# Spatial Profile of the Dead Core for the Fast Diffusion Equation with Dependent Coefficient

Authors: Zhengce Zhang; Biao Wang
Journal: International Journal of Differential Equations
 (2011)

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## Abstract

We consider the dead-core problem for the fast diffusion equation with
spatially dependent coefficient and obtain precise estimates on the single-point
final dead-core profile. The proofs rely on maximum principle and
require much delicate computation.


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## Body

## 1. Introduction

In this paper, we study the porous medium equation with the following initial boundary condition:(1.1)ut=(um)xx-xqup,(x,t)∈(0,1)×(0,T),ux(0,t)=0,u(1,t)=k,t∈(0,T),u(x,0)=u0(x),x∈[0,1],
where 0<p<m<1 and -2<q<0. Assume k>0 and that the initial data u0 satisfies(1.2)u0∈C([0,1]),u0>0in[0,1],u0(0)=0,u0(1)=k.
Moreover, we denote(1.3)α=11-p.
Here we are mainly interested in the asymptotic behavior of nonnegative and global classical solutions. However, Problem (1.1) is singular at x=0 for -2<q<0. In fact, the solutions can be approximated, if necessary, by the ones satisfying the following equation ut=(um)xx-(x+ϵ)qup with the same initial-boundary value conditions and taking the limit ϵ→0. We set(1.4)θ(t):=min⁡0≤x≤1u(x,t)
and denote(1.5)T=T(u0):=inf⁡{t>0;θ(t)=0}>0.
For suitable initial data, we will show that T(u0)<∞ (see Theorem 1.1). We say that the solution develops a dead core in finite time, and T is called the dead-core time.In the past few years, much attentions have been taken to the dead-core problems. For the semilinear case of0<p<m=1 and q=0, the temporal dead-core profile was investigated in [1] by Guo and Souplet. For the quasilinear case of 0<p<m<1 and q=0, Guo et al. [2] firstly investigated the solution which develops a dead core in finite time; then they obtained the spatial profile of the dead core and also studied the non-self-similar dead-core rate of the solution. Numerous related works have been devoted to some of the regularity and the corresponding problems such as blowup, quenching, and gradient blowup; we refer the interested reader to [3–11] and the references therein.Our aim of this paper is to study the dead-core problem for the fast diffusion with strong absorption. In view of the observation concerning the interaction of diffusion and absorption, this question is of interest since the effect of fast diffusion, as compared with linear diffusion, is much stronger near the levelu=0. Although our strategy of proof is close to that in [2], the proof is technically much more difficult due to the presence of a nonlinear operator and spatially dependent absorption coefficient.The paper is organized as follows. In Section2, we prove that the solution of the porous medium equation develops a dead core in finite time. In Section 3, firstly, we obtain the spatial profile of the dead-core upper bound estimate by the initial monotone assumption; then we construct auxiliary function and derive the lower bound estimate by maximum principle.Our first result gives sufficient conditions under which the solution of Problem (1.1) develops a dead core in finite time. To formulate this, let us first recall some well-known facts: (1.1) admits a unique steady state Uk∈C2((0,1]) under the condition -2<q<0 for each given k>0. Moreover, Uk is an even and nondecreasing function of x, and it is a nondecreasing function of k. Furthermore, there exists k0=k0(m,p)>0 such that if k∈(0,k0) then Uk vanishes on an interval of positive length, if k=k0 then Uk vanishes only at x=0, and if k>k0 then Uk is positive.Theorem 1.1.
Assume0<p<m<1,-2<q<0 and (1.2). (i)
Let0<k<k0. Then T(u0)<∞ for any u0.(ii)
Letk≥k0. For any η,M>0 there exists δ=δ(η,M)>0 such that T(u0)<∞ whenever ∥u0∥≤M and u0≤δ on a subinterval of (0,1] of length.For our main results on the spatial profile of the dead-core problem, we will assume thatu0 satisfies the conditions(1.6)u0∈C2([0,1]),(u0m)′′≤xqu0pin(0,1],u0isevenandnondecreasinginxandT(u0)<∞.
It then follows from the strong maximum principle that ut<0 in QT:=(0,1)×(0,T),u(-x,t)=u(x,t) for (x,t)∈(-1,1)×(0,T) and ux>0 in (0,1)×(0,T).Our main goal in this paper is thus to obtain the following precise estimates on the single-point final dead-core profile nearx=0.Theorem 1.2.
Letk>0 and assume  0<p<m<1,p+m>1,-1<q<0, (1.2), and (1.6), then there exist  C1,C2>0 such that
(1.7)C1x(q+2)/(m-p)≤u(x,T)≤C2x(q+2)/(m-p),0≤x≤1,
where C1=[ε(m-p)/m(q+2)]1/(m-p),C2=[(m-p)/m(q+1)(q+2)]1/(m-p), and ε≤(p+m-1)/(2p+m-1)(q+1) is an arbitrary positive constant.Remark 1.3.
Due to the technical difficulty, we cannot prove that the coefficients of the upper and lower bounds in Theorem1.2 are not identical. Also, it is very interesting whether Problem (1.1), even for the case q>0, exists the non-self-similar dead-core rate similar to that in [1, 2]. We leave these open questions to the interested readers.


## 2. Quenching in Finite Time

Proof of Theorem1.1.
Step 1.
We look for a supersolutionu¯ of ut-(um)xx+xqup=0inQT:=(0,1)×(0,T), which develops a dead core at time T. For any T∈(0,T0), we will construct u¯ under the following self-similar form:
(2.1)u¯(x,t)=ε(T-t)αV(y),y=x(T-t)-β,V(y)=(1+y2)γ,
where
(2.2)0<β<α(m-p)2=m-p2(1-p)
and γ,ε,T0>0 will be determined. Note that u¯(0,T)=0. Computations yield
(2.3)Pu¯=u¯t-(u¯m)xx+xqu¯p=ε(T-t)α-1(-αV+βyV′)-εm(T-t)αm-2β(Vm)′′+εpxq(T-t)αpVp=ε(T-t)αp{εp-1xqVp-αV+βyV′-εm-1(T-t)λ(Vm)′′}
for (x,t)∈QT, where λ=α(m-p)-2β>0. Assuming T≤T0(ε):=ε(1-m)/λ, we see that
(2.4)Pu¯≥ε(T-t)αp{εp-1xq-h(y)},whereh(y)=αV-βyV′+|(Vm)′′|.
Next taking γ>α/(2β) and using |(Vm)′′|~C|y|2mγ-2 as |y|→∞, we observe that
(2.5)h(y)~(α-2βγ)|y|2γ⟶-∞,as|y|⟶∞.
It follows that supy∈ℝh(y)<∞ and choosing ε=ε(m,p,β,γ)>0 sufficiently small, we conclude that Pu¯≥0 in QT. For further reference we also note that
(2.6)u¯(x,t)≥ε|x|2γT-μinQT,whereμ=2βγ-α>0.
Step 2 (we prove assertion (ii)).
Fixη,M>0 and x0∈[η/2,1-η/2]. Let u¯,T0 be as in Step 1 and set v¯(x,t)=u¯(x-x0,t). Taking T≤min(T0,T1), where T1=T1(η,M)>0 is sufficiently small, and using (2.6), we see that v¯(x,t)≥M for |x-x0|≥η/2 and t∈(0,T), hence in particular v¯(±1,t)≥k (here, we deal with the symmetry case in one dimension). Next put δ:=min|x-x0|≥η/2v¯(x,0). Then assuming ∥u0∥∞≤M and u0≤δ for |x-x0|≥η/2, we get u0≤v¯(x,0), and it follows from the comparison principle that u≤v¯ in QT; hence T(u0)≤T<∞. This proves conclusion (ii).
Step 3 (we prove assertion (i)).
First observe that assertion (ii) is actually true for anyk>0 in view of Step 2. On the other hand, by standard energy arguments, one can show that u(x,t) converges to Uk in L∞(0,1) as t→∞. Since Uk=0 on [0,η/2] for some η>0, it follows that for t0 large, the new initial data ũ0:=u(x,t0) satisfies the assumptions of part (ii) with M=k+1. The conclusion follows.


## 3. Dead-Core Profile Upper and Lower Bound

In this section, we will derive some a prior estimates for solutions of (1.1). Since ut<0 in QT and ux>0 in (0,1)×(0,T), we have (um)xx<xqup. Let v=um. Then from vx=mum-1ux,0<u≤k in QT and(3.1)vxx(x,t)<xqvr(x,t),(x,t)∈(0,1)×(0,T),
it follows that vx and ux are bounded in QT.Integrating the inequalityvxx(x,t)<xqvr(x,t) using vx≥0, we obtain(3.2)vx(x,t)=∫0xvxx(y,t)dy≤∫0xyqvr(y,t)dy≤vr(x,t)∫0xyqdy≤xq+1q+1vr(x,t);
hence v1-r(x,t)-v1-r(0,t)≤((1-r)/(q+1)(q+2))xq+2. Consequently(3.3)u(x,T)=v1/m(x,T)≤Cx(q+2)/m(1-r)=Cx(q+2)/(m-p),
where C=[(m-p)/m(q+1)(q+2)]1/(m-p). Together with the following lower bound lemma, we obtain Theorem 1.2.Lemma 3.1.
Let0<p<m<1,  p+m>1 and -1<q<0.  Let (1.2), and (1.6) be in force and fix t0∈(0,T).  Then there exists ε>0 such that the auxiliary function
(3.4)J:=(um)x-εxq+1up
satisfies J≥0 in [0,1]×(t0,T). In particular, there exists Cε>0 such that
(3.5)u(x,t)≥Cεx(q+2)/(m-p),(x,t)∈(0,1)×(t0,T),
where Cε=[ε(m-p)/m(q+2)]1/(m-p) and 0<ε≤(p+m-1)/(2p+m-1)(q+1).Proof.
The equation in (1.1) can be written under the form
(3.6)ut-auxx=m(m-1)um-2(ux)2-xqup,
with a=mum-1. For (x,t)∈(0,1)×(0,T), we compute
(3.7)(xq+1up)t=pxq+1up-1ut,(xq+1up)x=(q+1)xqup+pxq+1up-1ux(xq+1up)xx=(q+1)qxq-1up+2(q+1)pxqup-1ux+p(p-1)xq+1up-2(ux)2+pxq+1up-1uxx.
Therefore
(3.8)(xq+1up)t-a(xq+1up)xx=pxq+1up-1(ut-auxx)-a((q+1)qxq-1up+2(q+1)pxqup-1ux+p(p-1)xq+1up-2(ux)2)=pm(m-1)xq+1up+m-3ux2-px2q+1u2p-1-2m(q+1)pxqum+p-2ux-mq(q+1)xq-1um+p-1-mp(p-1)xq+1um+p-3ux2=-pm-1(-m+p)xq+1up-m-1(um)x2-2p(q+1)xqup-1(um)x-px2q+1u2p-1-mq(q+1)xq-1um+p-1.
Using
(3.9)(um)x=J+εxq+1up,
we deduce that
(3.10)(xq+1up)t-a(xq+1up)xx=b1J-px2q+1u2p-1{1+2ε(q+1)+ε2m-1(p-m)xq+2up-m}-mq(q+1)xq-1um+p-1
with b1=pm-1(m-p)xq+1up-m-1(um)x2(J+2εxq+1up)-2p(q+1)xqup-1.
On the other hand, we have(3.11)(um)xt-a(um)xxx=(aux)t-a(um)xxx=a(ut-(um)xx)x+atux=-aqxq-1up-apxqup-1ux+m(m-1)um-2utux=-mqxq-1um+p-1-pxqup-1(um)x+(m-1)u-1ut(um)x=-mqxq-1um+p-1-pxqup-1(J+εxq+1up)+(m-1)u-1ut(J+εxq+1up)=b2J+εxq+1u2p-1{-pxq+(m-1)u-put}-mqxq-1um+p-1=b2J+εxq+1u2p-1{(1-m-p)xq+(m-1)u-p(um)xx}-mqxq-1um+p-1,
where b2=-pxqup-1+(m-1)u-1ut.
Since(3.12)(um)xx=(J+εxq+1up)x=Jx+ε[(q+1)xqup+pxq+1up-1ux]=Jx+εxqup[q+1+pm-1xu-m(um)x]=Jx+εxqup[q+1+pm-1xu-m(J+εxq+1up)]=Jx+b3J+εxqup[q+1+εpm-1xq+2up-m]
with b3=εpm-1xq+1up-m being a smooth function on [0,1]×(0,T), it follows that(3.13)(um)xt-a(um)xxx=b2J+εxq+1u2p-1{(Jx+b3J+εxqup[q+1+εpm-1xq+2up-m])(1-m-p)xq+(m-1)u-p×(Jx+b3J+εxqup[q+1+εpm-1xq+2up-m])}-mqxq-1um+p-1=b4J+b5Jx+εxq+1u2p-1{[q+1+εpm-1xq+2up-m](1-m-p)xq+ε(m-1)xq×[q+1+εpm-1xq+2up-m]}-mqxq-1um+p-1,
where
(3.14)b4=b2+ε(m-1)xq+1up-1b3,b5=ε(m-1)xq+1up-1isasmoothfunctionon⁡[0,1]×(0,T).
Combining (3.10) and (3.13), we obtain
(3.15)b4J+b5Jx-mqxq-1um+p-1+εxq+1u2p-1{(1-m-p)xq+ε(m-1)xq[q+1+εpm-1xq+2up-m]}=Jt-aJxx+εb1J-εmqxq-1um+p-1-εx2q+1u2p-1{1+2ε(q+1)+ε2m-1(p-m)xq+2up-m}.
Namely
(3.16)Jt-aJxx-(b5+(m-1)u-1)Jx-b7J=εx2q+1u2p-1{(1-m-p)+ε(m-1)[q+1+εpm-1xq+2up-m]}+pεx2q+1u2p-1{1+2ε(q+1)+ε2m-1(p-m)xq+2up-m}-mq(1-ε(q+1))xq-1um+p-1=εx2q+1u2p-1{1-m+ε(m-1)[q+1+εpm-1xq+2up-m]}+pεx2q+1u2p-1{2ε(q+1)+ε2m-1(p-m)xq+2up-m}-mq(1-ε(q+1))xq-1um+p-1
with b6=b5+(m-1)u-1 being a smooth function on [0,1]×(0,T).
In order to makeb7≤0 in force, we require ε≤(p+m-1)/(2p+m-1)(q+1) and p+m>1.
Sincem<1, by choosing 0<ε≤ε0 with ε0=ε0(m,p)>0 small enough, it follows that
(3.17)Jt-aJxx-b6Jx-b7J≥εx2q+1u2p-1{1-m2-p(1-p)mε2xq+2up-m}=1-m2εx2q+1u3p-m-1{um-p-κε2xq+2},
where κ:=2p(1-p)/[m(1-m)]>0. Now observe that
(3.18)[um-p-m-p(q+2)mεxq+2]x=(m-p)[um-p-1ux-m-1εxq+1]=m-pmu-p[(um)x-εxq+1up]=m-pmu-pJ,
hence
(3.19)um-p-m-p(q+2)mεxq+2≥m-pmu-p∫0xu-pJ(y,t)dy.
Thus, taking ε0 possible smaller, we get
(3.20)Jt-aJxx-b6Jx-b7J≥1-m2εx2q+1u3p-m-1{um-p-m-p2mεxq+2},
hence
(3.21)Jt-aJxx-b6Jx-b7J≥Cεx2q+1u3p-m-1∫0xu-pJ(y,t)dy,
with C=(1-m)(m-p)/2m>0. Now for any 0<t0<t1<T, it follows from the maximum principle that J attains its minimum in Q=[0,1]×[t0,t1] on the parabolic boundary of Q (see [1, 2]).
It is thus sufficient to check thatJ≥0 on the parabolic boundary of Q for ε small. Clearly J=0 for x=0. Since ux is bounded on QT,u(x,t)≥η>0 in [1-η,1]×(t0,T) for some small constant δ>0. Therefore u extends to a classical solution on [1-η,1]×(t0,T], and Hopf’s Lemma implies that ux(1,t)≥δ̃>0 for t0<t<T; hence J(1,t)≥0 for t0<t<T if ε is chosen small enough. Moreover, also as a consequence of Hopf’s Lemma, we have ux(x,t0)≥cxq+1 in [0,1] for some c>0. Again decreasing ε if necessary, we deduce that J(x,t0)≥0 in [0,1]. The lemma follows.



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*Source: 751969-2011-12-04.xml*