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.

1. Introduction

In this paper, we study the porous medium equation with the following initial boundary condition:

(1.1)

where 0 < p < m < 1 and −2 < q < 0. Assume k > 0 and that the initial data u₀ satisfies

(1.2)

Moreover, we denote

(1.3)

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 u_t = (u^m) _xx − (x + ϵ) ^qu^p with the same initial-boundary value conditions and taking the limit ϵ → 0. We set

(1.4)

and denote

(1.5)

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 of 0 < 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 level u = 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 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 U_k ∈ C²((0,1]) under the condition −2 < q < 0 for each given k > 0. Moreover, U_k 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 U_k vanishes on an interval of positive length, if k = k₀ then U_k vanishes only at x = 0, and if k > k₀ then U_k is positive.

Theorem 1.1. Assume 0 < p < m < 1, − 2 < q < 0 and (1.2).

(i)
Let 0 < k < k₀. Then T(u₀) < ∞ for any u₀.
(ii)
Let k ≥ 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 that u₀ satisfies the conditions

(1.6)

It then follows from the strong maximum principle that u_t < 0 in Q_T : = (0,1)×(0, T), u(−x, t) = u(x, t) for (x, t)∈(−1,1)×(0, T) and u_x > 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 near x = 0.

Theorem 1.2. Let k > 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)

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 Theorem 1.2 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 Theorem 1.1.

Step 1. We look for a supersolution of u_t − (u^m) _xx + x^qu^p = 0 in Q_T : = (0,1)×(0, T), which develops a dead core at time T. For any T ∈ (0, T₀), we will construct under the following self-similar form:

(2.1)

where

(2.2)

and γ, ε, T₀ > 0 will be determined. Note that

. Computations yield

(2.3)

for (x, t) ∈ Q_T, where λ = α(m − p) − 2β > 0. Assuming T ≤ T₀(ε): = ε^(1−m)/λ, we see that

(2.4)

Next taking γ > α/(2β) and using |(V^m) ^′′ | ~ C | y|^2mγ−2 as |y | → ∞, we observe that

(2.5)

It follows that sup _y∈ℝ h(y) < ∞ and choosing ε = ε(m, p, β, γ) > 0 sufficiently small, we conclude that

in Q_T. For further reference we also note that

(2.6)

Step 2 (we prove assertion (ii)). Fix η, M > 0 and x₀ ∈ [η/2,1 − η/2]. Let be as in Step 1 and set . Taking T ≤ min (T₀, T₁), where T₁ = T₁(η, M) > 0 is sufficiently small, and using (2.6), we see that for |x − x₀ | ≥ η/2 and t ∈ (0, T), hence in particular (here, we deal with the symmetry case in one dimension). Next put . Then assuming ∥u₀∥_∞ ≤ M and u₀ ≤ δ for |x − x₀ | ≥ η/2, we get , and it follows from the comparison principle that in Q_T; hence T(u₀) ≤ T < ∞. This proves conclusion (ii).

Step 3 (we prove assertion (i)). First observe that assertion (ii) is actually true for any k > 0 in view of Step 2. On the other hand, by standard energy arguments, one can show that u(x, t) converges to U_k in L^∞(0,1) as t → ∞. Since U_k = 0 on [0, η/2] for some η > 0, it follows that for t₀ large, the new initial data 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 u_t < 0 in Q_T and u_x > 0 in (0,1)×(0, T), we have (u^m) _xx < x^qu^p. Let v = u^m. Then from v_x = mu^m−1u_x, 0 < u ≤ k in Q_T and

(3.1)

it follows that v_x and u_x are bounded in Q_T.

Integrating the inequality v_xx(x, t) < x^qv^r(x, t) using v_x ≥ 0, we obtain

(3.2)

hence v^1−r(x, t) − v^1−r(0, t) ≤ ((1 − r)/(q + 1)(q + 2))x^q+2. Consequently

(3.3)

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. Let 0 < 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)

satisfies J ≥ 0 in [0,1]×(t₀, T). In particular, there exists C_ε > 0 such that

(3.5)

where C_ε = [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)

with a = mu^m−1. For (x, t)∈(0,1)×(0, T), we compute

(3.7)

Therefore

(3.8)

Using

(3.9)

we deduce that

(3.10)

with

On the other hand, we have

(3.11)

where b₂ = −px^qu^p−1 + (m − 1)u⁻¹u_t.

Since

(3.12)

with b₃ = εpm⁻¹x^q+1u^p−m being a smooth function on [0,1] × (0, T), it follows that

(3.13)

where

(3.14)

Combining (3.10) and (3.13), we obtain

(3.15)

Namely

(3.16)

with b₆ = b₅ + (m − 1)u⁻¹ being a smooth function on [0,1]×(0, T).

In order to make b₇ ≤ 0 in force, we require ε ≤ (p + m − 1)/(2p + m − 1)(q + 1) and p + m > 1.

Since m < 1, by choosing 0 < ε ≤ ε₀ with ε₀ = ε₀(m, p) > 0 small enough, it follows that

(3.17)

where κ : = 2p(1 − p)/[m(1 − m)] > 0. Now observe that

(3.18)

hence

(3.19)

Thus, taking ε₀ possible smaller, we get

(3.20)

hence

(3.21)

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 that J ≥ 0 on the parabolic boundary of Q for ε small. Clearly J = 0 for x = 0. Since u_x is bounded on Q_T, 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 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 u_x(x, t₀) ≥ cx^q+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.

Acknowledgments

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

Abstract

1. Introduction

2. Quenching in Finite Time

3. Dead-Core Profile Upper and Lower Bound

Acknowledgments

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

Abstract

1. Introduction

2. Quenching in Finite Time

3. Dead-Core Profile Upper and Lower Bound

Acknowledgments

References

Citing Literature

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