Support properties of solutions to a degenerate equation with absorption and variable density

Nonlinear Analysis 68 (2008) 1940–1953 www.elsevier.com/locate/na

Support properties of solutions to a degenerate equation with absorption and variable density Zhaoyin Xiang a,∗ , Chunlai Mu b , Xuegang Hu c a School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu 610054, PR China b Department of Mathematics, Chongqing University, Chongqing 400044, PR China c Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing 400065, PR China

Received 14 September 2005; accepted 17 January 2007

Abstract In this paper, we study the properties of solutions to a degenerate parabolic equation with variable density and absorption. We first obtain a critical exponent, which distinguishes the localization of solutions from the positivity of them. When positivity prevails, we obtain the other critical exponent with respect to the decay of the variable density, which separates the global existence of interfaces from the disappearance of them. Moreover, the long time behavior of interfaces is characterized. c 2007 Elsevier Ltd. All rights reserved.

MSC: 35K15; 35K60; 35B40 Keywords: Localization; Positivity; Global existence of interfaces; Disappearance of interfaces

1. Introduction In this paper, we study the properties of solutions to the following problem:   ρ(x)u t = u m−1 |u x |λ−1 u x − c0 u p , x ∈ R, t > 0, x

u(x, 0) = u 0 (x),

(1.1)

x ∈ R,

where m > 1, λ ≥ 1, p > 0 and c0 > 0 are constants, the nontrivial initial data u 0 (x) is nonnegative continuous and compactly supported, while the variable function ρ(x) is positive, bounded and smooth enough. As an example of this situation, we mention the equation   1 m−1 λ−1 = u u |u | u − c0 u p , x ∈ R, t > 0, t x x x (1 + |x|)k which has been studied by many authors for some special k, m, λ, c0 (see [1–5,7–11,14–17,22–27] and references therein). ∗ Corresponding author. Tel.: +86 02881945470; fax: +86 02883202631.

E-mail address: [email protected] (Z. Xiang). c 2007 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2007.01.021

Z. Xiang et al. / Nonlinear Analysis 68 (2008) 1940–1953

1941

Equation in (1.1) arises in a wide range of physical contexts, including those noted in [10,12,13,16,25], where a more detailed physical background can be found. For instance, problem (1.1) can be used to model the flow of compressible fluids through porous media. In this case, the function ρ(x) can be interpreted as the density of the medium where the diffusive–absorptive process occurs. Under the assumption that p > 1 and ρ(x) ≡ ρ0 (constant), it is well known that the solution u to the heat equation, which is the special case of Eq. (1.1) with m = λ = 1, satisfies u(x, t) > 0 for any x ∈ R, t > 0 provided that u 0 (x) is a nontrivial and nonnegative function. In the case of m > 1, however, for instance the porous medium equation (PME), the above property ceases to be true. In efforts to understand these facts, in the past few years, many authors have investigated the properties of solutions to problem (1.1) and their variants. Hereafter, when we say a solution of problem (1.1), we mean it satisfies the equation in the weak sense (the precise definition of weak solution is given in Section 2). For instance, if p ≥ m > 1, λ = 1, ρ = ρ0 (constant), a weaker positivity property still holds: ∀ x ∈ R, ∃ T (x) ≥ 0

such that ∀ t ≥ T, u(x, t) > 0,

while if p < m, the localization property arises, namely, ∃L >0

such that supp u(., t) ⊂ [−L , L], ∀ t ≥ 0

(see [3,4,14]). We shall say the solution u has the positivity property for the former and u has the localization property for the latter, respectively. From the point of view of physics, if u represents the density of a gas and has the positivity property, it means that the gas eventually spreads throughout the entire domain. On the other hand, u with localization means that the gas will remain confined to a bounded region of space for all time (see [3]). The effect of variable ρ(x) and new interesting phenomena were observed in [9,15,16,25], where Cauchy problem (1.1) is considered for the case m > 1, λ = 1, c0 = 0. To be precise, we let ζ + (t) = sup{x : u(x, t) > 0},

ζ − (t) = inf{x : u(x, t) > 0}

denote the interfaces of solution u of (1.1) at time t. If the density function ρ(x) decays at infinity like ρ(x) ∼ |x|−k (k > k0∗ := 2), there exists T : 0 < T < ∞ such that |ζ ± (t)| → ∞ as t → T − . This phenomenon is described as the interfaces disappearing in finite time. Conversely, if 0 < k < k0∗ , then the solution remains compactly supported for all t > 0 and the interfaces will move out to infinity as t → ∞. Therefore, as far as the problem (1.1) is considered, one might question whether or not the above pictures are structurally stable with respect to the parameter c0 ≥ 0. For the case m > 1, λ = 1, Kersner et al. in [17,23] give a complete answer to this question. They proved that there is localization of the solution if p < m := pc1 , and positivity if p > pc1 . In the case of positivity, they also showed that the interfaces will disappear in finite time for p−1 k > k1∗ := 2 p−m , and the solution remains compactly supported for any t ≥ 0 if k < k1∗ . Recently, by using an energy method, Tedeev [27] considered the problem (1.1) (in several space dimensions) with c0 = 0 and gave the conditions for disappearance of interfaces or of compactly supported solutions for general λ > 0. For more problems concerned with the interfaces of solution to degenerate parabolic equations, we refer the reader to [1,11,22,26,29] and references therein. We also mention the recent work [6] of Ferreira et al., where the authors investigate the interfaces of inhomogeneous porous medium equations with convection. The purpose of the present paper is to extend the results of [27] to c0 ≥ 0 and those of [17,23] to λ ≥ 1. Our methods are based on the comparison arguments and some of them are motivated by [17,23,24]. We find that the critical exponent with respect to positivity or localization of the solution to problem (1.1) is pc := m + λ − 1 and the critical exponent of the decay density with respect to global existence or disappearance of interfaces is k ∗ := ( p − 1)(λ + 1)/( p − (m + λ − 1)). Clearly, if λ = 1, then pc and k ∗ are consistent with pc1 and k1∗ , respectively. In order to understand the values of the critical exponents pc and k ∗ , we may consider the equation   |x|−k u t = u m−1 |u x |λ−1 u x − c0 u p , (1.2) x

which has a singular weight ρ(x) ˜ = |x|−k . If p 6= pc = m + λ − 1,

k 6= k ∗ =

( p − 1)(λ + 1) , p − (m + λ − 1)

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then we observe that the function u(x, t) = t α f (x/t β ) with α := −

(k ∗

λ+1 , − k)( p − (m + λ − 1))

β :=

k∗

1 −k

is a self-similar solution to problem (1.2). If p > m +λ−1, k = k ∗ , the function u(x, t) = eαt f (eβt x) is a self-similar solution to problem (1.2) for any α, β ∈ R such that α( p − (m + λ − 1)) = β(λ + 1) (see [8,9,17]). Now we state our results as follows. Theorem 1.1 (Localization). Let p < m + λ − 1. Assume u(x, t) is a solution to problem (1.1). Then there exists a positive constant L, which depends only on m, λ, p, c0 and ku 0 k∞ , such that |ζ ± (t)| ≤ L

for any t ≥ 0.

Theorem 1.1 implies that the solution u is localized if p < m + λ − 1. Conversely, the following theorem suggests that u possesses the positivity property if p > m + λ − 1. Thus, pc = m + λ − 1 is the critical value of p such that u has the localization or positivity property. Theorem 1.2 (Positivity). Let p > m + λ − 1. Assume u(x, t) is a solution to problem (1.1). Then there exist constants a, b > 0, which depend only on m, λ, p, kρk∞ , c0 and u 0 , such that 1

|ζ ± (t)| ≥ b (log(at + 3)) 2λ

for any t ≥ 0.

Next, in the case of positivity p > m + λ − 1, we investigate how the decay of ρ(x) influences the behavior of interfaces of the solution u. In the rest of this paper, we define k ∗ := ( p − 1)(λ + 1)/( p − (m + λ − 1)) for p > m + λ − 1. Theorem 1.3 (Global Existence of Interfaces). Let p > m + λ − 1, k < k ∗ . Assume u(x, t) is a solution to problem (1.1). If ρ(x) satisfies the condition ρ1 ≤ ρ(x) ≤ ρ2 , (1 + |x|)k

x ∈ R,

(1.3)

where ρ1 , ρ2 are positive constants, then there exist constants A, B > 0, which depend only on m, λ, p, k, ρ1 , c0 and u 0 , such that 1

|ζ ± (t)| ≤ B (At + 1) k ∗ −k

for any t ≥ 0.

Theorem 1.4 (Disappearance of Interfaces). Let p > m + λ − 1, k > k ∗ . Assume u(x, t) is a solution to problem (1.1). If ρ(x) satisfies the condition ρ(x) ≤

ρ2 , (1 + |x|)k

x ∈R

(1.4)

where ρ2 is a positive constant, then there exist constants T : 0 < T < ∞ (depending only on m, λ, p, k, ρ and u 0 ) such that u(x, t) > 0,

for any x ∈ R, t ≥ T.

Remark 1.1. Rather than the usual assumption that ρ1 /(1 + |x|)k ≤ ρ(x) ≤ ρ2 /(1 + |x|)k , only the LHS restriction is needed for Theorem 1.3 and only the RHS one is needed for Theorem 1.4. Moreover, if we restrict ourselves to reference densities of the form ρ/(1 ¯ + |x|)k , then we see that k ∗ is the critical exponent between global existence and disappearance of interfaces, by combining Theorem 1.3 with Theorem 1.4. In fact, in the critical case, we have the following global existence of interfaces, although the characteristic is not very precise.

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Theorem 1.5 (Global Existence of Interfaces). Let p > m + λ − 1, k = k ∗ . Assume u(x, t) is a solution to problem (1.1). If ρ(x) satisfies the condition (1.3), then there exist constants B, β > 0, which depend only on m, λ, p, k, ρ1 , c0 and u 0 , such that |ζ ± (t)| ≤ Beβt

for any t ≥ 0.

Remark 1.2. Theorems 1.2, 1.3 and 1.5 characterize the long time behavior of interfaces. For the case of λ = 1 and c0 = 0, Galaktionov et al. [7] give the more precise asymptotic behavior of interfaces. Finally, we give a brief outline of the rest of this paper. In Section 2, we first introduce some definitions and notation. Subsequently, in Section 3, we deal with the localization and positivity of the solution (Theorems 1.1 and 1.2). The global existence (Theorems 1.3 and 1.5) and disappearance (Theorem 1.4) of interfaces will be investigated in Sections 4 and 5, respectively. 2. Preliminaries In this section, we give some preliminaries for completeness. By the necessity of applying the comparison principle to non-cylindrical domains, we first give some notation and assumptions, which are similar to those of [24]. In the sequel, we put S T := R × (0, T ], (T > 0). Given two functions: ηi : [0, T ] → R (i = 1, 2), such that η10 (t), η20 (t) ∈ Lip([0, T ]) and η1 (t) < η2 (t) for every t ∈ [0, T ], we consider subsets of S T of the following three types: D T := {(x, t) ∈ S T : η1 (t) < x < η2 (t)},

T D+ := {(x, t) ∈ S T : x > η1 (t)},

T D− := {(x, t) ∈ S T : x < η2 (t)}.

Since the equation in (1.1) is degenerate at points where u = 0 or u x = 0, there is no classical solution in general, and we therefore consider its weak solutions. Let ψi ∈ C[0, T ] be nonnegative functions (i = 1, 2), ψ0 (resp. ψ0+ , ψ0− ) be nonnegative, continuous and bounded functions defined on [η1 (0), η2 (0)] (resp. [η1 (0), ∞), (−∞, η2 (0)]), and the compatibility conditions hold. We introduce the following definition. Definition 2.1. A bounded weak solution to the first boundary value problem   ρ(x)u t = u m−1 |u x |λ−1 u x − c0 u p , (x, t) ∈ D T , x

u(x, 0) = ψ0 (x), u(ηi (t), t) = ψi (t),

x ∈ [η1 (0), η2 (0)],

(2.1)

t ∈ [0, T ], (i = 1, 2)

is a nonnegative continuous function u ∈ L ∞ (D T ) ∩ L 2 (0, T ; W 1, p (η1 (T ), η2 (T ))) such that | (u σ )x |λ+1 ∈ L 1 (D T ), (σ = (m + λ − 1)/λ) and u satisfies Z η2 (τ ) Z η2 (0) ρ(x)u(x, τ )ϕ(x, τ )dx − ρ(x)ψ0 (x)ϕ(x, 0)dx η1 (τ )

η1 (0)

ZZ − Dτ

ρuϕt + u m−1 |u x |λ−1 u x ϕx + c0 u p ϕdxdt = 0,

(2.2)

for any ϕ ∈ C 1 ( D¯ T ), ϕ ≥ 0 such that ϕ(ηi (t), t) = 0 in [0, T ] and any τ ∈ (0, T ). In order to obtain the definition of a bounded weak subsolution, we only need to replace the equality sign = in (2.2) with an inequality sign ≤, and additionally assume ϕ ≥ 0. T . In fact, Similarly, we can define a bounded weak solution (subsolution) to the Cauchy–Dirichlet problem in D+ τ τ we only need to replace D in (2.2) with D+ ∩ {x < r + η2 (t), 0 < t ≤ T } for every r > 0. Now, we give the definition of a weak solution to problem (1.1).

Definition 2.2. A bounded weak solution (subsolution) to problem (1.1) is a nonnegative bounded function u ∈ C(R×(0, ∞)) which is a solution (subsolution) to problem (2.1) with ψ0 = u 0 |[−r,r ] , η1 = −r , η2 = r, ψ1 = u(−r, t) and ψ2 = u(r, t) in any rectangle (−r, r ) × (0, T ], where r, T > 0.

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When 0 < p < 1, the equations in (1.1) and (2.1) have strong absorption and the map s → s p is not Lipschitz continuous. Bertsch [2] introduced a more restrictive concept of a supersolution to (2.1). We adopted it to the present paper. By a bounded weak supersolution to the above problems, we mean a weak solution to the corresponding problem with a additional nonnegative bounded function on the right-hand side of the equation. Now, by using similar arguments to [21,24,28] (see also [5,18]), we see that there exists a unique solution T , D T and S T . to problem (1.1) and the usual comparison principle holds in the non-cylindrical domains D T , D+ − 1 Moreover, if ζ (t) ∈ C [0, T ] satisfies η1 (t) < ζ (t) < η2 (t), and Lu ≤ 0 in the classical sense in D T \ ζ (t), then u is a subsolution of (2.1) in D T provided that the usual initial-boundary comparison is true (see next paragraph for the definition of L). As far as the supersolution is considered, we need to restrict additionally for 0 ≤ Lu ≤ M some M > 0 besides the usual assumptions. The proof is standard (see [6,24]). Similar results are true for the other domains mentioned above. When u > 0 or u ≡ 0, we define   Lu = ρ(x)u t − u m−1 |u x |λ−1 u x + c0 u p . x

Throughout this paper, we define the set S+ = {(x, t) : x > 0, t > 0}. 3. Localization and positivity of solutions In this section, we investigate the asymptotic behavior, localization and positivity properties of solutions to problem (1.1). In the following any statements concerning interfaces will be proved only for the right interface. The proof of localization is comparatively easy. Proof of Theorem 1.1. Since p < m + λ − 1, we let λ+1  x  m+λ−1− p u(x, t) = A 1 − , x > 0, t > 0, L + where A, L > 0 are to be determined later. We claim u(x, t) ≤ u(x, t),

(x, t) ∈ [0, ∞) × [0, T ]

for any T > 0, which implies supp u(., t) ⊆ supp u(., t) ⊆ (−∞, L]. To prove this claim, we set SL = {(x, t) : 0 < x < L , t > 0}. A direct computation gives !  λ p(λ+1)  x  m+λ−1− p Am+λ−1 (λ( p + 1) + m − 1) λ+1 p 1− Lu = − + c0 A L(m + λ − 1 − p) L(m + λ − 1 − p) L in SL . Taking A = ku 0 k∞ + 1 and choosing L large enough, we have u(0, t) = A > u(0, t),

t > 0,

u(x, 0) ≥ u 0 (x),

x > 0,

0 ≤ Lu ≤ c0 A ,

in SL .

p

Note that Lu = 0 in S \ SL . Therefore, by using the comparison principle, we have proved the claim.



Now we turn to considering the positivity of the solution. Hereafter, without loss of generality, we assume u 0 (0) > 0. We need the following lemma to prove Theorem 1.2. Lemma 3.1. Let p ≥ m + λ − 1. Assume u(x, t) is a solution to problem (1.1). Then there exist constants a1 , b1 > 0, which depend only m, λ, p, c0 , ρ and u 0 , such that 1

u(0, t) ≥ b1 (a1 t + 1)− m+λ−2

for any t ≥ 0.

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Proof. Define the following auxiliary function: λ

1

u(x, t) = (at + 1)− m+λ−2 (b2 − x 2 ) m+λ−2 ,

(x, t) ∈ [−b, b] × [0, ∞),

where a, b > 0 are to be determined. We shall apply the comparison principle to [−b, b] × [0, ∞) and then obtain 2λ 1 u(x, t) ≤ u(x, t) in [−b, b] × [0, ∞). In particular, u(0, t) ≥ u(0, t) = b m+λ−2 (at + 1)− m+λ−2 for any t ≥ 0, which is desired. To do this, we first consider the initial-boundary values. Noting that u 0 (0) > 0, we may choose b small enough such that u(x, 0) < u 0 (x),

x ∈ [−b, b].

Clearly, there holds t > 0.

u(±b, t) = 0 ≤ u(±b, t),

On the other hand, a series of computations yield  λ  λ+1 2λ aρ(x) 2λ + λ|x|λ−1 − |x|λ+1 (b − x 2 )−1 Lu = − m+λ−2 m+λ−2 m+λ−2 ! + c0 (at + 1)−

p−(m+λ−1) m+λ−2

aρ(x) − + m+λ−2

≤



λ( p−1)

2λ m+λ−2

λ

− p−(m+λ−1) m+λ−2

λ|x|λ−1 + c0 (at + 1)

λ

m+λ−1

λ

m+λ−1

(at + 1)− m+λ−2 (b2 − x 2 ) m+λ−2

(b2 − x 2 ) m+λ−2

× (at + 1)− m+λ−2 (b2 − x 2 ) m+λ−2 ≤ 0,

(b2 − x 2 )

λ( p−1) m+λ−2

!

in (−b, b) × (0, ∞),

provided that aρ(x) ≥ m+λ−2



2λ m+λ−2

λ

λ( p−1)

λb(λ−1) + c0 b m+λ−2 .

(3.1)

Here we use |x| ≤ b and p ≥ m + λ − 1 > 0. Inequality (3.1) holds in (−b, b) if we take a satisfying !  λ−1 λ( p−1) 2λb 1 2 2λ a≥ + (m + λ − 2)c0 b m+λ−2 . min ρ(x) m+λ−2 [−b,b]

It follows from the comparison principle that the conclusion of Lemma 3.1 is true. Proof of Theorem 1.2. We claim there exist constants a, b > 0 such that u(x, t) ≥ u(x, t),

(x, t) ∈ S+ ,

where u is defined as follows:   λ 1 1 m+λ−2 u(x, t) = (at + 3)− m+λ−2 b − x log− 2λ (at + 3) , +

x > 0, t > 0.

Then the conclusion of Theorem 1.2 follows from h i 1 S ∩ supp u(., t) ⊇ S ∩ supp u(., t) = 0, b log 2λ (at + 3) , for any t ≥ 0. Now we prove the above claim. For simplicity, we set   1 τ = at + 3, ξ = b − x log− 2λ (at + 3) . +

Firstly, it follows from Lemma 3.1 that 1

1

u(0, t) = b(at + 3)− m+λ−2 ≤ b1 (a1 t + 3)− m+λ−2 ≤ u(0, t),

t > 0,



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if we take a = a1 , b < b1 . Moreover, since u 0 (0) > 0, we may further choose b small enough such that u(x, 0) ≤ u 0 (x),

x > 0.

Next, we consider the operator L in set S1 := S+ ∩ {ξ > 0}. A direct computation gives m+λ−1

λ

Lu = τ − m+λ−2 ξ m+λ−2  −

λ m+λ−2

m+λ−1

2λ+1 a a ρ(x) + xρ(x)ξ −1 log− 2λ τ m+λ−2 2(m + λ − 2) ! λ+1 p−(m+λ−1) λ( p−1) − −1 − λ+1 ξ log 2λ τ + c0 τ m+λ−2 ξ m+λ−2

−

2λ+1 a xρ(x)ξ −1 log− 2λ τ 2(m + λ − 2)

λ

≤ τ − m+λ−2 ξ m+λ−2  −

λ m+λ−2

m+λ−1

λ+1

ξ

−1

− λ+1 2λ

log

λ

:= τ − m+λ−2 ξ m+λ−2 (I1 + I2 + I3 ),

τ + c0 τ

− p−(m+λ−1) m+λ−2

ξ

λ( p−1) m+λ−2

!

in S1 .

Then Lu ≤ 0 if we have proved 1 I2 ≤ 0, in S1 ; 2 1 (ii) I3 + I2 ≤ 0, in S1 . 2 The inequality (i) is equivalent to  λ 1 λ axρ(x) ≤ λ log 2 τ, m+λ−2 (i)

I1 +

(x, t) ∈ S1 .

1

Recall that 0 < x < b log 2λ τ in S1 . Then the above inequality holds provided that we choose b > 0 (notice that we have fixed a = a1 ) such that λ  λ−1 λ λ b≤ log 2λ 3. a1 kρ(x)k∞ m + λ − 2 Inequality (ii) is satisfied if  λ+1 λ( p−1) λ 1 τ ξ m+λ−2 +1 ≤ 2c0 m + λ − 2

p−(m+λ−1) m+λ−2

log−

λ+1 2λ

τ,

in S1 .

(3.2)

Since p > m + λ − 1, there exist τ0 (depending on m, λ, p) such that f (τ ) := τ

p−(m+λ−1) m+λ−2

log−

λ+1 2λ

≥ f (τ0 ),

for any τ ≥ 3.

Since ξ < b in S1 , (3.2) holds if we also choose b such that  λ+1 λ( p−1) 1 λ b m+λ−2 +1 ≤ f (τ0 ). 2c0 m + λ − 2 Summarizing the above arguments, we have proved Lu ≤ 0 in S1 . Clearly, Lu = 0 in S \ S1 . The comparison principle implies that our claim is true.  At the end of this section, we give a theorem which describes the asymptotic behavior and is at variance with the case c0 = 0. Theorem 3.1 (Asymptotic Behavior). Let u(x, t) be a solution to problem (1.1). Then we have ku(., t)k∞ → 0,

as t → ∞.

Moreover, if p ∈ (0, 1), there exists T ∈ (0, ∞) such that u ≡ 0 in R × (T, ∞).

Z. Xiang et al. / Nonlinear Analysis 68 (2008) 1940–1953

1947

Proof. We consider the following auxiliary functions:  − 1    A(1 + t) p−1 , if p > 1, −bt if p = 1, u(x, t) = Ae ,  1   A [1 − bt] 1− p , if 0 < p < 1, +

where A, b are constants to be determined. After a simple computation, we see that u defined as above is a supersolution in each cases. The conclusion follows from the comparison principle. The proof is standard and easy, so we omit the details here.  4. Global existence of interfaces In this and next section, we further analyze the properties of interfaces when positivity of solution arises. First of all, to establish the global existence of interfaces, we first give the following lemma. Lemma 4.1. Let p > m + λ − 1. Assume u(x, t) is a solution to problem (1.1). Then there exists B0 > 0, which depends only m, λ, p, c0 and u 0 , such that λ+1 − p−(m+λ−1)

u(x, t) ≤ B0 (1 + x)

,

for any x ≥ 0, t ≥ 0.

Proof. We define the following auxiliary function: λ+1 − p−(m+λ−1)

u(x, t) = B0 (1 + x)

,

x > 0, t > 0,

where B0 is to be determined. We easily get ! λ  λ( p + 1) + m − 1 λ+1 p − p(λ+1) m+λ−1 (1 + x) p−(m+λ−1) . Lu = c0 B0 − B0 p − (m + λ − 1) p − (m + λ − 1) Since p > m + λ − 1, we can choose B0 large enough such that p

0 ≤ Lu ≤ c0 B0 ,

x > 0, t > 0.

On the other hand, since u 0 (x) is compactly supported, we may further take B0 sufficiently large such that λ+1 − p−(m+λ−1)

u(x, 0) = B0 (1 + x)

≥ u 0 (x),

u(0, t) = B0 ≥ ku 0 k∞ + 1 > u(0, t),

x > 0,

t > 0.

Then, it follows from the comparison principle that the lemma is proved. For technical reasons, we introduce the generalized pressure (while u is known as the density of a gas) v(x, t) =

m+λ−2 λ u λ (x, t). m+λ−2

As pointed out in [19,20], this pressure variable is appropriate for studying the properties related to interface behavior and geometry. Thus, we rewrite the problem (1.1) equivalently as follows:  m+λ−2  ρ(x)vt = v |vx |λ−1 vx + |vx |λ+1 − cv q , x ∈ R, t > 0, x λ (4.1) v(x, 0) = v0 (x), x ∈ R, where c =



m+λ−2 λ

 m+λp−2 m+λ−2

c0 , q =

m+λp−2 m+λ−2 , v0 (x)

=

m+λ−2 λ λ . m+λ−2 u 0 (x)

 m+λ−2  v |vx |λ−1 vx − |vx |λ+1 + cv q . x λ Then the comparison principle is still valid for operator L1 .  L1 v = ρ(x)vt −

We define

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Proof of Theorem 1.3. We investigate the following auxiliary function:   ∗ 2 − k −(λ+1) , x > 0, t > 0, v(x, t) = (At + 1) (k ∗ −k)λ B 2 − x 2 (At + 1)− k ∗ −k +

where A, B are to be determined. For simplicity, we define   2 k ∗ − (λ + 1) . γ = > 0, τ = At + 1, ξ = B 2 − x 2 τ − k ∗ −k ∗ + (k − k)λ If we set n o 1 S2 = (x, t) : x > θ Bτ k ∗ −k , t > 0 , q ∗ (k −k)γ where θ = 1+(k ∗ −k)γ < 1, we claim that v(x, t) ≤ v(x, t)

in S2 ,

which implies the desired result: 1

supp u(., t) ∩ [0, ∞) = supp v(., t) ⊆ [0, B(At + 1)− k ∗ −k ),

for any t ≥ 0.

Now we prove the above claim. Firstly, we consider the operator L1 in S1 ∩ S2 . Note that S1 ∩ S2 := {(x, t) : x > 0, t > 0, ξ > 0} ∩ S2 o n 1 1 = (x, t) : θ Bτ k ∗ −k < x < Bτ k ∗ −k , t > 0 . Observe as a preliminary that in S1 ∩ S2 there holds L1 v = −γ Aρ(x)τ −γ −1 ξ +

2Aρ(x)x 2 −γ −1− ∗2 −γ (λ+1)− k ∗2λ−k k −k + (m + λ − 2)2λ x λ−1 τ τ ξ ∗ k −k

− 2λ+1 x λ+1 τ −γ (λ+1)− ≥ −γ Aρ(x)τ −γ −1 ξ +

2(λ+1) k ∗ −k

+ cτ −qγ ξ q

2(λ+1) 2 2A ρ(x)x 2 τ −γ −1− k ∗ −k − 2λ+1 x λ+1 τ −γ (λ+1)− k ∗ −k −k

k∗

:= I1 + I2 + I3 .

(4.2)

1

Since x < Bτ k ∗ −k , from (4.2) we have 2AB 2 kρ(x)k∞ + (m + λ − 2)2λ B λ+1 + cB 2q , in S1 ∩ S2 . k∗ − k Let us show that L1 v ≥ 0 in S1 ∩ S2 . Firstly,     1 1 −γ −1 2 2 − k ∗2−k I1 + I2 = γ Aρ(x)τ −B + 1 + ∗ x τ ≥ 0, 2 (k − k)γ L1 v ≤

in S1 ∩ S2 .

We only need to show that 1 I2 + I3 ≥ 0, in S1 ∩ S2 , 2 which is equivalent to ρ(x) ≥

2λ+1 (k ∗ − k) λ−1 −γ λ+1− ∗2λ k −k , x τ A k ∗ −(λ+1) (k ∗ −k)λ

Recalling γ = provided that

1

Bτ k ∗ −k 1 + Bτ

1 k ∗ −k

in S1 ∩ S2 .

and using the definition of S1 ∩ S2 and condition (1.3), we see that the above inequality holds

!k ≥

2λ+1 (k ∗ − k) k+λ−1 B . Aρ1

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Z. Xiang et al. / Nonlinear Analysis 68 (2008) 1940–1953

Since τ ≥ 1 for any t ≥ 0, !k  1 k Bτ k ∗ −k B . ≥ 1 1+ B 1 + Bτ k ∗ −k Then by taking A, B such that Aρ1 ≥ 2λ+1 (k ∗ − k)B λ−1 (1 + B)k , we have proved L1 v ≥ 0 in S1 ∩ S2 . Clearly, L1 v = 0 in S \ S1 . Now we compare v with v on the boundary ∂ S2 of set S2 . Since u 0 (x) are compactly supported, we see that v(x, 0) ≤ v(x, 0),

if we take B sufficiently large. 1

On the other hand, it follows from Lemma 4.1 that on the lateral boundary x = θ Bτ k ∗ −k ,  − ((λ+1)(m+λ−2)  m+λ−2  1 1 λ p−(m+λ−1))λ v θ Bτ k ∗ −k , t ≤ B0 λ 1 + θ Bτ k ∗ −k m+λ−2 − ((λ+1)(m+λ−2) m+λ−2  1 λ p−(m+λ−1))λ θ Bτ k ∗ −k ≤ B0 λ m+λ−2 m+λ−2 λ ∗ = B0 λ (θ B)−(k −k)γ τ −γ m+λ−2 B2 ≤ τ −γ 1 + (k ∗ − k)γ   1 = v θ Bτ k ∗ −k , t provided that   ∗1 2+(k −k)γ λ(1 + (k ∗ − k)γ ) m+λ−2 ∗ B≥ B0 λ θ −(k −k)γ . m+λ−2 Therefore, we can apply the comparison principle to S2 to prove our claim.



Proof of Theorem 1.5. We define the following auxiliary function:   v(x, t) = e−γ t B 2 − x 2 e−2βt := e−γ t ξ, x > 0, t > 0, where γ =

k ∗ −(λ+1) β, λ

while B, β > 0 are to be determined. A direct computation yields

Lv = −γρ(x)e−γ t ξ + 2βρ(x)x 2 e(γ +2β)t + λ2λ x λ−1 e−(γ +2β)λt−γ t − 2λ+1 x λ+1 x λ+1 e−(γ +2β)(λ+1)t + ce−γ q ξ q ≥ −γρ(x)e−γ t ξ + 2βρ(x)x 2 e(γ +2β)t − 2λ+1 x λ+1 x λ+1 e−(γ +2β)(λ+1)t := I1 + I2 + I3 , in S1 = {(x, t) : x > 0, t > 0, ξ > 0}.  q n o  q γ k ∗ −(λ+1) βt βt If we set S2 := (x, t) : x > β+γ Be , t > 0 = (x, t) : x > k ∗ −1 Be , t > 0 , then we see that I1 +

1 I2 ≥ 0, 2

in S1 ∩ S2 .

We want to show that 12 I2 + I3 ≥ 0 in S1 ∩ S2 , which is equivalent to βρ(x) ≥ 2λ+1 x λ−1 e−(γ +2β)λt ,

in S1 ∩ S2 .

(4.3)

Using similar arguments to the proof of Theorem 1.3, we have (4.3) provided that βρ1 ≥ 2λ+1 B λ−1 (1 + B)k .

(4.4)

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Therefore, if (4.4) holds, we have Lv ≥ 0 in S1 ∩ S2 . Clearly, Lv = 0 in S+ \ S1 . On the boundary of ∂ S2 , it follows from Lemma 4.1 and the arguments of Theorem 1.3 that ! ! r r k ∗ − (λ + 1) βt k ∗ − (λ + 1) βt v Be , t ≤ v Be , t , v(x, 0) ≤ v(x, 0), k∗ − 1 k∗ − 1 if we take B sufficiently large. So if we first fix B large enough and then take β satisfying (4.4), the desired conclusion follows from the comparison principle.  5. Disappearance of interfaces In this section, we investigate the phenomena of disappearance of interfaces. We first give the following lemma. Lemma 5.1. Let p > m + λ − 1. Assume u(x, t) is a solution to problem (1.1) with supp u 0 ⊇ [−R0 , R0 ] , (R0 > 1). 1 Define δ = min[−1,1] u 0 > 0, ρ0 = min[−1,1] ρ > 0. Then for any given l > 0, if a0 ≤ C0 (m, λ, p, ρ0 , l)T −l− m+λ−2 and T > T0 (m, λ, p, ρ0 , l), we have λ

u(x, t) ≥ a0 (T − t)l (1 − x 2 )+m+λ−2 , In particular, u(0, t) ≥ a0 (T

− t)l ,

(x, t) ∈ [−1, 1] × [0, T ).

t ∈ [0, T ).

Proof. We consider the following auxiliary function: λ

u(x, t) ≥ a0 (T − t)l (1 − x 2 )+m+λ−2 ,

(x, t) ∈ [−1, 1] × [0, T ),

where a0 is to be determined. For convenience, we define τ = T − t, ξ = (1 − x 2 )+ . Then it is easy to check that  λ λ 2λ l−1 m+λ−2 Lu = −la0 ρ(x)τ ξ + a0m+λ−1 τ l(m+λ−1) m+λ−2   λp 2−m λ 2λ p λ+1 m+λ−2 λ−1 m+λ−2 |x| ξ + c0 a0 τ lp ξ m+λ−2 × λ|x| ξ − m+λ−2 λ  λp λ λ 2λ p ≤ −la0 ρ(x)τ l−1 ξ m+λ−2 + λa0m+λ−1 |x|λ−1 τ l(m+λ−1) ξ m+λ−2 + c0 a0 τ lp ξ m+λ−2 m+λ−2 := I1 + I2 + I3 , (x, t) ∈ [−1, 1] × [0, T ). We want to show that Lu ≤ 0 in (−1, 1) × (0, T ). To do this, we impose 1 I1 + I2 ≤ 0, (x, t) ∈ (−1, 1) × (0, T ), 2 which is equivalent to  λ 2λ 2λa0m+λ−2 |x|λ−1 τ l(m+λ−2)+1 ≤ lρ(x), m+λ−2 (i)

(x, t) ∈ (−1, 1) × (0, T ),

(5.1)

and 1 I1 + I3 ≤ 0, 2 which is equivalent to (ii)

p−1 l( p−1)+1

2c0 a0

τ

(x, t) ∈ (−1, 1) × (0, T ),

λ( p−1)

ξ m+λ−2 ≤ lρ(x),

(x, t) ∈ (−1, 1) × (0, T ).

(5.2)

It follows from condition (1.3) that inequality (5.1) holds if we choose a0 satisfying a0 T

1 l+ m+λ−2

 ≤

lρ0 2λ



1 m+λ−2



m+λ−2 2λ



λ m+λ−2

:= C1 (m, λ, ρ0 , l).

(5.3)

Z. Xiang et al. / Nonlinear Analysis 68 (2008) 1940–1953

1951

Similarly, inequality (5.2) holds if we further choose a0 satisfying   1 1 lρ0 p−1 l+ p−1 := C2 ( p, c0 , ρ0 , l), a0 T ≤ 2c0 which holds under assumption (5.3) and T

1 1 p−1 − m+λ−2

=T

p−(m+λ−1) − ( p−1)(m+λ−2)

≤

C2 . C1

(5.4)

As long as the initial and lateral data are considered, we have u(x, 0) ≤ a1 T l ≤ u 0 (x),

if a0 T l ≤ δ,

which holds under assumption (5.3) and 1

T − m+λ−2 ≤

δ . C1

(5.5)

Clearly, u(±1, t) = 0 ≤ u(0, t),

t ∈ [0, T ].

We ensure that (5.3)–(5.5) are satisfied for suitable T, a0 . We first fix T0 (m, λ, p, ρ0 , l, δ) such that for T > T0 (5.4) 1 and (5.5) hold, and then choose a0 ≤ C0 (m, λ, p, ρ0 , l)T −l− m+λ−2 . Consequently, the lemma follows our choice.  Proof of Theorem 1.4. We investigate the following auxiliary function:
λ u(x, t) = a(T − t)γ b − x(T − t)β +m+λ−2 , in Q T := {(x, t) : x > 0, t ∈ (0, T )} , where a, b > 0 are to be determined and γ =

λ+1 > 0, (k − k ∗ )( p − (m + λ − 1))

β=

1 . k − k∗

We claim: There exist a, b > 0 such that u(x, t) ≤ u(x, t),

(x, t) ∈ Q T ,

which implies supp u(., t) ⊇ supp u(., t) = [0, b(T − t)−β ]. Since the last set tends to [0, ∞) as t → T , the theorem has been proved. Now, we prove the claim. For convenience, we define
τ = T − t, ξ = b − xτ β + . A direct computation gives γ 2−m λaβ Lu = −aγρ(x)τ γ −1 ξ m+λ−2 + xρ(x)τ γ +β−1 ξ m+λ−2 m+λ−2  λ+1 λp 2−m λ − a m+λ−1 τ γ (m+λ−1)+β(λ+1) ξ m+λ−2 + c0 a p τ γ p ξ m+λ−2 m+λ−2 := I1 + I2 + I3 + I4 , in ST := Q T ∩ {ξ > 0}.

To obtain Lu ≤ 0 in ST we divide the set ST into two subsets ST1 = ST ∩ {I1 + I2 < 0} and ST2 = ST \ ST1 . To be precise, we have   (m + λ − 2)γ 1 β ST = (x, t) : 0 < xτ < b, t > 0 , λβ + (m + λ − 2)γ   (m + λ − 2)γ 2 β ST = (x, t) : b < xτ < b, t > 0 . λβ + (m + λ − 2)γ

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We shall prove that there exist a, b > 0 such that 1 I3 + I4 ≤ 0, 2 which is equivalent to (i)

2c0 a p−(m+λ−1) ξ

in ST ,

m+λp−2 m+λ−2

 ≤

λ m+λ−2

λ+1

,

in ST ,

(5.6)

and 1 I3 ≤ 0, in ST2 , 2 which is equivalent to  λ λ 1 a m+λ−2 τ γ (m+λ−2)+βλ+1 , xρ(x) ≤ 2β m + λ − 2 (ii)

I2 +

in ST2 .

(5.7)

Here we use the fact that γ ( p − (m + λ − 1)) − β(γ + 1) = 0. Combining (i) with (ii) and using the definition of ST1 , we see that Lu ≤ 0 in ST . So we must ensure that (5.6) and (5.7) are satisfied by a suitable choice of a, b. Since ξ ≤ b, (5.6) is implied by a p−(m+λ−1) b

m+λp−2 m+λ−2

≤ C3 (m, λ, c0 ).

(5.8)

Using the definition of ST2 and condition (1.4), we see that (5.7) holds if a m+λ−2 bk−1 ≥ C4 (m, λ, p, k)ρ2 .

(5.9)

Clearly, Lu = 0 in Q T \ ST . So we have proved Lu ≤ 0 in ST provided that (5.8) and (5.9) hold. Let us further consider the initial and lateral data of u on ∂ ST . From Theorem 1.2, without loss of generality, we may assume supp u 0 (x) is sufficiently large, say, [−K − 1, K + 1] ⊂ supp u 0 , where K is to be determined later.
λ Then, u(x, 0) = aT γ b − x T β +m+λ−2 ≤ u 0 (x) if bT −β ≤ K ,

λ

ab m+λ−2 T γ ≤ δ K := min u 0 (x). [−K ,K ]

(5.10)

Next, it follows from Lemma 5.1 with l = γ that γ

u(0, t) = a(T − t)γ b m+λ−2 ≤ a0 (T − t)γ ≤ u(0, t), if there holds γ

ab m+λ−2 := a0 < C0 T

1 − m+λ −γ 2

.

(5.11)

Now, we choose suitable a, b > 0 to satisfy the above requirement. To this purpose, we first take b = K T β and a m+λ−2 = b1−k . With this choice, we see that (5.9) reads ρ2 ≤ C(m, λ, p, k). We remark that this condition is not essential. In fact, scaling u µ (x, t) = u(x, µt), we see that u µ (x, t) satisfies   λ−1 ρµ (x)u µt = u m−1 |u | u − c0 u µp , x ∈ R, t > 0, (5.12) µx µx µ x

where ρµ (x) = ρ(x)/µ. Applying the previous arguments to (5.12), we see that (5.9) is satisfied for ρµ if µ is large enough. For such a and b, determined by the above arguments, we also see that (5.11) holds if K λ+1−k ≤ C0m+λ−2 . This can been done by taking K sufficiently large (notice that k > k ∗ implies λ − 1 + k < 0). Moreover, replacing a, b with K , T in (5.8) and (5.10), we see that there exists T1 (m, p, λ, k, c0 ) : T1 > T0 such that for T > T1 inequality (5.8) and the second inequality in (5.10) hold. To summarize, the proof of Theorem 1.4 is complete. 

Z. Xiang et al. / Nonlinear Analysis 68 (2008) 1940–1953

1953

Acknowledgements The authors are grateful to the referees for their careful reading of this paper. Also, the authors would like to thank Professor A. de Pablo for his kind help. This work was supported by the National Natural Science Foundation of China (10571126), the Program for New Century Excellent Talents in University, and the Youth Foundation of Science and Technology of UESTC. References [1] D. Andreucci, A.F. Tedeev, M. Ughi, The Cauchy problem for degenerate parabolic equations with source and damping, Ukrainian Math. Bull. 1 (2004) 1–23. [2] M. Bertsch, A class of degenerate diffusion equations with a singular nonlinear term, Nonlinear Anal. 7 (1983) 117–127. [3] M. Bertsch, R. Kersner, L.A. Peletier, Positivity versus localization in degenerate diffusion equations, Nonlinear Anal. 9 (1985) 987–1008. [4] X.-Y. Chen, H. Matano, M. Mimura, Finite-point extinction and continuity of interfaces in a nonlinear diffusion equation with strong absorption, J. Reine Angew. Math. 459 (1995) 1–36. [5] C. Ebmeyer, J.M. Urbano, The smoothing property for a class of doubly nonlinear parabolic equations, Trans. Amer. Math. Soc. 357 (2005) 3239–3253. [6] R. Ferreira, A. de Pablo, G. Reyes, A. S´anchez, The interfaces of an inhomogeneous porous medium equation with convection, Comm. Partial Differential Equations 31 (2006) 497–514. [7] V.A. Galaktionov, S. Kamin, R. Kersner, On compactly supported solutions for nonlinear filtration equation in an inhomogeneous medium, 1996. Unpublished manuscript. [8] V.A. Galaktionov, S. Kamin, R. Kersner, J.L. Vazquez, Intermediate asymptotics for inhomogeneous nonlinear heat conduction, J. Math. Sci. 120 (2004) 1277–1294. [9] V.A. Galaktionov, J.R. King, On the behaviour of blow-up interfaces for an inhomogeneous filtration equation, IMA J. Appl. Math. 57 (1996) 53–77. [10] M. Guedda, D. Hilhorst, M.A. Peletier, Disappearing interfaces in nonlinear diffusion, Adv. Math. Sci. Appl. 7 (1997) 695–710. [11] C. Gui, X. Kang, Localization for a porous medium type equation in high dimensions, Trans. Amer. Math. Soc. 356 (2004) 4273–4285. [12] D.D. Joseph, L. Preziosi, Heat waves, Rev. Modern Phys. 61 (1989) 41–73. [13] D.D. Joseph, L. Preziosi, Addendum to the paper “Heat waves”, Rev. Modern Phys. 62 (1990) 375–391. [14] A.S. Kalashnikov, Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations, Russian Math. Surveys 42 (1987) 169–222. [15] S. Kamin, R. Kersner, Disappearance of interfaces in finite time, Meccanica 28 (1993) 117–120. [16] S. Kamin, P. Rosenau, Propagation of thermal waves in an inhomogeneous medium, Comm. Pure Appl. Math. XXXIV (1981) 831–852. [17] R. Kersner, G. Reyes, A. Tesei, On a class of parabolic equations with variable density and absorption, Adv. Differential Equations 7 (2002) 155–176. [18] R. Kersner, A. Tesei, Well-posedness of initial value problems for singular parabolic equations, J. Differential Equations 199 (2004) 47–76. [19] K.A. Lee, J.L. V´azquez, Geometrical properties of solutions of the porous medium equation for large times, Indiana Univ. Math. J. 5 (2003) 991–1016. [20] K.A. Lee, A. Petrosyan, J.L. V´azquez, Large-time geometric properties of solutions of the evolution p-Laplacian equation, J. Differential Equations 229 (2006) 389–411. [21] F.-C. Li, C.-H. Xie, Global and blow-up solutions to a p-Laplacian equation with nonlocal source, Comput. Math. Appl. 46 (2003) 1525–1533. [22] M.A. Peletier, Problems in degenerate diffusion, Ph.D. Thesis, 1997. [23] G. Reyes, A. S´anchez, Disappearance of interfaces for the porous medium equation with variable density and absorption, Asymptot. Anal. 36 (2003) 13–20. [24] G. Reyes, A. Tesei, Basic theory for a diffusion–absorption equation in an inhomogeneous medium, Nonlinear Differential Equations Appl. 10 (2003) 197–222. [25] P. Rosenau, S. Kamin, Nonlinear diffusion in a finite mass medium, Comm. Pure Appl. Math. XXXV (1982) 113–127. [26] A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov, A.P. Mikhailov, Blow-up in Problems for Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, 1995, Nauka, Moscow, 1987 (in Russsian). [27] A.F. Tedeev, Conditions for the time global existence and nonexistence of a compact support of solutions to the Cauchy problem for quasilinear degenerate parabolic equations, Siberian Math. J. 45 (2004) 155–164. [28] M. Tsutsumi, On solutions of some doubly nonlinear degenerate parabolic equations with absorption, J. Math. Anal. Appl. 132 (1988) 187–212. [29] J.L. V´azquez, The Porous Medium Equation: Mathematical Theory, in: Oxford Math. Monogr., Oxford University Press, Oxford, 2007.