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International Journal of Differential Equations
Volume 2011 (2011), Article ID 751969, 9 pages
Spatial Profile of the Dead Core for the Fast Diffusion Equation with Dependent Coefficient
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China
Received 7 May 2011; Accepted 25 October 2011
Academic Editor: Donal O'Regan
Copyright © 2011 Zhengce Zhang and Biao Wang. 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.
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.
In this paper, we study the porous medium equation with the following initial boundary condition: where and . Assume and that the initial data satisfies Moreover, we denote Here we are mainly interested in the asymptotic behavior of nonnegative and global classical solutions. However, Problem (1.1) is singular at for . In fact, the solutions can be approximated, if necessary, by the ones satisfying the following equation with the same initial-boundary value conditions and taking the limit . We set and denote For suitable initial data, we will show that (see Theorem 1.1). We say that the solution develops a dead core in finite time, and 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 of and , the temporal dead-core profile was investigated in  by Guo and Souplet. For the quasilinear case of and , Guo et al.  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 level . Although our strategy of proof is close to that in , 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 Section 2, 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 under the condition for each given . Moreover, is an even and nondecreasing function of , and it is a nondecreasing function of . Furthermore, there exists such that if then vanishes on an interval of positive length, if then vanishes only at , and if then is positive.
Theorem 1.1. Assume and (1.2). (i)Let . Then for any .(ii)Let . For any there exists such that whenever and on a subinterval of of length.
For our main results on the spatial profile of the dead-core problem, we will assume that satisfies the conditions It then follows from the strong maximum principle that in for and in .
Our main goal in this paper is thus to obtain the following precise estimates on the single-point final dead-core profile near .
Remark 1.3. Due to the technical difficulty, we cannot prove that the coefficients of the upper and lower bounds in Theorem 1.2 are not identical. Also, it is very interesting whether Problem (1.1), even for the case , 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 Theorem 1.1. Step 1. We look for a supersolution of , which develops a dead core at time . For any , we will construct under the following self-similar form:
and will be determined. Note that . Computations yield
for , where . Assuming , we see that
Next taking and using as , we observe that
It follows that and choosing sufficiently small, we conclude that in . For further reference we also note that
Step 2 (we prove assertion (ii)). Fix and . Let be as in Step 1 and set . Taking , where is sufficiently small, and using (2.6), we see that for and , hence in particular (here, we deal with the symmetry case in one dimension). Next put . Then assuming and for , we get , and it follows from the comparison principle that in ; hence . This proves conclusion (ii).
Step 3 (we prove assertion (i)). First observe that assertion (ii) is actually true for any in view of Step 2. On the other hand, by standard energy arguments, one can show that converges to in as . Since on for some , it follows that for large, the new initial data satisfies the assumptions of part (ii) with . 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 in and in , we have . Let . Then from in and it follows that and are bounded in .
Integrating the inequality using , we obtain hence . Consequently where . Together with the following lower bound lemma, we obtain Theorem 1.2.
Proof. The equation in (1.1) can be written under the form
with . For , we compute
we deduce that
On the other hand, we have where .
Since with being a smooth function on , it follows that where Combining (3.10) and (3.13), we obtain Namely with being a smooth function on .
In order to make in force, we require and .
Since , by choosing with small enough, it follows that where . Now observe that hence Thus, taking possible smaller, we get hence with . Now for any , it follows from the maximum principle that attains its minimum in on the parabolic boundary of (see [1, 2]).
It is thus sufficient to check that on the parabolic boundary of for small. Clearly for . Since is bounded on in for some small constant . Therefore extends to a classical solution on , and Hopf’s Lemma implies that for ; hence for if is chosen small enough. Moreover, also as a consequence of Hopf’s Lemma, we have in for some . Again decreasing if necessary, we deduce that in . The lemma follows.
The authors thank Professor Bei Hu for helpful suggestions. The paper is supported by Youth Foundation of NSFC (no. 10701061) and the Fundamental Research Funds for the Central Universities of China.
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