# Blow-Up Analysis for a Quasilinear Degenerate Parabolic Equation with Strongly Nonlinear Source

Authors: Pan Zheng; Chunlai Mu; Dengming Liu; Xianzhong Yao; Shouming Zhou
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We investigate the blow-up properties of the positive solution of the Cauchy problem for a quasilinear degenerate parabolic equation with strongly nonlinear sourceut=div(|∇um|p−2∇ul)+uq,(x,t)∈RN×(0,T), where N≥1, p>2 , and m, l,q>1, and give a secondary critical exponent on the decay asymptotic behavior of an initial value at infinity for the existence and nonexistence of global solutions of the Cauchy problem. Moreover, under some suitable conditions we prove single-point blow-up for a large class of radial decreasing solutions.


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## Body

## 1.     Introduction

In this paper, we consider the following Cauchy problem to a quasilinear degenerate parabolic equation with strongly nonlinear source(1.1)ut=div⁡(|∇um|p-2∇ul)+uq,(x,t)∈RN×(0,T),u(x,0)=u0(x),x∈RN,
where N≥1, p>2,  m ,  l ,  q>1, and the initial data u0(x) is nonnegative bounded and continuous.Equation (1.1) has been suggested as a mathematical model for a variety of physical problems (see [1]). For instance, it appears in the non-Newtonian fluids and is a nonlinear form of heat equation. Moreover, it can also be used to model the nonlinear heat propagation in a reaction medium (see [2]).One of the particular features of problem (1.1) is that the equation is degenerate at points where u=0 or ∇u=0. Hence, there is no classical solution in general and we introduce the following definition of weak solution (see [3, 4]).Definition 1.1.
A nonnegative measurable functionu(x,t) defined in RN×(0,T) is called a weak solution of the Cauchy problem (1.1) if for every bounded open set Ω with smooth boundary ∂Ω, u∈Cloc(Ω×(0,T)),  um,ul∈Llocp(0,T;W1,p(Ω)), and
(1.2)∫Ω‍uφdx+∫t0t‍∫Ω‍(-uφt+|∇um|p-2∇ul⋅∇φ)dxdt=∫t0t‍∫Ω‍uqφdxdt+∫Ω‍u(x,t0)φ(x,t0)dx
for all 0≤t0≤t≤T and all test functions  φ∈C01(Ω×(0,T)). Moreover,
(1.3)lim⁡t→t0∫Ω‍u(x,t)η(x)dx=∫Ω‍u(x,t0)η(x)dx
for any η(x)∈C01(Ω).Under some suitable assumptions, the existence, uniqueness and regularity of a weak solution to the Cauchy problem (1.1) and their variants have been extensively investigated by many authors (see [5–7] and the references therein).The first goal of this paper is to study the blow-up behavior of solutionu(x,t) of (1.1) when the initial data u0(x) has slow decay near x=∞. For instance, in the following case
(1.4)u0(x)≅M|x|-awithM>0,a≥0,
we investigate the existence of global and nonglobal solutions for the Cauchy problem (1.1) in terms of M and a. In recent years, many authors have studied the properties of solutions to the Cauchy problem (1.1) and their variants (see [8–17] and the references therein). In particular, J.-S. Guo and Y. Y. Guo [18] obtained the secondary critical exponent for the following porous medium type equation in high dimensions:
(1.5)ut=Δum+up,(x,t)∈RN×(0,T),u(x,0)=u0(x),x∈RN,
where p>1, m>1 or max{0,1-(2/N)}<m<1, u0(x) is nonnegative bounded and continuous, and proved that for p>pm*=m+(2/N), there exists a secondary critical exponent a*=2/(p-m) such that the solution u(x,t) of (1.5) blows up in finite time for the initial data u0(x), which behaves like |x|-a at x=∞ if a∈(0,a*), and there exists a global solution for the initial data u0(x), which behaves like |x|-a at x=∞ if a∈(a*,N). Here, we say that the solution blows up in finite time; it means that there exists T∈(0,+∞) such that ∥u(·,t)∥L∞<∞ for all t∈[0,T), but limt→T-sup∥u(.,t)∥L∞=∞.Mu et al. [19] studied the secondary critical exponent for the following p-Laplacian equation with slow decay initial values:
(1.6)ut=div⁡(|∇u|p-2∇u)+uq,(x,t)∈RN×(0,T),u(x,0)=u0(x),x∈RN,
where p>2, q>1, and showed that, for q>qc*=p-1+(p/N), there exists a secondary critical exponent ac*=(p/(q+1-p)) such that the solution u(x,t) of (1.6) blows up in finite time for the initial data u0(x) which behaves like |x|-a at x=∞ if a∈(0,ac*), and there exists a global solution for the initial data u0(x), which behaves like |x|-a at x=∞ if a∈(ac*,N).Recently, Mu et al. [20] also investigated the secondary critical exponent for the doubly degenerate parabolic equation with slow decay initial values and obtained similar results.On the other hand, in this paper, we will also consider single-point blow-up for the Cauchy problem (1.1). It is interesting to study the set of blow-up points and the behavior of the solution u(x,t) at the blow-up point.In order to investigate single-point blow-up for the Cauchy problem (1.1), we introduce the concept of the blow-up point.Definition 1.2.
A pointx∈Ω is called a blow-up point if there exists a sequence (xn,tn) such that xn→x, tn→T- and u(xn,tn)→∞ as  n→∞, where T is blow-up time.In recent years, some authors also studied single-point blow-up for the Cauchy problem to nonlinear parabolic equations (see [21, 22] and the references therein) by different methods. In particular, when p=2, l=1 and N=1, the Cauchy problem (1.1) has been investigated by Weissler in [23], and the author obtained that the solution blows up only at a single point. Galaktionov and Posashkov [24] studied the single-point blow-up and gave the upper and lower bound near the blow-up point for the Cauchy problem (1.1) when p>2 and m=l=1. Recently, when p>2 and m=l, Mu and Zeng [25] extended Galaktionov's results to the doubly degenerate parabolic equation. For more works about single-point blow-up, we refer to [26, 27], where the parabolic systems have been considered.Motivated by the above works, based on a modification of the energy methods, comparison principle, and regularization methods used in [15, 19, 21, 24], we investigate the secondary critical exponent and single-point blow-up for the Cauchy problem (1.1). Before stating the results of the secondary critical exponent, we start with some notations as follows.LetCb(RN) be the space of all bounded continuous functions in RN. For a≥0, we define
(1.7)Φa={ϕ(x)∈Cb(RN)∣ϕ(x)≥0,lim⁡|x|→∞sup⁡|x|aϕ(x)<∞},Φa={ϕ(x)∈Cb(RN)∣ϕ(x)≥0,lim⁡|x|→∞inf⁡|x|aϕ(x)>0}.
Moreover, we denote
(1.8)qc*=l+m(p-2)+pN,ac*=pq-l-m(p-2).
Our main results of this paper are stated as follows.Theorem 1.3.
ForN≥2, p>2, m>1, l>1, and  q>qc*=l+m(p-2)+(p/N), suppose that u0(x)∈Φa for some a∈(0,ac*); then the solution u(x,t) of the Cauchy problem (1.1) blows up in finite time.Theorem 1.4.
ForN≥2, p>2, m>1, l>1, and q>qc*=l+m(p-2)+(p/N), suppose that u0(x)=λφ(x) for some λ>0 and φ(x)∈Φa for some a∈(ac*,N); then there is λ0=λ0(φ)>0 such that the solution u(x,t) of the Cauchy problem (1.1) exists globally for all t>0, and if λ<λ0, one has
(1.9)||u(x,t)||∞≤Ct-aβ,∀t>0,
where β=1/(a(l+m(p-2)-1)+p), C>0.Remark 1.5.
Whenp>2, N≥2 and q>qc*, we have qc*>1 and 0<ac*<N.Remark 1.6.
It follows from Theorems1.3 and 1.4 that the number ac*=p/(q-l-m(p-2)) gives another cut-off between the blow-up case and the global existence case. Therefore, the number ac* is a new secondary critical exponent of the Cauchy problem (1.1). Unfortunately, in the critical case a=ac*, we do not know whether the solution of (1.1) exists globally or blows up in finite time.Remark 1.7.
Whenm=l=1 or m=l>1, the results of Theorems 1.3 and 1.4 are consistent with those in [19, 20], respectively.Remark 1.8.
In [28], Afanas’eva and Tedeev also established the Fujita type results for (1.1) with m=l. In particular, if u0(x)~|x|-a, 0<a<N, they obtained that if q<m(p-1)+(p/a), then every nontrivial solution blows up in finite time, and if q>m(p-1)+(p/a), then the solution exists globally for a small initial data u0(x). We note that when m=l in (1.1), if q>m(p-1)+(p/N) and 0<a<p/(q-m(p-1)), then 0<a<N and q<m(p-1)+(p/a), while if q>m(p-1)+(p/N) and p/(q-m(p-1))<a<N, then q>m(p-1)+(p/a). Therefore, the results of Theorems 1.3 and 1.4 coincide with those in [28].Finally, we also consider single-point blow-up for a large number of radial decreasing solutions of the Cauchy problem (1.1) and give upper bound of the radial solution u(r,t) in a small neighborhood of the point (x,t), where x=0, t=T. We assume that the initial data u0(x)=u0(r) satisfies the following condition:(H)
u0(x)=u0(r)≥0 for r>0, u0(0)>0, and  u0(r)∈C1(R+1), u0′(0)=0, and u0′(r)≤0 for r>0, M0=supu0(r)<+∞, K0=sup|u0′(r)|<+∞.Theorem 1.9.
LetN≥1,p>2,m>1,l>1,andq>l+m(p-2), and let condition (H) hold. In addition, assume that the initial function u0(x)=u0(r) satisfies
(1.10)u0q(r)⋅rN=o(1)asr⟶∞,(1.11)λ0=inf⁡r>0,u0(r)>0{-|(u0m)′|p-2(u0l)′ru0q}∈(0,p-2(p-2)(N+1)+2].
Let T be the blow-up time; then one has
(1.12)u(r,t)≤Cr-p/(q-l-m(p-2)),(r,t)∈R+1×(0,T),
where
(1.13)C=[q-l-m(p-2)(lmp-2)1/(p-1)pλ01/(p-1)]-((p-1)/(q-l-m(p-2)))>0,
that is, there is single-point blow-up at point x=0.Remark 1.10.
By (1.11), the best upper estimate (1.12) obtained by our method has the following form:
(1.14)u(r,t)≤[q-l-m(p-2)p(lmp-2)1/(p-1)(p-2(p-2)(N+1)+2)1/(p-1)]-((p-1)/(q-l-m(p-2)))⋅r-(p/(q-l-m(p-2)))
in R+1×(0,T). But, we do not give the lower bound estimate of the radial solution u(r,t) in a small neighborhood of the point (x,t), where x=0, t=T.Remark 1.11.
Whenm=l=1 or m=l>1, the results of Theorem 1.9 are consistent with those in [24, 25], respectively. For 1<q<l+m(p-2), in [29], the authors obtained the results of global blow-up to arbitrary compactly supported initial data.Remark 1.12.
From Theorem1.9, we obtain the same decay exponent as that of Theorem 1.2 in [29] by different methods. Moreover, it is interesting to see that the decay exponent of the upper estimate of Theorem 1.9 is also the same as the secondary critical exponent of Theorems 1.3 and 1.4.This paper is organized as follows. In Section2, by using the energy method, we will obtain a blow-up condition and prove Theorem 1.3. In Section 3, using the comparison principle, we can construct a global supersolution to prove Theorem 1.4. Finally, we consider the single-point blow-up under some suitable conditions and prove Theorem 1.9 in Section 4.


## 2.   Blow-Up Case

By using the energy method, we will obtain a blow-up condition corresponding to (1.1). Therefore, we need to select a suitable test function as follows:
(2.1)ψε(x)=Aεe-ε|x|,Aε=1∫RN‍e-ε|x|dx=εN∫ωN‍∫0∞‍e-rrN-1drds,∇ψε(x)=-εψεx|x|.Proof of Theorem1.3.
Suppose thatu(x,t) is the solution of the Cauchy problem (1.1) and T is the blow-up time. Let
(2.2)E(t)=1s∫RN‍us(x,t)ψε(x)dx,t∈[0,T),
where 0<s<1/p,p>2, Then, E(t)∈C[0,T)⋂C1(0,T) and
(2.3)E′(t)=∫RN‍us-1ψεutdx=∫RN‍us-1ψεdiv⁡(|∇um|p-2∇ul)dx+∫RN‍uq+s-1ψεdx=-l(s-1)∫RN‍us+l-3ψε|∇um|p-2|∇u|2dx+ε∫RN‍us-1ψε|∇um|p-2∇ulx|x|dx+∫RN‍uq+s-1ψεdx≥lmp-2(1-s)∫RN‍ul+m(p-2)+s-p-1ψε|∇u|pdx-lmp-2ε∫RN‍ul+m(p-2)+s-pψε|∇u|p-1dx+∫RN‍uq+s-1ψεdx.
Using Young's inequality, we have
(2.4)ε∫RN‍ul+m(p-2)+s-pψε|∇u|p-1dx≤p-1p∫RN‍ul+m(p-2)+s-p-1ψε|∇u|pdx+εpp∫RN‍ul+m(p-2)+s-1ψεdx.
Since 0<s<(1/p), it follows from (2.3) and (2.4) that
(2.5)E′(t)≥∫RN‍uq+s-1ψεdx-lmp-2εpp∫RN‍ul+m(p-2)+s-1ψεdx.
By q>qc*=l+m(p-2)+(p/N)>l+m(p-2)>1, ∫RN‍ψε(x)dx=1, and Hölder’s inequality, we obtain
(2.6)∫RN‍ul+m(p-2)+s-1ψεdx=∫RN‍ul+m(p-2)+s-1ψε(l+m(p-2)+s-1)/(q+s-1)ψεq-[l+m(p-2)]/(q+s-1)dx≤[∫RN‍uq+s-1ψεdx](l+m(p-2)+s-1)/(q+s-1).
Therefore, by (2.5) and (2.6), we have
(2.7)dEdt≥[∫RN‍uq+s-1ψεdx](l+m(p-2)+s-1)/(q+s-1)×[(∫RN‍uq+s-1ψεdx)(q-l-m(p-2))/(q+s-1)-lmp-2εpp].
Applying Jensen’s inequality, we obtain
(2.8)(∫RN‍uq+s-1ψεdx)(q-l-m(p-2))/(q+s-1)≥(∫RN‍usψεdx)(q-l-m(p-2))/s.
Thus, it follows from (2.7) and (2.8) that
(2.9)dEdt≥12(∫RN‍usψεdx)(q+s-1)/s=12s(q+s-1)/sE(q+s-1)/s(t)
as long as
(2.10)E(t)≥1s(2lmp-2εpp)s/(q-l-m(p-2))∀t∈[0,T).
Hence, if E(0) satisfies
(2.11)E(0)≥1s(2lmp-2εpp)s/(q-l-m(p-2))=C0,
then E(t) increases and remains below C0 for all t∈[0,T).
And by (2.9) we have
(2.12)E(t)≥(Eq-1/s(0)-C1t)-s/(q-1),whereC1=q-12s(q-1)/s>0.
Therefore, from (2.11) and (2.12), we obtain that u(x,t) blows up in finite time T=(1/C1)Eq-1/s(0): and get an estimate on the blow-up time T of the solution u(x,t) as follows:
(2.13)T≤2q-1(2lmp-2εpp)(1-q)/(q-l-m(p-2)).
Finally, it remains to verify the blow-up condition (2.11). Since u0(x)∈Φa for some a∈(0,ac*), there exist two positive constants M and R0>1 such that u0(x)≥M|x|-a for all |x|≥R0, and we have
(2.14)E(0)=1s∫RN‍u0s(x)ψε(x)dx>MsAεs∫|x|≥R0‍|x|-ase-ε|x|dx=MsAεsε-N+as∫|y|≥εR0‍|y|ase-|y|dy.
By the definition of Aε, 0<a<ac*, we can choose 0<ε≤(1/R0) so small such that (2.11) holds. The proof of Theorem 1.3 is complete.


## 3. Global Existence

In this section, we shall prove Theorem1.4 by constructing a global supersolution. To do this, we introduce the radially symmetric self-similar solution UM,a(x,t) to the following Cauchy problem:
(3.1)ut=div⁡(|∇um|p-2∇ul),(x,t)∈RN×(0,+∞),(3.2)u(x,0)=u0(x)=M|x|-a,x∈RN.
It is well known that the existence and uniqueness of the solution of (3.1) have been well established (see [7]). By symmetric properties of (3.1), the solution UM,a(x,t) is given by the following form
(3.3)UM,a(x,t)=t-aβfM(r),withr=|x|tβ,β=1a(l+m(p-2)-1)+p,
where the positive function fM is the solution of the problem
(3.4)(|(fMm)′|p-2(fMl)′)′+N-1r|(fMm)′|p-2(fMl)′(r)+βrfM′(r)+aβfM(r)=0,r>0,fM(r)≥0,r≥0,fM′(0)=0,lim⁡r→+∞rafM(r)=M.
We shall prove the existence of solution fM(r) to (3.4) by the following ordinary differential equation, and furthermore we obtain the nonincreasing property of the solution fM(r).Firstly, given a fixedη>0, we consider the following Cauchy problem:
(3.5)(|(gm)′|p-2(gl)′)′+N-1r|(gm)′|p-2(gl)′(r)+βrg′(r)+aβg(r)=0,r>0,g(0)=η,g′(0)=0.
According to the standard of the Cauchy problem for ODE and the methods used in [7, 30], we can obtain that the solution g(r) of the Cauchy problem (3.5) is positive, and g(r)→0 as r→∞; moreover,
(3.6)lim⁡r→+∞rag(r)=M
for some M=M(η)>0.Secondly, we shall prove that there exists a one-to-one correspondence betweenM∈(0,+∞) and η∈(0,+∞). Indeed, this can be seen from the following relation:
(3.7)gη(r)=ηg1(ησr),σ=1-l-m(p-2)p,
where g1(r) is the solution of (3.5) for η=1. Then,
(3.8)M(η)=η1-aσM(1),withM(1)=lim⁡r→+∞rag1(r).
Therefore, we can deduce that, for each M>0, there exists a positive, bounded, and global solution fM(r) satisfying (3.4).Finally, we shall prove that the solutiong(r) is non-increasing, that is, fM(r) is also non-increasing. To do this, we need the following lemmas.Lemma 3.1.
Letg(r) be the solution of (3.5); then
(3.9)lim⁡r→0N|(gm)′(r)|p-2(gl)′(r)r=-aβg(0).Proof.
Integrating the (3.5) over (0,ε) with ε>0, we have
(3.10)(|(gm)′|p-2(gl)′)(ε)+∫0ε‍N-1r|(gm)′|p-2(gl)′dr+∫0ε‍βrg′dr+∫0ε‍aβgdr=0.
Dividing by ε and taking ε→0 in (3.10), we obtain
(3.11)lim⁡ε→0[|(gm)′(ε)|p-2(gl)′(ε)ε+N-1ε|(gm)′(ε)|p-2(gl)′(ε)]=-lim⁡ε→0aβg(ε),
which implies that (3.9) holds. The proof of Lemma 3.1 is complete.Lemma 3.2.
If there existsr0∈[0,+∞) such that g(r0)=0, then g(r)=0 for all r≥r0.Proof.
We shall prove by contradiction. Assuming that Lemma3.2 does not hold, it is easy to see that there exists ε>0 such that
(3.12)g(r)>0,g′(r)>0in(r0,r0+ε).
Multiplying (3.5) by rN-1 and integrating over (r0,r) with r∈(r0,r0+ε), we obtain
(3.13)rN-1|(gm)′|p-2(gl)′+βrNg(r)=∫r0r‍NβrN-1g(r)dr-∫r0r‍aβrN-1g(r)dr.
It follows from (3.12) and (3.13) that
(3.14)βrNg(r)≤∫r0r‍NβrN-1g(r)dr-∫r0r‍aβrN-1g(r)dr≤(N-a)βNg(r)(rN-r0N),
equivalently,
(3.15)1≤N-aN(1-(r0r)N).
Letting  r→r0 in (3.15), we obtain the inequality 1≤0, which is a contradiction. The proof of Lemma 3.2 is complete.Lemma 3.3.
The solutiong(r) of (3.5) is monotone nonincreasing in [0,+∞).Proof.
Our method is based on the contradiction argument. Suppose that, for somer0>0, g′(r0)>0, by Lemma 3.1, there exists r1∈(0,r0) such that
(3.16)g′(r1)=0,(|(gm)′|p-2(gl)′)′(r1)≥0.
By Lemma 3.2, we have g(r1)>0. Using the similar argument in Lemma 3.1, we obtain
(3.17)lim⁡r→r1N|(gm)′(r)|p-2(gl)′(r)r-r1=-aβg(r1)<0,
which is a contradiction with (3.14). The proof of Lemma 3.3 is complete.Next, we apply the monotone properties offM(r) to obtain the condition on the global existence of the solution to (1.1).Proof of Theorem1.4.
We prove Theorem1.4 by the following steps.
Step1. Since φ(x)∈Φa, there exists a constant K>0 such that
(3.18)φ(x)≤K(1+|x|)-a∀x∈RN.
Taking M>K and the self-similar solution UM,a(x,t) of (3.1) defined as (3.3), since limr→+∞rafM(r)=M>K, there exists a positive constant R0 such that
(3.19)rafM(r)>Kforr≥R0.
Setting γ=fM(R0)=min{fM(r)∣r∈[0,R0]}>0, it is easy to verify that φ(x)≤UM,a(x,t) for all x∈RN, where t0∈(0,1) and t0-aβγ>∥φ∥∞.
Letλ>0; then w(x,t)=λUM,a(x,λl+m(p-2)-1t+t0) is the solution of the following problem
(3.20)wt=div⁡(|∇wm|p-2∇wl),(x,t)∈RN×(0,+∞),w(x,0)=λUM,a(x,t0)≥λφ(x),x∈RN.
Taking η=fM(0) and noting that fM(r) is non-increasing, we have
(3.21)‖w(x,t)‖∞=ηλ(λl+m(p-2)-1t+t0)-aβ.
Step2.Set v(x,t)=A(t)w(x,B(t)), where A(t) and B(t) are solutions of the following problem:
(3.22)A′(t)=ηq-1λq-1[λl+m(p-2)-1B(t)+t0]-(a(q-1)/(a(l+m(p-2)-1)+p))Aq(t),t∈(0,+∞),B′(t)=Al+m(p-2)-1(t)t∈(0,+∞),A(0)=1,B(0)=0.
By a direct calculation, we obtain that v(x,t) satisfies
(3.23)vt≥div⁡(|∇vm|p-2∇vl)+vq,(x,t)∈RN×(0,+∞),v(x,0)=w(x,0)=λUM,a(x,t0)≥λφ(x),x∈RN.
Step  3. We shall prove that there exists a positive constant λ0=λ0(φ) such that the problem (3.22) has a global solution (A(t),B(t)) with A(t) bounded in [0,T) if λ∈[0,λ0). According to the standard theory of ODE, the local existence and uniqueness of solution (A(t),B(t)) of (3.22) hold. By (3.22), we have A′(t)>0,A(t)>1 for t>0; furthermore, the solution is continuous as long as the solution exists and A(t) is finite.
From (3.22), when A(t) exists in [0,t], then B(t) is uniquely defined by
(3.24)B(t)=∫0t‍Al+m(p-2)-1(s)ds.
Since p>2 and A(t) is increasing, we obtain
(3.25)B(s)=∫0s‍Al+m(p-2)-1(τ)dτ≥Al+m(p-2)-1(0)s=s∀s∈[0,t].
By (3.22), (3.25), and a>ac*=p/(q-l-m(p-2)), it follows that
(3.26)1-A1-q(t)=(q-1)ηq-1λq-1∫0t‍[λl+m(p-2)-1B(s)+t0]-(a(q-1)/(a(l+m(p-2)-1)+p))ds≤(q-1)ηq-1λq-1∫0t‍(λl+m(p-2)-1s+t0)-(a(q-1)/(a(l+m(p-2)-1)+p))ds≤(q-1)ηq-1λq-l-m(p-2)((a(q-1))/(a(l+m(p-2)-1)+p))-1t01-(a(q-1)/(a(l+m(p-2)-1)+p)).
Let λ0=λ0(φ) be a positive constant defined by
(3.27)(q-1)[a(l+m(p-2)-1)+p]a(q-l-m(p-2))-pηq-1λ0q-l-m(p-2)t01-(a(q-1)/(a(l+m(p-2)-1)+p))=12.
Then from (3.26), q>qc*>l+m(p-2)>1 and a>ac*=p/(q-l-m(p-2)), we have 1≤A(t)≤21/q-1 for any λ∈(0,λ0], as long as A(t) exists globally.
On the other hand, by (3.22) and (3.25), we have
(3.28)t≤B(t)≤2(p-2)/(q-1)t∀t≥0.
Therefore, B(t) is also global.
Step  4. For any λ∈(0,λ0], where λ0=λ0(φ) is defined as (3.27), the solution u(x,t) of (1.1) with initial value u0(x)=λφ(x) exists globally, and u(x,t)≤v(x,t) in RN×(0,+∞). Moreover, there exists a positive constant C such that
(3.29)‖u(⋅,t)‖∞≤‖v(.,t)‖∞≤21/(q-1)ηλ(λl+m(p-2)-1B(t)+t0)-aβ≤Ct-aβ∀t>0.
The proof of Theorem 1.4 is complete.


## 4. Single Point Blow-Up

In this section, under some suitable assumptions, we shall prove that the blow-up set consists of the single pointx=0. Moreover, we also give the upper estimate of the solution u(x,t) in a small neighborhood of the point (x,t), where x=0, t=T.First, we suppose that the solution is radially symmetric, that is, depending only onr=|x| at a given time t>0. Therefore, we study the following problem:
(4.1)ut=r-(N-1)[rN-1|(um)r|p-2(ul)r]r+uq,(r,t)∈(0,+∞)×(0,+∞),u(r,0)=u0(r),r∈(0,+∞),rN-1|(um)r|p-2(ul)r=0,(r,t)∈{0}×[0,+∞).Proof of Theorem1.9.
It is based on the method in [21]. The main idea is to apply the maximum principle to the auxiliary function ωi(r,t), which is defined in (4.7), and to show that ωi is small enough in (0,i)×(0,T). Then by integrating the obtained inequality and taking limit as i→∞, one can get upper bound of the solution u(r,t). Therefore, we divide the proof into the following steps.
Step1. Since problem (1.1) has no classical solution, we will construct the weak solution by means of regularization of the degenerate equation.
Now define a strictly monotone sequence{εi}, εi>0 for all i=1,2,3,…, such that
(4.2)εi⟶0,asi⟶+∞.
Then, the weak solution u(r,t) is the limit function of the solution of the following regularized problem (see [31]):
(4.3)(ui)t=r-(N-1)[rN-1(((uim)r)2+εi2)(p-2)/2((uil)r+εi)]r+uiq,(r,t)∈(0,i)×(0,T),ui(r,0)=u0(r),r∈(0,i),ui(i,t)=u0(i),t∈(0,T),rN-1(((uim)r)2+εi2)(p-2)/2((uil)r+εi)=0,(r,t)∈{0}×(0,T).
By the standard methods used in [1, 32], the uniform estimates for the passage to the limit which do not depend on εi are established. Therefore, for any fixed εi>0, we may assume that, for all sufficiently large i, the function ui(r,t) satisfies the following conditions:
(4.4)|ui|≤M1,|(ui)r|≤M2inQi,T=(0,i)×(0,T),
where M1, M2 do not depend on i, and
(4.5)(ui(r,t))r∣r=0=0∀t∈(0,T).
Moreover, by using condition (H) and the maximum principle in [33], we have
(4.6)(ui(r,t))r≤0inQi,T.
Step2. Set
(4.7)wi(r,t)=rN-1(((uim)r)2+εi2)(p-2)/2(uil)r+λ0rNuiq,
where λ0 is given in (1.11).
By a direct calculation, we find thatwi(r,t) satisfies the following parabolic equation:
(4.8)(wi)t=ai(wi)rr+bi(wi)r+ciwi+di+ei,
where
(4.9)ai=luil-1(((uim)r)2+εi2)(p-2)/2+m(p-2)uim-1(((uim)r)2+εi2)(p-4)/2(uil)r(uim)r,(4.10)bi=(((uim)r)2+εi2)(p-2)/2[l(1-N)uil-1r+l(l-1)uil-2(ui)r]+(p-2)(uil)r(uim)r⋅(((uim)r)2+εi2)(p-4)/2[m(1-N)uim-1r+m(m-1)uim-2(ui)r],(4.11)ci=quiq-1[(p-1)-λ0((p-2)(N+1)+2)],(4.12)di=AqrN-1[1-λ0(N+1)]uiq-1,withA≡-(p-2)εi2(uil)r[((uim)r)2+εi2](p-4)/2,(4.13)ei=(p-2)rN-1(((uim)r)2+εi2)(p-4)/2(uil)r(uim)r×[-(m+q-2)mqrλ0uiq+m-3((ui)r)2+((m-1)(1-λ0N)+q(1-λ0(N+1)))muq+m-2(ui)r]+rN-1(((uim)r)2+εi2)(p-2)/2[-(l+q-2)lqrλ0uiq+l-3((ui)r)2+((l-1)(1-λ0N)+q(1-λ0(N+1)))luq+l-2(ui)r]+λ0q[λ0((p-2)(N+1)+2)-(p-2)]rNui2q-1.
Since N≥1,p>2,m>1,l>1,andq>l+m(p-2), it follows from (1.11) and (4.6) that ei≤0.
Now we consider the coefficientsci and di in Qi,T. By (4.6) and p>2, we have
(4.14)A=-(p-2)εi2(uil)r[((uim)r)2+εi2](p-4)/2≥0.
It follows from (4.4) that |A|=O(εiν) as i→∞, where ν=min{2,p-1}, and we obtain the following estimate:
(4.15)sup⁡ci≤M3,sup⁡di≤M4iN-1εiνinQi,T,
where M3, M4 are positive constants, which are independent of i.
Therefore, we have the parabolic differential inequality(4.16)(wi)t≤ai(wi)rr+bi(wi)r+ciwi+di,inQi,T,
where ci, di satisfy (4.15).
Next, we consider the functionwi(r,t) on the parabolic boundary of Qi,T. At first, it is easy to see that wi(0,t)=0 for all t∈(0,T). By (1.10), we have wi(i,t)≤λ0iNu0q(i)=o(1), as i→∞ for all t∈(0,T). Finally, it follows from (1.11) that
(4.17)wi(r,0)=rN-1(((u0m)r)2+εi2)(p-2)/2(u0l)r+λ0rNu0q≤rN-1[|(u0m)r|p-2(u0l)r+λ0ru0q]≤0∀r∈[0,i].
Hence, for all sufficiently largei, there exists γi=supwi>0 on the parabolic boundary of Qi,T and γi=o(1) as i→∞.
In order to estimatewi(r,t) in Qi,T, we study the following ODE:
(4.18)dw¯idt=M3w¯i+M4iN-1εiν,t>0,w¯i(0)=γi,
which has the solution
(4.19)w¯i(t)=(M4iN-1εiνM3+γi)eM3t-M4iN-1M3εiν,t>0.
Taking the sequence εi such that
(4.20)iN-1εiν⟶0asi⟶∞,
it is obvious that
(4.21)w¯i(t)≤w¯i(T)=αi∀t∈(0,T),
where
(4.22)αi⟶0asi⟶∞.
Setting
(4.23)z(r,t)=wi(r,t)-w¯i(t),
we have z(r,t)≤0 on the parabolic boundary, and z(r,t) satisfies the following parabolic inequality:
(4.24)zt≤aizrr+bizr+ciz-(M3-ci)w¯i+[di-M4iN-1εiν],inQi,T.
It follows from (4.15) that
(4.25)zt≤aizrr+bizr+ciz,inQi,T.
By the maximum principle (Chapter II, [33]), we obtain that z(r,t)≤0 in Qi,T, that is,
(4.26)wi(r,t)≤w¯i(t)≤w¯i(T)=αiinQi,T.
Step3. For large i and (ui(r,t))r∈[-M2,0], we have the following estimate
(4.27)(((uim)r)2+εi2)(p-2)/2(uil)r≥|(uim)r|p-2(uil)r-δi,
where
(4.28)0<δi=O(εiν)⟶0asi⟶∞.
By (4.7) and (4.26), we have
(4.29)|(uim)r|p-2(uil)ruiq≤r1-Nαi+δiuiq-λ0r,
namely,
(4.30)lmp-2ui(l-1)+(m-1)(p-2)-q(ui)r|(ui)r|p-2≤r1-Nαi+δiuiq-λ0r.
Letting  F(b)=b|b|-((p-2)/(p-1)), we deduce that
(4.31)(lmp-2)1/(p-1)ui((l+m(p-2)-q-p+1)/(p-1))(ui)r≤F[r1-Nαi+δiuiq-λ0r].
For arbitrary t0∈(0,T), r0∈supp u(r,t0)≡{r>0∣u(r,t0)>0}, we denote μ0=u(x0,t0)>0. By (4.6) and the uniform convergence ui(r,t0)→u(r,t0) as i→∞, r∈[0,r0], we have ui(r,t0)≥(μ0/2),   r∈[0,r0] for all sufficiently large i≥i0. Therefore, from (4.31) we obtainly for t=t0,i>i0,
(4.32)(lmp-2)1/(p-1)ui((l+m(p-2)-q-p+1)/(p-1))(ui)r≤F[2q(r1-Nαi+δi)μ0q-λ0r].
Integrating the above inequality over interval [r1,r0], where r1>0, we obtain
(4.33)-(lmp-2)1/(p-1)(p-1)q-l-m(p-2)ui-((q-l-m(p-2))/(p-1))|r1r0≤∫r1r0‍F[2q(r1-Nαi+δi)μ0q-λ0r]dr.
Letting i→∞, by (4.22), (4.28), and the uniform convergence ui(r,t0)→u(r,t0) as i→∞, r∈[0,r0], we have the following estimate:
(4.34)-(lmp-2)1/(p-1)(p-1)q-l-m(p-2)[u-((q-l-m(p-2))/p-1)(r0,t0)-u-((q-l-m(p-2))/p-1)(r1,t0)]≤lim⁡i→∞∫r1r0‍F[2q(r1-Nαi+δi)μ0q-λ0r]dr=-λ01/(p-1)p-1p(r0p/(p-1)-r1p/(p-1)).
Setting r1→0, from (4.34), we obtain the upper estimate
(4.35)u(r0,t0)≤[q-l-m(p-2)p(lmp-2)1/p-1λ01/p-1]-((p-1)/(q-l-m(p-2)))r0-(p/(q-l-m(p-2))),
where (r0,t0)∈R+1×(0,T). The proof of Theorem 1.9 is complete.



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*Source: 109546-2012-07-18.xml*