Volume 2009, Article ID 415709,16pages doi:10.1155/2009/415709
Research Article
Existence and Uniqueness of Very Singular
Solution of a Degenerate Parabolic Equation with Nonlinear Convection
Zhong Bo Fang, Daxiong Piao, and Jian Wang
School of Mathematical Sciences, Ocean University of China, Qingdao, 266-071, China
Correspondence should be addressed to Zhong Bo Fang,[email protected] Received 22 October 2008; Revised 24 February 2009; Accepted 8 April 2009 Recommended by Ugur Abdulla
We here investigate the existence and uniqueness of the nontrivial, nonnegative solutions of a nonlinear ordinary differential equation:|f|p−2fβrfαffq0 satisfying a specific decay rate: limr→ ∞rα/βfr 0 withα: p−1/pq−2p2andβ: q−p1/pq−2p2. Herep >2 and q > p−1. Such a solution arises naturally when we study a very singular self-similar solution for a degenerate parabolic equation with nonlinear convection termut |ux|p−2uxx uqxdefined on the half line0,∞.
Copyrightq2009 Zhong Bo Fang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
In this paper, we consider a quasilinear degenerate diffusion equation with nonlinear convection term defined on the half line as
ut |ux|p−2uxx uqx, x, t∈R×R, 1.1
with homogeneous Neumann boundary condition
ux0, t 0, 1.2 wherep >2, q > p−1.
Equation1.1 sometimes called the non-Newtonian filtration equation orp-Laplacian equationarises in the study of the compressible fluid flows in a homogeneous isotropic rigid porous medium, combustion of solid fuels and has various other applications; see, 1, 2.
From a mathematical point of view, we note that 1.1 is a quasilinear equation which is nonuniform parabolic, and it degenerates on the set{ux 0} ifq 1, 1.1reduces to the standardp-Laplacian by an easy change of variables.
We are mostly interested in nonnegative solutions of1.1having the form ux, t t−αf
xt−β
:t−αfr, 1.3
whereα, βare positive numbers. We substitute1.3into1.1and find
α:
p−1
pq−2p2, β:
q−p1
pq−2p2, 1.4
andf,as a function ofr xt−β,solves an ordinary differential equation fp−2f
βrfαf fq
0. 1.5
We observe that ifux, tsolves1.1then the rescaled functions uρx, t ρα/βu
ρx, ρβt
, ρ >0, 1.6
define a one parameter family of solutions to1.1. A solutionux, tis said to be self-similar whenuρx, t ux, t, for everyρ > 0.It can be easily verified thatux, tis a self-similar solution to 1.1 if and only if u has the form 1.3. We also remark that the self-similar solutions play an important role in the study of large time behaviors of general solutions see3–5,and the evolution of interfaces of compactly supported solution of the diffusion- convection eqautionsee6,7.
Every nonnegative, bounded solution of1.5has exactly one critical point and since we here apply the shooting method, led to solve a more general initial value problem,
fp−2f
βrfαf fq
0, 1.7
forr >0 with initial conditions
f0 0, f0 λ, 1.8
whereλmay be any positive number.
Using Shauder’s fixed point theorem or Banach contraction theorem, we find that initial value problem has a unique solution which we denote byfr;λ.In many cases, it turns out that the limit
Lλ lim
r→ ∞rα/βfr, 1.9
exists and we distinguish between fast and slow orbits according to whetherLλ 0 or not, respectively. The fast orbit will bring out a very singular solution of1.1. The very singular solution has a stronger singularity at the origin than the singular solution of that equation. By a singular solution we mean a nonnegative and nontrivial solution which satisfies the equation and vanishes outside any open neighborhood of the origin ast → 0.A singular solution is called a very singular solution if the integral ofux, tover any open neighborhood of the origin becomes unbounded ast → 0,which is equivalent to, ifuis given by1.3,
rlim→ ∞rα/βfr 0. 1.10
Furthermore, if 0 < β < αand a solutionf of1.5satisfies1.10, thenux, tgiven explicitly by1.3becomes a very singular self-similar solution of1.1.
Our goal is to find values ofq and initial data λ which insure that f·, λ is a fast decaying solution and to give an exact asymptotic behavior of solutions near infinity. More precisely, our main results include the following.
iIfα≤βi.e., q≥2p−1, then there does not exist any fast orbit and indeed, only exists slow orbits for anyλ >0.
iiIfα > βi.e.,p−1< q < 2p−1, then there existsλ1such that ifr;λis changing sign forλ∈0, λ1;
iifr;λis a slow orbit having the behavior
fr;λ∼Lλr−α/β, 1.11
near infinity forλ∈λ1,∞, withLλ>0;
iii fr;λ1 is the only fast orbit having the compact support with interface relation
rlim→R−
fp−2/p−1
r − p−2
/ p−1
β1/p−1R1/p−1, 1.12
for some 0< R <∞.
There have been many works dealing with the existence, uniqueness,and qualitative behavior of self-similar solutions to a class of parabolic equations with absorptionor source, convection term. For instance, it is thoroughly treated on the p-Laplacian equation with absorption term:
utdiv
|∇u|p−2∇u
−uq, x, t∈RN×R, 1.13
withp >1, q >1.Forp2 linear diffusion case; see8–10, forp >2 slow diffusion case;
see11and for 1< p < 2fast diffusion case; see3. Recently some papers have studied for a class of heat equation with nonlinear convection term. They derived some estimates and used scaling, convergence of rescaled solutions to self-similar ones and thus obtained the asymptotic of general solutions; see9,12,13for details. Similar arguments have been used
in the case of the porous medium equation; see4. In addition, classification of the singular self-similar solutions is found for the linear diffusion equation with convection on half line under the homogeneous Neumann boundary condition; see14–17and another important application aries in the problem about the evolution of interfaces of compactly supported solutions of the fast diffusion equation with absorption which motivated our investigation;
see6,7.
The plan of the paper is the following. InSection 2, we derive basic properties off which will be useful in the proof of the main results. InSection 3, we show that there does not exist any fast orbit and thus no very singular solution whenq≥2p−1.InSection 4, we find a fast decaying solution whenp−1< q <2p−1.InSection 5, we prove the uniqueness of the fast orbit.
2. Preliminary Results
In this section we will derive some properties off which will be useful in the proof of the main results.
We first show that the sign offdepends on the sign ofα, andfdecreases as long as it is positive.
Lemma 2.1. Assume that α > 0, β > 0, and λ > 0. Let f be a solution to 1.5, 1.8 on 0, R,the maximal existence interval of positive solution withRpossibly infinity. Thenf decreases monotonically in0, R.
Proof. By1.5and1.8we obtain|f|p−2f0 −αλ < 0.Thus, the functionf is strictly decreasing forr near 0.Suppose that there existsr0 < Rsuch thatfr < 0 on0, r0and fr0 0.From1.5one sees|f|p−2fr1<0,which is impossible.
ByLemma 2.1,fr<0 in0, Rfor anyλ >0,and we find that ifR <∞, thenfR 0 andfR−≤0. We next show that iffR− 0, thenfvanishes identically afterR.
Lemma 2.2. Assume thatα >0 andλ >0. Letfbe any solution of 1.5withfR fR 0 for R >0. Thenf 0 for allr≥R.
Proof. By convention,1.5is rewritten as fp−2f
βrfαffq−1f
0. 2.1
Thus, without loss of generality, we may assume thatfr>0 andfr>0 forrnearRwith r > R. For suchr, we find easily from2.1that|f|p−2fr<0. Integrating overR, r, we see that forr > R,|f|p−2fr<0, which contradicts to the assumption.
Lemma 2.3. Assume thatα >0, β >0, andλ >0.Letf be a solution of 1.5,1.8for allr >0.
Then
ilimr→ ∞fr 0, iilimr→ ∞fr 0.
Proof. Sincefis strictly decreasing and bounded below by 0,there exists
rlim→ ∞fr l∈0, λ. 2.2
If we define the energy functionEr p−1/p|f|pα/2f2,then we obtain d
drEr −
f2
βrqfq−1
≤0, 2.3
forr≥0.Thus,Erdecreases monotonically to a limit and there also exists the limit
rlim→ ∞fr −l1, l1∈0,∞. 2.4
In particularl1must be zero. Otherwisefbecomes negative for some positiver.
We now prove thatl0. Suppose to the contrary thatl >0.We find thatαf → αland fq → 0.From1.5, one gets
fp−2f
βrf≤ −α
2l 2.5
at near infinity. Letw|f|p−2f,thenf−|w|1/p−1and we have from2.5that
wβr|w|2−p/p−1w≤ −α
2l. 2.6
Multiplying this by an integrating factor
ρx er0βs|ws|2−p/p−1ds, 2.7
and integrating from 0 tor,we obtain rfp−1
≤ −α 2lrp−1r
0ρrdτ
ρr , 2.8
which in turn implies with the use of L’Hopital theorem
lim sup
r→ ∞ rf≤ − p−1
α
2β l. 2.9
This leads to a contradiction.
3. The case β ≥ α q ≥ 2p − 1
In this section, we show that there does not exist any fast orbit for the problem1.5and1.8 and thus there is no very singular solution for1.1when 0< α≤β.
Theorem 3.1. Assume thatβ≥αq≥2p−1.For eachλ >0,letfr;λbe the solution of1.5, 1.8. ThenR∞, and lim infr→ ∞rα/βfr;λ>0.
Proof. We assume thatR <∞,on the contrary, and integrate1.5over0, Rto get fp−2fR
α−βR
0
frdr−λq0, 3.1
which is impossible. Thusfis positive for allr ≥0 andR∞.
Moreover, we have forr >0,
rα/β−1fp−2fβrα/βf
rα/β−1fp−2f
α/β−1
r fp−2fαfβrf
. 3.2
By1.5, we get
rα/β−1fp−2fβrα/βf
rα/β−1
α/β−1
r fp−2f− fq
>0, 3.3
by the conditionβ ≥ αandf < 0.If we define the functionFr : rα/β−1|f|p−2fβrα/βf, then we see thatF0 0 andFris strictly increasing for allr > 0.Sincefis a decreasing function, one must have lim infr→ ∞rα/βfr;λ>0.
We will see later that the limit limr→ ∞rα/βfr;λexists for eachλ >0. Thus we may conclude together withTheorem 3.1that there exist slow orbits only.
4. The Case α > β p − 1 < q < 2 p − 1
In this section, we first show that the solution changes sign for smallλand we next show that the solution becomes a slow orbit for suitably largeλ.We then find a fast orbit between them.
The slow orbits will be shown to be ordered, and the minimal one becomes the fast orbit as we have seen in many cases; see10,16.
Define the following three sets for any initial valueλ >0, S2
λ >0;R <∞, f R−, λ
<0 , S2
λ >0;R <∞, f R−, λ
0 , S3
λ >0;R∞, fr, λ>0 .
4.1
Obviously, these sets are disjoint andS1∪ S2∪ S3 0,∞.
We first show that the problem1.5,1.8has changed sign for “small”λ >0.
Theorem 4.1. The setS1/∅and open.
Proof. By integrating1.5, one has
fp−2fβrfφr:−
α−βr
0
fdr−fqλq. 4.2
One easily finds thatφ0 0,φr −α−βf−qfq−1f, andφ0 −α−βλ <0.
Suppose thatφr<0 and thus
fp−2fβrf <0, 0< r < r0, 4.3 for somer0to be determined later. An integration of4.3yields
fp−2/p−1r< λp−2/p−1−
p−12 β1/p−1 p
p−2 rp/p−1. 4.4
Thus ifr0 > R0 : pp−2/p−12β1/p−1λp−2/p−1p−1/p, thenfmust change sign and we are done. Otherwise, we may assume thatφr0 0 for somer0≤R0. From definition, we obtainfr0 −β1/p−1r01/p−1f1/p−1r0and
φr0 − α−β
fr0−qfq−1fr0≥0. 4.5
Combining these, we have
0< α−β≤qβ1/p−1r01/p−1fq−1−p−2/p−1. 4.6 Sincefis a decreasing solution, we also havefr0≤λand
α−β≤qβ1/p−1
pp−2 p−12β1/p−1
1/p
λp−2/pp−1q−1−p−2/p−1. 4.7
The inequality4.7does not hold for all sufficiently smallλ, which proves the first part of the theorem. The continuous dependence of solutions on the initial values implies thatS1is an open set.
We next prove that the problem1.5,1.8has a global positive decaying solution for all suitably largeλ.
Lemma 4.2. Letα > β, then for anyR0 there existsλ0such thatfr fr, λ>0 for 0< r < R0
andfR0 |fR0|p−2fR0>0 for allλ≥λ0.
Proof. We definefλt 1/λfr, λ, trλδwithδ q−p1/p−1>0.Thenfλsatisfies fλ0 0, fλ0 1, and the following equation:
fλp−2fλ
λ−qp−2p−1/p−1
βtfλ αfλ
fλq
0. 4.8 By integrating the above equality over0, t,we obtain
fλp−2fλλ−qp−2p−1/p−1
α−βt
0
fλdτλ−qp−2p−1/p−1βtfλ fλq−1
0. 4.9
Sincefλis bounded by 1,for any >0 there isλ0such that wheneverλ≥λ0,
1− <fλp−2fλ fλq <1 , 4.10 fort∈0,qp−2p−1/p−1− ,which implies the lemma.
We also prove the next key observation.
Proposition 4.3. Assume thatα >0,β >0, μ > 0 andf be any globally positive solution to1.5, 1.8. Consider the functionEcr:cfrfforc >0. Then
iifc > α/β, thenEcris eventually positive;
iiifc < α/β, thenEcris eventually negative.
Proof. By direct calculation and1.5, we obtain p−1fp−2 Ecr c1
p−1fp−2f−βr2f−αrf−qrfq−1f, 4.11 and at anyr r0for whichEcr0 0,we have
p−1fp−2 Ecr0 −c1 p−1
cp−1
f/r0p−1
βc−α
r0fqcfq. 4.12 Since the middle term on the right-hand side of4.12dominates the others for all sufficiently larger0, the sign ofEcr0is only decided by the sign ofcβ−αand thusEcrbecomes of the same sign eventually.
In order to provei, we suppose that there existsr1such thatEcr<0 for allr ≥r1. From1.5andLemma 2.1we deduce that
fp−2f
− βc−α
f−βEcr− fq
>0, 4.13
forr≥r1.Multiplying the previous inequality byfand integrating fromrtoτ withr1≤r ≤ τ,we have
p−1
pfpτ−c1f2τ≤
p−1
pfpr−c1f2r, 4.14
wherec1 : βc−α/2.Lettingτ → ∞and using Lemmas 2.1 and2.3, we get the following inequality:
−ff−2/p≥c2 >0, r ≥r1. 4.15
Integrating the previous inequality fromr1tor≥r1, we obtain p/
p−2
fp−2/pr1−p/
p−2
fp−2/pr≥c2r−r1. 4.16
Lettingr → ∞,we get a contradiction.
We proveiisimilarly. Suppose that there existsr2such thatEcr >0 for allr ≥ r2. From1.5and assumption,
fp−2f
αf −βrfαf−qfq−1f≤βcfqc/rfq. 4.17
Sincefdecreases, we may rewrite this as fp−2f
≤ −c2f < c2rf
c , 4.18
where we definec2 α−cβ cq/r2λq−1 and assume it to be positive by retakingr2. The inequality4.18is rewritten asp−1/p−2|f|p−2f≤ −c3rfor some positive constantc3 and an integration fromrr2tor ∞yields a contradiction, which completes the proof. We rewrite the problem1.5,1.8as the following system:
f|h|−p−2/p−1h,
h−βr|h|−p−2/p−1h−αf−qfq−1|h|−p−2/p−1h.
4.19
Given anyδ >0,we denote Lδ:
f, h
: 0< f ≤1, 0> h >−δf
, 4.20
then we obtain the following lemma.
Lemma 4.4. For given δ > 0 there exists arδ : δ αδ−1/p−1/β such that Lδ is positively invariant forr ≥ rδ.That is,fr0, hr0 ∈ Lδforr0 ≥ rδimplies that the orbitfr, hrof 4.19remains in triangle regionLδfor allr≥rδ.
Proof. We will show that givenδ >0 there existsrδ>0 such that ifr≥rδthen the vector field determined by4.19points intoLδ, except at the critical point0, 0. It is easy to see this fact
on the toph0 and the linef1, and it is enough to verify this only on the lineh−δf.By the system4.19, we have
h
f −βr|h|−p−2/p−1h−αf−qfq−1|h|−p−2/p−1h
|h|−p−2/p−1h −βrα
δ|h|p−2/p−1−qfq−1<−βrαδ−1/p−1fp−2/p−1≤ −δ,
4.21
ifr ≥rδ: δαδ−1/p−1/β.
As a consequence, we can prove the existence of globally positive solutions.
Theorem 4.5. The setS3/∅and open.
Proof. From Lemma 4.2, we can find r0 such that f > 0 for 0 ≤ r ≤ r0 and fr0
|fr0|p−2fr0 > 0 for all sufficiently large λ. Thus fr0,|f|p−2fr0 ∈ L1 and by Lemma 4.4,fis positive for allr >0, which proves the first part of the theorem.
We next prove that S3 is an open set. Set λ0 ∈ S3 and then by Proposition 4.3, E1r frfbecomes positive for all larger.Thus there exists sufficiently larger0such that fr0,|f|p−2fr0 ∈ L1.Then by continuous dependence of solutions on the initial value there is a neighborhoodNofλ0 such thatfr;λ > 0 andfr0;λ,|f|p−2fr0;λ ∈ L1 for anyr, λ∈0, r0×N.ByLemma 4.4, we deduce that the orbits remain inL1for anyr > r0, which implies in particular thatfr, λ>0 for anyr > r0andλ∈N.Therefore,fr;λ>0 for anyr >0 andλ∈NandS3is open.
We are now going to find exact decay-rates for globally positive solutions.
Theorem 4.6. For any givenλ >0,letfbe any solution to1.5,1.8such thatf >0 for anyr >0.
Then limr→ ∞rα/βfr;λ Lλ>0 exists.
Proof.
Step 1. By Lemmas 2.1 and 2.3 we know that fr < 0 for r > 0 and limr→ ∞fr 0,limr→ ∞fr 0.Moreover,we have seen that if c < β/α,then Ecr cf rf < 0 for all sufficiently larger; say,r > r0.We easily find that
fr≤fr0r−c, r > r0. 4.22
We also recall that ifd > α/β,thenEdr dfrf>0 and thus
−fr< dfr
r , r > r1, 4.23
for somer1>0.
Step 2. From1.5, we get
rα/β−1fp−2fβrα/βf
rα/β−1 α/β−1
/rfp−2f− fq
, 4.24
and integrating over0, r,we see that rα/β−1fp−2fβrα/βf
α/β−1r
0
fp−2fsα/β−2dsq r
0
fq−1fsα/β−1ds. 4.25 Using4.22and4.23, we find that two integrals of the right hand side of4.25converge and limr→ ∞rα/β−1|f|p−2f0.Therefore, the limit Lλ limr→ ∞rα/βfr, λexists and finite.
Step 3. We now show that Lλ>0.Assume that Lλ 0.Integrating4.24overr,∞,we have
rα/β−1fp−2fβrα/βf
1−α/β∞
r
fp−2fsα/β−2ds−q ∞
r
fq−1fsα/β−1ds.
4.26
Again using4.23, we see that Lλ limr→ ∞rα/βfr, λexists and finite. On the other hand, by4.23,
fr≥fr1r−d, r > r1. 4.27
These conflictions implies that Lλ>0.
Remark 4.7. Obviously, the limit value Lλ 0 is achieved only whenf has the compact support andProposition 4.3andTheorem 4.6remain true for the caseα≤β.
We finally show that there exists a fast orbit.
Theorem 4.8. The setS2/∅and closed. Moreover, the interface relation holds
rlim→R−
fp−2/p−1 r −
p−2
p−1 β1/p−1R1/p−1, 4.28
for anyλ∈ S2.
Proof. By Theorems 4.1 and4.5, we immediately see thatS2is nonempty and closed set. From Lemma 2.2, any solutionf fr, λwith λ ∈ S2 has a compact support; say,0, Randf satisfies conditionfR 0, fR 0.Integrating the equation1.5fromrtoR, we get
fp−2fr −βrfr
α−βR
r
fsds−fqr. 4.29
Dividing byf,we have
fp−2fr/fr −βr
α−βR
r
fsds/fr−fq−1r. 4.30
Sincefis strictly decreasing, we find that
0≤ R
r
fsds≤frR−r. 4.31
Hence
rlim→R−
R
r
fsds/fr 0. 4.32
Lettingr → R−in4.30, then we obtain
rlim→R−
fp−2fr/fr −βR, 4.33
and which is equivalent to the second result of the theorem.
In addition, we show the monotonicity of the solutions of the problem1.5,1.8with respect toλin the sense that two positive orbits do not intersect each other.
Theorem 4.9. Assume thatα >0, β >0 andfiare solutions of problem1.5,1.8on0, Riwith initial datafi0 λi >0, i1,2; where0, Ridenote the maximal existence interval offi andRi are possible infinity. Then
λ2> λ1⇒f2r> f1r, ∀0≤r≤R:min{R1, R2}. 4.34 Proof. Suppose contrarily that there existsR0 ∈ 0, Rsuch thatf1r < f2rforr ∈ 0, R0 andf1R0 f2R0.We define
gkr:k−p/p−2f1kr, r ∈0, R1/k 4.35
fork >0 and thengkrsolves gkp−2gk
βrgkαgkkpq−2p−1/p−2 gkq
0. 4.36
By Lemma 2.1we know thatf1is strictly decreasing on0, R1and sogkis strictly decreasing with respect tok. In particular, limk→0gkr ∞for anyr ∈0, R.Thus there exists a small k0>0 such that
gkr> f2r, forr∈0, R, k∈0, k0, 4.37
and the set
I :
k∈0, k0;gkr> f2r,forr ∈0, R0
4.38 is nonempty and open. Settingl : supI,we see thatl <1,l /∈Iand there existsr0 ∈0, R0 such thatglr0 f2r0.
Ifr0R0, thenglR0 l−p/p−2f1lR0 f2R0. Sincef1R0 f2R0andglis strictly decreasing with respect tol,we conclude thatl1 and which contradicts to the hypothesis.
Ifr0∈0, R0,thenglmust touchf2atrr0from the above. But in this case we deduce from 1.5that
glp−2gl
r0−f2p−2f2 r0
1−lpq−2p−1/p−2 f2q
r0<0, 4.39
which obviously violates the strong maximum principle. Thusglmust touchf2atr0 from the above. But also from1.5, we find|f2|p−2f20 −αλ2and|f2|p−2f20 −f2q0
−qλq−12 f20.Similarly forgl, we obtain, glp−2gl
0−f2p−2f2 0
lpq−2p−1/p−2−1
qλq−12 f20<0, 4.40 which leads to another contradiction and completes all the proofs.
5. Uniqueness
In this section, we show that there exists only one fast decaying solution for the problem1.5, 1.8.
Recall that such a solution has compact support0, Rand has an interface relation
rlim→R−
fp−2/p−1 r −
p−2
p−1β1/p−1R1/p−1, 5.1
byTheorem 4.8.
Theorem 5.1. The setS2consists of only one element.
Proof. Let F and f be any two fast orbits with compact supports 0, Ri, for i 1,2, respectively and satisfyF0> f0.We define
fkr kf k−γr
, γ
p−2
/p, 5.2
and thenfkwill be larger thanFon0, R2for sufficiently largek.We now define τ min
k≥1;fkr≥Fr,0≤r≤R2
. 5.3
The uniqueness proof is now reduced to showing thatτ is not greater than 1.Suppose that τ > 1,on the contrary. We will show that there exists an > 0 such thatfτ− r ≥ Frfor everyr ∈ 0, R2.Indeed, we are going to show thatfτrdoes not touchFrin compact support0, R2by dividing into three cases:
iin the interior of the support;
iiat the origin;
iiiatR2.
In fact,fτrsolves fτp−2fτ
βrfτ αfτ fτq
−τ
1−τq−γ−1 fq
. 5.4
iIffτtouchesFatr0∈0, R2,thenfτr0 Fr0, fτr0 Fr0<0 and fτp−2fτ
r0<Fp−2F
r0, 5.5
butfτr≥Frnearrr0,which obviously violates the strong maximum principle.
iiIffτ touchesF atr0 0,thenfτ0 F0> 0, fτ0 F0 0 and|fτ|p−2fτ
−αfτ0 |F|p−2F0<0.Differentiating the equations1.5and5.4, we reduce that
fτp−2fτ
0−Fp−2F
0 −τ
1−τq−γ−1 fq
0<0. 5.6
Thus, we have
fτp−2fτ
r−Fp−2F
r≤0 5.7
nearr0 0,which leads to a contradiction.
iiiFor the final case, we define the functionsu,Uτ corresponding toFandfτby ux, t :t−αFr,
Uτx, t :t−αfτr :τt−αf τ−γr
, 5.8
whereγ p−2/p, rrt−βas defined before. Thenux, tis a solution of1.1and Uτis a supersolution . Indeed, a straightforward computation shows that
Uτt−
|Uτx|p−2x
Uτx− Uτq
xτ
τq−γ−1−1
fq≥0, for τ >1. 5.9
Following directly the proof of Lemma 10 in18, we can show that for fixedt >0 and all sufficiently smallδ>0, there exists aθθδ∈0,1such thatUτx, t≤Uτx, tδ,if xsatisfiesθR2≤xt−βτ−γ ≤R2and limδ↓0 θδ θ0 ∈0,1.In the proof, we use the interface relation5.1cruciallysee18for details. In particular, we have
Uτx,1≤Uτ
x,1δ
, 5.10
for θR2τγ ≤ x < R2τγ1δβ.In other words, we found a separation near the right end rR2.
On the other hand, as previously proved, fτ cannot touchF at r0 ∈ 0, R2, which implies for any 1>0,there existsκκ 1∈0,1such thatFx≤κfτx,that is,
ux,1≤κUτx,1. 5.11
We choose 1 >0 so that 0< 1<1−θ0and findδ0δ0 1such that 1− 1> θ
δ
, 5.12
forδ∈0, δ0.By continuity ofUτ,there existsδ1δ1 1∈0, δ0such that κUτx,1≤Uτ
x,1δ
, 5.13
for anyδ∈0, δ1and 0 ≤x <1− 1R2τγ.Combining5.10,5.11, and5.13and using again the continuity ofUτ,we deduce that forδ ∈ δ, δ1,whichδ−δ small enough, we have
Fx< Uτx,1δ τ1δ−αf
x1δ−βτ−γ
, 5.14
for anyx≥0.Furthermore, from the continuity with respect toτ,there existsτ1∈0, τsuch that
ux,1 Fx≤τ11δ−αf
x1δ−βτ1−γ
Uτ1x,1δ, 5.15
for anyx≥0.By parabolic maximum principle, we haveux, t≤Uτ1x, tδ,that is, t−αF
xt−β
≤τ1tδ−αf
xtδ−βτ1−γ
, 5.16
for anyt≥1 andx≥0.Rewriting5.16of the form Fr≤τ1t/tδ−αf
rt/tδ−βτ1−γ
, 5.17
and lettingt → ∞,we find that
Fr≤τ1f rτ1−γ
, 5.18
which contradicts the fact thatτis the smallest constant with that property. Thusfτdoes not meet atr0 R2.
Hence we may find >0 so that
fτ− r≥Fr, for everyr ∈0, R2, 5.19 which means that we can slightly reduce the factorτ.Hence we may conclude thatτ 1 but it is obviously impossible.
References
1 E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer, New York, NY, USA, 1993.
2 Z. Wu, J. N. Zhao, J. Yin, and H. Li, Nonlinear Diffusion Equations, World Scientific, River Edge, NJ, USA, 2001.
3 M. Kwak and K. Yu, “Asymptotic behaviour of solutions of a degenerate parabolic equation,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 45, no. 1, pp. 109–121, 2001.
4 Ph. Laurenc¸ot and F. Simondon, “Long-time behaviour for porous medium equations with convection,” Proceedings of the Royal Society of Edinburgh: Section A, vol. 128, no. 2, pp. 315–336, 1998.
5 J. N. Zhao, “The asymptotic behaviour of solutions of a quasilinear degenerate parabolic equation,”
Journal of Differential Equations, vol. 102, no. 1, pp. 33–52, 1993.
6 U. G. Abdulla, “Evolution of interfaces and explicit asymptotics at infinity for the fast diffusion equation with absorption,” Nonlinear Analysis: Theory, Methods & Applications, vol. 50, no. 4, pp. 541–
560, 2002.
7 U. G. Abdulla and J. R. King, “Interface development and local solutions to reaction-diffusion equations,” SIAM Journal on Mathematical Analysis, vol. 32, no. 2, pp. 235–260, 2000.
8 H. Brezis, L. A. Peletier, and D. Terman, “A very singular solution of the heat equation with absorption,” Archive for Rational Mechanics and Analysis, vol. 95, no. 3, pp. 185–209, 1986.
9 M. Escobedo, O. Kavian, and H. Matano, “Large time behavior of solutions of a dissipative semilinear heat equation,” Communications in Partial Differential Equations, vol. 20, no. 7-8, pp. 1427–1452, 1995.
10 M. Kwak, “A semilinear heat equation with singular initial data,” Proceedings of the Royal Society of Edinburgh: Section A, vol. 128, no. 4, pp. 745–758, 1998.
11 L. A. Peletier and J. Y. Wang, “A very singular solution of a quasilinear degenerate diffusion equation with absorption,” Transactions of the American Mathematical Society, vol. 307, no. 2, pp. 813–826, 1988.
12 M. Escobedo and E. Zuazua, “Large time behavior for convection-diffusion equations inRN,” Journal of Functional Analysis, vol. 100, no. 1, pp. 119–161, 1991.
13 M. Escobedo, J. L. V´azquez, and E. Zuazua, “A diffusion-convection equation in several space dimensions,” Indiana University Mathematics Journal, vol. 42, no. 4, pp. 1413–1440, 1993.
14 P. Biler and G. Karch, “A Neumann problem for a convection-diffusion equation on the half-line,”
Annales Polonici Mathematici, vol. 74, pp. 79–95, 2000.
15 M. Guedda, “Self-similar solutions to a convection-diffusion processes,” Electronic Journal of Qualitative Theory of Differential Equations, no. 3, pp. 1–17, 2000.
16 G. Leoni, “On the existence of fast-decay solutions for a quasilinear elliptic equation with a gradient term,” Atti del Seminario Matematico e Fisico dell’Universit`a di Modena, vol. 46, supplement, pp. 827–846, 1998.
17 L. A. Peletier and H. C. Serafini, “A very singular solution and other self-similar solutions of the heat equation with convection,” Nonlinear Analysis: Theory, Methods & Applications, vol. 24, no. 1, pp. 29–49, 1995.
18 S. Kamin and L. V´eron, “Existence and uniqueness of the very singular solution of the porous media equation with absorption,” Journal d’Analyse Math´ematique, vol. 51, pp. 245–258, 1988.
