p,q)-
parameterregionintothreeclasses:(i)p<max{m,q};(ii)p=max{m,q};(iii)p>max{m,q}.
Theorem. Ifp<max{m,q},thenallsolutionsof(.)arebounded.
Let φ(x)bethefirsteigenfunctionof
– φ (x)= λφ(x)in , φ(x)= in ∂ (.)
withthefirsteigenvalue λ,normalizedby
φ
=,then λ >and φ>in . ∞ 
Theorem. Assumethatp=max{m,q}.Thenallsolutionsareglobalif λ ≥,andthere
existbothglobalandnonglobalsolutionsif λ <.
Theorem. Ifp>max{m,q},thenthereexistbothglobalandnonglobalsolutionsto(.).
To obtain the blow-up rate of blow-up solutions to (.), we need an extra assumption
N that =B ()= {x ∈R : |x|<R}andu =u (r),u (r) ≤,herer= |x|.BytheassumptionR   
and comparison principle, we know that u is radially decreasing in r with max u(x,t)=
u(,t).
Theorem . Suppose that p> max{m,q}.Ifthesolutionu(x,t) of (.) blows up in finite
timeT,thenthereexistsapositiveconstantcsuchthat
–p–maxu(x,t) ≥c(T –t) ast →T.
DuandLiBoundaryValueProblems2012,2012:84 Page3of11
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Furthermore, if p > m ≥ q, then we have also the upper estimate, that is, there exists a
positiveconstantC suchthat
–
p–maxu(x,t) ≤C(T –t) ast →T.

NWe remark that in =R ,Liang[] studied the blow up rate of blow-up solutions to
thefollowingCauchyproblem
m p Nu = u +u,(x,t) ∈R ×(,T)(.)t
N+with the bounded initial function,  < m < p < m ,andobtainedthat
u
<∞ NL (R )(N–)+
–p–C(T –t) fort ∈(,T).Byusingthesamescalingargumentsinthispaper,wecanfind
thatTheorem.iscorrectfor(.)withp>m.
Now,wepayattentiontothenonextinctionpropertyofsolutionsandhavethefollowing
result.
Theorem. Anysolutionof(.)doesnotgoextinctinfinitetimeforanynontrivialand
nonnegativeinitialvalueu (x)withmeas{x ∈ ;u (x)>}>. 
Therestofthispaperisorganizedasfollows.Inthenextsection,wediscusstheglobal
existence and nonexistence of solutions, and prove Theorems .-..Subsequently,in
Sects.and,we considerthe estimateoftheblow-uprateandstudythenonextinction
phenomenon for the problem (.). The weak comparison principle is stated and proved
intheAppendix.
2 Globalexistenceandnonexistence
Proof of Theorem . If m ≥q,thatis p<m, then by the comparison principle, we have
u ≤w,wherewsatisfies
m pw = w +w , x ∈ ,t>,t
w(x,t)=, x ∈ ∂ ,t>, (.)
w(x,)=u (x), x ∈ .
Weknowfrom[,]thatwisbounded.
Ifm<q,wehavep<q.Itisobviousthatu=max{,
u
}isatime-independentupper ∞
solutionto(.).
Proof of Theorem . Since p =q and p=max{m,q} imply p=m>q. Due to the fact that
thesolutionof(.)isanuppersolutionof(.),theconclusionsfor λ ≥isobvioustrue;
see[,].
Nowconsider λ <withsmallinitialdata.Let ψ(x)betheuniquesolutionof
– ψ (x)= in , ψ(x)= on ∂ , (.)
 – m m
mand h(t)solves h(t)=–δh(t) with h()=h ,where< δ≤ ψ
.Set u=h(t)ψ (x). ∞
Then
qm m q m m m q
m mu – u –u +u =–δh ψ +h –h ψ +h ψt
 q m–qm q m–q
m m m=h – δψ +h ψ –h ψ ≥DuandLiBoundaryValueProblems2012,2012:84 Page4