DEAD-CORE RATES FOR THE POROUS MEDIUM EQUATION WITH A STRONG ABSORPTION

We study the dead-core rate for the solution of the porous medium equation with a strong absorption. It is known that solutions with certain class of initial data develop a dead-core in finite time. We prove that, unlike the cases of semilinear heat equation and fast diffusion equation, there are solutions with the self-similar dead-core rate. This result is based on the construction of a Lyapunov functional, some a priori estimates, and a delicate analysis of the associated re-scaled ordinary differential equation.


Introduction
In this paper, we study the following initial boundary value problem (P): where k > 0, 0 < p < 1, m > p, and the initial data u 0 satisfies (1.4) Problem (P) with m = 1 arises from, e.g., the modeling of an isothermal reaction-diffusion process in which u represents the concentration of the reactant.The reactant is injected with a fixed amount k on the boundary and the parameter p is the order of the reaction (cf.[2,10]).We have a strong absorption in (1.1), since p ∈ (0, 1) implies that u p−1 → ∞ as u → 0. When m > 1, (1.1) is called the slow diffusion (or porous medium) equation with a strong absorption.
It is clear that (1.1)-(1.3)admits a unique local (in time) positive classical solution u.We denote We say that the solution u develops a dead-core in finite time, if T = T (u 0 ) < ∞.The time T is called the dead-core time.In fact, the existence of finite-time dead-core of (P) for the semilinear case (0 < p < m = 1) can be found in [2,10,3].For the fast diffusion case (0 < p < m < 1), we refer the reader to the work [5].Although some dead-core criteria for the slow diffusion case (0 < p < 1 < m) was already given in [1], we provide here the following theorem on the occurrence of a dead-core for the sake of completeness.We would like to dedicate this work to our Ph.D. thesis advisor, Professor Avner Friedman, on the occasion of his 80th birthday.
(2) There exist positive constants ε 0 = ε 0 (p, m) and t 0 = t 0 (p, m) such that if The main purpose of this paper is to study the asymptotic behavior of the solution as t → T − when T < ∞.Recently, there were many works on the question of temporal dead-core rates, see, e.g., [7,8,9,5,6].In particular, it was shown in [7,5] that for 0 < p < m ≤ 1 the dead-core rate is always non-self-similar, i.e. its order is not the same as for the corresponding ODE y ′ = −y p .More precisely, it holds for solutions with monotone symmetric initial data.For m = 1, by using the special solutions with different dead-core rates constructed in [8,9] and the braid group theory, the authors in [6] were able to determine the dead-core rates for the solutions with general initial data in both one spatial dimensional and radially symmetric higher dimensional cases.
Actually there are countably infinite many different dead-core rates which are different from the self-similar rate.The non-self-similar dead-core rate for the fast diffusion equation is proved in [5].The goal of this paper is to study the dead-core rate in the case of slow diffusion (m > 1).
In the sequel, in addition to (1.4) we shall assume that the dead-core time T = T (u 0 ) < ∞ and u 0 satisfies Then we can easily derive that (1.7) by the maximum principle.It is clear that u(0, t) = min |x|≤1 u(x, t) and u(0, t) → 0 as t ↑ T .
To study the temporal dead-core rate, we introduce the following self-similar variables where the exponent α is defined by Then it is easy to see that z satisfies (1.11) where The associated ODE with (1.9) on the whole real line R is given by (1.12) Then we have the following Liouville's type theorem.
Theorem 1.2.Let 0 < p < 1 < m and let Z ∈ C 2 (R) be a solution of (1.12) such that where With some a priori estimates and Theorem 1.2, we can prove the following dead-core rate estimate for certain class of initial data.
and assume that Let u be a solution of (P) with u 0 satisfying (1.4) and (1.6).In addition, we assume that the initial datum u 0 satisfies In the case m + p < 2, the ω-limit set is not empty.That is, for any sequence {t j } ↑ T , there is a subsequence where Z is a solution of the ODE (1.12) with Z ′ (0) = 0 and Z(0) > 0.
Surprisingly, Theorem 1.3 shows that, unlike the semilinear and fast diffusion cases, we do have the self-similar singularity of dead-core rate for certain class of initial data in the slow diffusion case.We expect that the non-self-similar dead-core rates should also occur for some initial data.This is left open in the present study.Also, another interesting question is whether there are non-constant solutions with Z ′ (0) = 0 and Z(0 This paper is organized as follows.The proof of Theorem 1.1 shall be given in §2.The we give some a priori estimates in §3.In §4, we give a proof of Theorem 1.2.Finally, we prove the self-similar dead-core rate result (Theorem 1.3) in §5.

Some a priori estimates
In this section, we shall derive some a priori estimates for solutions of (P).Let u be a solution of (P) with the finite dead-core time T .Then we can easily derive the following estimate from (1.7).Lemma 3.1.Assume that u 0 satisfies (1.4) and (1.6).Then we have Proof.Indeed, from (1.7) we have v xx < v p/m , where v := u m .Multiplying by v x and integrating it over [0, x] for x ∈ (0, 1], the estimate (3.1) follows.
In terms of (r, s, z), (3.1) reads Next, similar to [5, Lemma 3.2], we have the following uniform bound for z.Since the proof is exactly the same as that in [5, Lemma 3.2], we omit it here.Lemma 3.2.Let q = p/m.Assume that u 0 satisfies (1.4) and (1.6).Then the corresponding global solution z of (1.9) satisfies z(0, s) ≤ κ for all s ≥ s 0 and for some constant c > 0.
The following key estimate exhibits a self-similar singularity of the dead-core rate for certain class of initial data.Lemma 3.3.Under the assumptions of Theorem 1.3, there exists a ĉ > 0 such that It is clear that, for any x 0 , the function (m−p) 2 2m(m+p) (x − x 0 ) 2 is an explicit solution of the above problem.The assumption (1.15) implies that v x (x, 0) ≤ (m − p)σ 0 x for 0 ≤ x ≤ 1.Note that (1.14) implies that k < k 0 so that we can take Recalling (1.6), the function v(x, 0) is a supersolution and therefore Applying maximum principle, we obtain It follows that, by (1.14) and (3.7), We now differentiate (3.6) in x and obtain Under the assumption (1.15), noting that 0 < p < 1, the function (m − p)σ 0 x is a supersolution and therefore or, equivalently, ( u 1−p (0, t) Recalling (1.6), the proof is completed.
In terms of z, we have

The Steady State of Scaled equation
In this section, we shall give a proof of Theorem 1.2.

Scaling. Recall the constants
For convenience, we introduce the following two extra constants: For some positive constants A and B to be chosen later, we introduce A direct substitution gives Substituting z = vν and dividing the resulting equation by Av ν−1 , we then obtain In particular, due to (4.4), we only consider any solution of (4.5) on [0, ∞) that satisfies
Lemma 4.2.Set J(y) = yV y (y) − V (y).For each y > y 0 , Proof.First, from (4.5) we have by the definition of ρ.This equation can be written as d dy Integrating this identity over (y, L) for fixed y > y 0 , sending L → ∞, and using This can be written as (4.10) a d dy Note that, from (4.5) and the fact that 2 a − σ > 0, Hence, by Lemma 4.1 we can integrate the identity (4.10) over [y, L] and sending L → ∞ to obtain (4.9).

The Case
Proof.As V /y is a decreasing function, we can define In addition, since J ≤ 0 implies V y ≤ V /y, we have On the other hand, from which after integration gives, for 0 < y 1 < y, V σ+1 (y).
Sending y 1 ↘ 0 along the sequence on which V y (y 1 ) → k, we obtain Next, we integrate Integrating over (0, y) again, we obtain Recalling that σ > −1 and dividing both side by y 2 /2 and sending y ↘ 0 we then obtain where the improper integral is convergent since σ > −1.It then follows from (4.8) that This implies that V 2 y ≡ 1 in (0, ∞).Thus, V (y) = y for all y ≥ 0.

Self-similar dead-core rate
This section is devote to the dead-core rate for solutions of (P) with initial data satisfying the assumptions of Theorem 1.3.To derive the asymptotic behavior of solution of (P) near the dead-core time, we shall construct a Lyapunov functional following a method of Zelenyak [11] (see also [4,5]).
In order to construct a Lyapunov function, we need to study the backward continuation of solution to the ODE associated with (1.9).For this, we take a smooth and non-increasing function ζ on R such that Recall the constant c defined in (3.15).We define where g(v) := βv γ − v q .Note that ĝ(v) = g(v) whenever v ≥ c, and g(+∞) = +∞ since γ > q.
Moreover, as in [5], we have the following identity Indeed, this identity can be derived by applying the uniqueness of the solutions to the initial value problems satisfied by two associated ODEs to both sides of (5.5).
Let z be the solution of (1.9)- (1.11).Since z(r, s) ≥ c in Ω, we see that z also satisfies Following a method of Zelenyak [11] (see also [4,5]), we define where R(s) = e s and Φ = Φ(r, v, w) will be specified later.Using (5.6), we calculate that for some positive constant c * independent of s.We also observe from (5.8) that 0 < ψ(ξ; r, v, 0) ≤ v < 2v for all ξ ∈ [0, r] whenever v ≥ κ.Hence (5.9) also holds for w = 0 for any s > s 0 .Now, from and R ′ (s) = e s , it follows that (5.10 for all s ≥ s 1 for some positive constant c 4 .Hence by (5.10) and (5.11) we obtain that J 1 is bounded by a function that decays exponentially fast.Thus we conclude that (5.12) where J 1 satisfies the property ∫ ∞ s 0 |J 1 (s)|ds < ∞.From (5.12), we obtain that We are ready to prove Theorem 1.3.Since the proof is very standard (cf.[5] for example), we only outline its proof here.
Proof of Theorem 1.3.Take any sequence s j with s j → ∞ as j → ∞ and define z j (r, s) = z(r, s + s j ) for all j ∈ N and (r, s) ∈ Ω.Using (3.14) and (3.3)-(3.5),applying the standard parabolic estimates and a diagonal process, there exist a subsequence {j l } and a function z ∞ ∈ C 2,1 (R 2 ) satisfying (1.9) in R 2 such that z j l (r, s) → z ∞ (r, s) as l → ∞ locally uniformly in C 2,1 (R 2 ).Note that (z ∞ ) r (0, s) = 0 for all s.
Note that z ∞ is monotone non-decreasing in r > 0 and z ∞ (0) ≥ c > 0. Hence the case m + p < 2 is proved.Moreover, it follows from Theorem 1.2 that z ∞ ≡ κ when m + p ≥ 2.
Since the sequence {s j } is arbitrary, (1.16) follows.Therefore, we have completed the proof of the theorem.