Radial symmetric solutions of the Cahn-Hilliard equation with degenerate mobility

Jingxue Yin & Changchun Liu

Department of Mathematics Jilin University, Changchun, Jilin 130012, P. R. China

                           yjx@mail.jlu.edu.cn

Abstract. In this paper we study the radial symmetric solutions of the two-dimensional Cahn-Hilliard equation with degenerate mobility. We adopt the method of parabolic regularization. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence and the nonnegativity of weak solutions. Keywords. Cahn-Hilliard equation, radial solution, degenerate mobility, nonnegativity. AMS Classification: 5G25, 35Q99, 35K9, 82B26

1 Introduction

This paper is devoted to the radial symmetric solutions of the Cahn-Hilliard equation with degenerate mobility

\frac{∂u}{∂t} + div[m(u)(k∇Δu − ∇A(u))] = 0,

with the boundary value conditions

\frac{∂u}{∂n}|_{∂B} =J^{→} · n^{→} |_{∂B} = 0,

and initial value condition

u|_{t=0} = u_{0}(x),

where B is the unit ball in ℝ^{2}, n^{→} is the outward unit normal to ∂B, k > 0,

J^{→}= m(u)(k∇Δu − ∇A(u)),

and m(s), A(s) are appropriately smooth and satisfy the following structure conditions

(H_{1})  0 ≤ m(s) ≤ C_{1}|s|^{p},

(H_{2})  H(s) = ∫_{0}^{s} A(r)dr ≥ −μ, |A′(s)| ≤ C_{2}|s|^{q} + C_{3},

for some positive constants p, q, μ, C′s. We note that a reasonable choice of A(s) is the cubic polynomial, namely

A(s) = γ_{1}s^{3} + γ_{2}s^{2} + γ_{3}s + γ_{4}, γ_{1} > 0,

which corresponds to the so called double-well potential

H(s) = \frac{1}{4}γ_{1}s^{4} + \frac{1}{3}γ_{2}s^{3} + \frac{1}{2}γ_{3}s^{2} + γ_{4}s.

The Cahn-Hilliard equation was introduced to study several diffusion processes, such as phase separation in binary alloys, see [1, 2]. During the past years, such an equation has been paid extensive attention. In particular, there are vast literatures on the investigation of the Cahn-Hilliard equation with constant mobility, for an overview we refer to [3, 4]. However, there are only a few works devoted to the equation with degenerate mobility, see [5, 6, 7, 8, 9, 10, 11, 12], among which Elliott and Garcke [6] was the first who established the basic existence results of weak solutions for space dimensions large than one.

In this paper, we study the radial symmetric solutions of the Cahn-Hilliard equation. We will study the problem in two-dimensional case, which has particular physical derivation of modeling the oil film spreading over a solid surface, see [13]. After introducing the radial variable r = |x|, we see that the radial symmetric solution satisfies

\frac{∂(ru)}{∂t} + \frac{∂}{∂r} \{rm(u) [k \frac{∂V}{∂r} − A′(u)\frac{∂u}{∂r}]\} = 0, rV = \frac{∂}{∂r} (r \frac{∂u}{∂r}) , (1.1)

\frac{∂u}{∂r}|_{r=0} = \frac{∂u}{∂r}|_{r=1} = J̃|_{r=0} = J̃|_{r=1} = 0, (1.2)

u|_{t=0} = u_{0}(r), (1.3)

where

J̃ = m(u) (k \frac{∂V}{∂r} − A′(u)\frac{∂u}{∂r}) .

It should be noticed that the equation (1.1) is degenerate at the points where r = 0 or u = 0, and hence the arguments for one-dimensional problem can not be applied directly. Because of the degeneracy, the problem does not admit classical solutions in general. So, we introduce the weak solutions in the following sense

Definition A function u is said to be a weak solution of the problem (1.1)– (1.3), if the following conditions are fulfilled:

(1) ru(r, t) is continuous in \overline{Q}_{T}, where Q_{T} = (0, 1) × (0, T);

(2) \sqrt{rm(u)}u_{rrr} ∈ L^{2}(P), where P = \overline{Q}_{T} \(\{u = 0\} ∪ \{t = 0\} ∪ \{r = 0\});

(3) For any φ ∈ C^{1}(\overline{Q}_{T}), the following integral equality holds

\begin{matrix}− ∫_{0}^{1} ru(r, T)φ(r, T)dr + ∫_{0}^{1} ru_{0}(r)φ(r, 0)dr + ∫∫_{Q_{T}} ru\frac{∂φ}{∂t}drdt \\ + ∫∫_{P} rm(u) [k \frac{∂V}{∂r} − A′(u)\frac{∂u}{∂r}] \frac{∂φ}{∂r}drdt = 0; \end{matrix}

(4) u satisfies the lateral boundary value condition (1.2) at the points where u ≠ 0.

We first investigate the existence of weak solutions. Because of the degeneracy, we will first consider the regularized problem. Based on the uniform estimates for the approximate solutions, we obtain the existence. Owing to the background, we are much interested in the nonnegativity of the weak solutions. For this purpose, we construct a suitable test function and discuss such a property under some conditions on the data. This paper is arranged as follows. We first study the regularized problem in Section 2, and then establish the existence in Section 3. Subsequently, we discuss the nonnegativity of weak solutions in the last Section.

2 Regularized problem

To discuss the existence, we adopt the method of parabolic regularization, namely, the desired solution will be obtained as the limit of some subsequence of solutions of the following regularized problem

\frac{∂(r_{ε}u)}{∂t} + \frac{∂}{∂r} \{r_{ε}m_{ε}(u) [k \frac{∂V}{∂r} − A′(u)\frac{∂u}{∂r}]\} = 0, r_{ε}V = \frac{∂}{∂r} (r_{ε} \frac{∂u}{∂r}) , (2.1)

\frac{∂u}{∂r}|_{r=0} = \frac{∂u}{∂r}|_{r=1} = J̃_{ε}|_{r=0} = J̃_{ε}|_{r=1} = 0, (2.2)

u|_{t=0} = u_{0}(r), (2.3)

where r_{ε} = r + ε, m_{ε}(s) = m(s) + ε and

J̃_{ε} = m_{ε}(u) [k \frac{∂V}{∂r} − A(u)\frac{∂u}{∂r}] .

Theorem 2.1 For each fixed ε > 0 and suitably smooth u_{0}, under the assumptions (H_{1}), (H_{2}), the problem (2.1)–(2.3) admits a unique classical solution u in the space C^{4+α,1+α/4}(\overline{Q}_{T}) for some α ∈ (0, 1).

To prove the theorem, we need some a priori estimates on the solutions. We first have

Lemma 2.1 For any α ∈ (0, \frac{1}{2}] and β < α, there is a constant M independent of ε such that

|r_{ε}^{α}u(r, t) − s_{ε}^{α}u(s, t)| ≤ M|r − s|^{β}

and

|r_{ε}u(r, t) − s_{ε}u(s, t)| ≤ M|r − s|^{β}

for all r, s ∈ (0, 1), where s_{ε} = s + ε.

Proof. We first introduce some notations. Let I = (0, 1) and for any fixed ε ≥ 0 denote by W_{∗,ε}^{1,2}(I) the class of all functions satisfying

∥u∥_{∗,ε} = (∫_{0}^{1}(r + ε)|u′(r)|^{2}dr)^{1/2} + (∫_{0}^{1}(r + ε)|u(r)|^{2}dr)^{1/2} < +∞.

It is obvious that W^{1,2}(I) ⊂ W_{∗,ε}^{1,2}(I), but the class W_{∗,ε}^{1,2}(I) is quite different from W^{1,2}(I). In particular, we notice that the functions in W_{∗,ε}^{1,2}(I) may not be bounded. However, it is not difficult to prove that for u ∈ W_{∗,ε}^{1,2}(I), the following properties hold:

(1) If 0 < α ≤ 1, then

sup_{0<r<1}((r + ε)^{α}|u(r)|) ≤ C∥u∥_{∗,ε},

where C is a constant depending only on α; (2) If 0 < α ≤ \frac{1}{2}, then for any β < α

|(r_{1} + ε)^{α}u(r_{1}) − (r_{2} + ε)^{α}u(r_{2})| ≤ C|r_{1} − r_{2}|^{β}∥u∥_{∗,ε},

where C is a constant depending only on α and β.

Now, we set

F_{ε}(t) = ∫_{0}^{1} [\frac{k}{2} r_{ε} (\frac{∂u}{∂r})^{2} + r_{ε}H(u) + r_{ε}μ] dr

and get from the equation (2.1)

\frac{dF_{ε}(t)}{dt}

= ∫_{0}^{1} [2\frac{k}{2} r_{ε} \frac{∂u}{∂r} \frac{∂}{∂t} (\frac{∂u}{∂r}) + r_{ε}A(u)\frac{∂u}{∂t}] dr
= ∫_{0}^{1} [kr_{ε} \frac{∂u}{∂r} \frac{∂}{∂t} (\frac{∂u}{∂r}) + r_{ε}A(u)\frac{∂u}{∂t}] dr
= ∫_{0}^{1} [−k \frac{∂}{∂r} (r_{ε} \frac{∂u}{∂r}) + r_{ε}A(u)] ·

· (−\frac{1}{r_{ε}} \frac{∂}{∂r} \{r_{ε}m_{ε}(u) [k \frac{∂V}{∂r} − A′(u)\frac{∂u}{∂r}]\}) dr

= ∫_{0}^{1} [\frac{k}{r_{ε}} \frac{∂}{∂r} (r_{ε} \frac{∂u}{∂r}) − A(u)] (\frac{∂}{∂r} \{r_{ε}m_{ε}(u) [k \frac{∂V}{∂r} − A′(u)\frac{∂u}{∂r}]\}) dr
= − ∫_{0}^{1} \frac{∂}{∂r} [kV − A(u)] r_{ε}m_{ε}(u) [k \frac{∂V}{∂r} − A′(u)\frac{∂u}{∂r}] dr
= − ∫_{0}^{1} rm_{ε} [k \frac{∂V}{∂r} − A′(u)\frac{∂u}{∂r}]^{2} dr
≤ 0,

which implies that

F_{ε}(t) ≤ F_{ε}(0)  (2.4)

and

∫_{0}^{1} r_{ε} (\frac{∂u}{∂r})^{2} dr ≤ C.  (2.5)

Integrating the equation (2.1) on Q_{t} = (0, 1) × (0, t), we have

∫_{0}^{1} r_{ε}u(r, t)dr = ∫_{0}^{1} r_{ε}u_{0}(r)dr.  (2.6)

For any ρ ∈ (0, 1),

\frac{1 + 2ε}{2}u(ρ, t) − ∫_{0}^{1} s_{ε}u(s, t)ds

= ∫_{0}^{1} s_{ε}[u(ρ, t) − u(s, t)]ds
= ∫_{0}^{1} ∫_{s}^{ρ} s_{ε} \frac{∂u}{∂r}(r, t)drds
= ∫_{0}^{ρ} ∫_{s}^{ρ} s_{ε} \frac{∂u}{∂r}(r, t)drds + ∫_{ρ}^{1} ∫_{s}^{ρ} s_{ε} \frac{∂u}{∂r}(r, t)drds

= ∫_{0}^{ρ} ∫_{0}^{r} s_{ε} \frac{∂u}{∂r}(r, t)dsdr + ∫_{ρ}^{1} ∫_{r}^{1} s_{ε} \frac{∂u}{∂r}(r, t)dsdr

= ∫_{0}^{ρ} (\frac{r^{2}}{2} + εr) \frac{∂u}{∂r}(r, t)dr + ∫_{ρ}^{1} [\frac{1}{2}(1 − r^{2}) + ε(1 − r)] \frac{∂u}{∂r}(r, t)dr

≤ ∫_{0}^{ρ} r_{ε}|\frac{∂u}{∂r}(r, t)|dr + 2 ∫_{ρ}^{1} |\frac{∂u}{∂r}(r, t)|dr.

Setting ρ_{ε} = ρ + ε and multiplying the above inequality with 2ρ_{ε}^{1/2}, we get

\begin{aligned}& |(1 + 2ε)ρ_{ε}^{1/2}u(ρ, t) − 2ρ_{ε}^{1/2} ∫_{0}^{1} s_{ε}u(s, t)ds| \\ ≤ & 2ρ_{ε}^{1/2} ∫_{0}^{ρ} r_{ε}|\frac{∂u}{∂r}(r, t)|dr + 4ρ_{ε}^{1/2} ∫_{ρ}^{1} |\frac{∂u}{∂r}(r, t)|dr \\ ≤ & 2ρ_{ε}^{1/2} ∫_{0}^{ρ} r_{ε}|\frac{∂u}{∂r}(r, t)|dr + 4 ∫_{ρ}^{1} r_{ε}^{1/2}|\frac{∂u}{∂r}(r, t)|dr \\ ≤ & C (∫_{0}^{1} r_{ε}|\frac{∂u}{∂r}(r, t)|^{2}dr)^{1/2} . \end{aligned}  (2.7)

From (2.5), (2.6) and (2.7), we see that r_{ε}^{1/2}u(r, t) is uniformly bounded on Q_{T}. Furthermore u(·, t) ∈ W_{∗,ε}^{1,2}(I) for any fixed t ∈ (0, T), with ∥u(·, t)∥_{∗,ε} bounded by a constant C independent of ε. The desired estimates then follow from the properties of W_{∗,ε}^{1,2}(I) mentioned above. The proof is complete.

Lemma 2.2 For any α > 0, there is a constant M independent of ε such that

r_{ε}^{α}|u(r, t)| ≤ M, ∥u∥_{∗,ε} ≤ M  (2.8)

and

|r_{ε}u(r, t_{2}) − r_{ε}u(r, t_{1})| ≤ M|t_{2} − t_{1}|^{1/16}  (2.9)

for all r ∈ (0, 1), t_{1}, t_{2} ∈ (0, T).

Proof. The first two estimates have already been seen from the arguments in Lemma 2.1. To prove (2.9), we need an integral estimate first. Multiplying the equation (2.1) by V and integrating with respect to r over (0, 1), we get

\begin{aligned}0 & = ∫_{0}^{1} \{\frac{∂r_{ε}u}{∂t}V + \frac{∂}{∂r} [r_{ε}m_{ε} (k \frac{∂V}{∂r} − A′ \frac{∂u}{∂r})] V \} dr \\ & = − ∫_{0}^{1} \{\frac{∂}{∂t} [r_{ε} (\frac{∂u}{∂r})^{2}] + r_{ε}m_{ε}k (\frac{∂V}{∂r} )^{2}\} dr + ∫_{0}^{1} r_{ε}m_{ε}A′ \frac{∂u}{∂r} \frac{∂V}{∂r} dr \end{aligned}

i.e.

\frac{d}{dt} ∫_{0}^{1} r_{ε} (\frac{∂u}{∂r})^{2} dr + ∫_{0}^{1} r_{ε}m_{ε}k (\frac{∂V}{∂r} )^{2} dr

≤ \frac{1}{2} ∫_{0}^{1} r_{ε}m_{ε}k (\frac{∂V}{∂r} )^{2} dr + \frac{1}{2k} ∫_{0}^{1} r_{ε}m_{ε}(A′)^{2} (\frac{∂u}{∂r})^{2} dr
≤ \frac{k}{2} ∫_{0}^{1} r_{ε}m_{ε}k (\frac{∂V}{∂r} )^{2} dr + \frac{1}{2k} ∫_{0}^{1} r_{ε}|u|^{p}(C_{1}|u|^{q} + C_{2})^{2} (\frac{∂u}{∂r})^{2} dr,

which, together with the first two estimates in this lemma, implies that

∫_{0}^{t} ∫_{0}^{1} r_{ε}m_{ε}(r) (\frac{∂V}{∂r} )^{2} drdt ≤ C.  (2.10)

Now, we begin to show (2.9). Without loss of generality, we assume that t_{1} < t_{2} and set Δt = t_{2} − t_{1}. Integrating both sides of the equation (2.1) over (t_{1}, t_{2}) × (y, y + (Δt)^{α}) and then integrating the resulting relation with respect to y over (x, x + (Δt)^{α}), we get

(Δt)^{α} ∫_{x}^{x+(Δt)^{α}} ∫_{0}^{1}(y + θ(Δt)^{α} + ε)·

·[u(y + θ(Δt)^{α}, t_{2}) − u(y + θ(Δt)^{α}, t_{1})]dθdy
= − ∫_{x}^{x+(Δt)^{α}} ∫_{y}^{y+(Δt)^{α}} ∫_{t_{1}}^{t_{2}} \frac{∂}{∂r} \{r_{ε}m_{ε} [k \frac{∂V}{∂r} − A′(u)\frac{∂u}{∂r}]\} dτdrdy
= − ∫_{x}^{x+(Δt)^{α}} ∫_{t_{1}}^{t_{2}} k [(y + (Δt)^{α} + ε)m_{ε}(u(y + (Δt)^{α} + ε, τ))·

·\frac{∂}{∂y} V (y + (Δt)^{α} + ε, τ) − (y + ε)m_{ε}(u(y + ε, τ))\frac{∂}{∂y} V (y + ε, τ)] dτdy

+ ∫_{x}^{x+(Δt)^{α}} ∫_{t_{1}}^{t_{2}} [−(y + ε)m_{ε}(u(y + ε, τ)A′(u(y + ε, τ))\frac{∂}{∂y} u(y + ε, τ)

+ (y + (Δt)^{α} + ε)m_{ε}(u(y + (Δt)^{α} + ε, τ))A′(u(y + (Δt)^{α} + ε, τ))·

· \frac{∂}{∂y} u(y + (Δt)^{α} + ε, τ)] dτdy.

By the mean value theorem, there exists x^{∗} = y^{∗} + θ^{∗}(Δt)^{α}, y^{∗} ∈ (x, x + (Δt)^{α}), θ^{∗} ∈ (0, 1) such that the left hand side of the above equality can be expressed by

(Δt)^{α} ∫_{x}^{x+(Δt)^{α}} ∫_{0}^{1}(y + θ(Δt)^{α} + ε)·
· [u(y + θ(Δt)^{α}, t_{2}) − u(y + θ(Δt)^{α}, t_{1})] dθdy
= (Δt)^{2α}(y^{∗} + θ^{∗}(Δt)^{α} + ε)[u(y^{∗} + θ^{∗}(Δt)^{α}, t_{2}) − u(y^{∗} + θ^{∗}(Δt)^{α}, t_{1})].

For the right hand side, we have

− ∫_{x}^{x+(Δt)^{α}} ∫_{t_{1}}^{t_{2}} k [(y + (Δt)^{α} + ε)m_{ε}(u(y + (Δt)^{α} + ε, τ)

\frac{∂}{∂y} V (y + (Δt)^{α} + ε, τ) − (y + ε)m_{ε}(u(y + ε, τ)\frac{∂}{∂y} V (y + ε, τ)] dτdy

+ ∫_{x}^{x+(Δt)^{α}} ∫_{t_{1}}^{t_{2}} [−(y + ε)m_{ε}(u(y + ε, τ)A′(u(y + ε, τ))\frac{∂}{∂y} u(y + ε, τ)

+ (y + (Δt)^{α} + ε)m_{ε}(u(y + (Δt)^{α} + ε, τ)A′(u(y + (Δt)^{α} + ε, τ))·

·\frac{∂}{∂y} u(y + (Δt)^{α} + ε, τ)] dτdy

= − ∫_{x+(Δt)^{α}}^{x+2(Δt)^{α}} ∫_{t_{1}}^{t_{2}} k [(r + ε)m_{ε}(u(r + ε, τ))\frac{∂}{∂r} V (r + ε, τ)] dτdr

+ ∫_{x}^{x+(Δt)^{α}}(r + ε)m_{ε}(u(r + ε, τ)\frac{∂}{∂r} V (r + ε, τ)dτdr

+ ∫_{x+(Δt)^{α}}^{x+2(Δt)^{α}} ∫_{t_{1}}^{t_{2}}(r + ε)m_{ε}(u(r + ε, τ)A′(u(r + ε, τ))\frac{∂}{∂r} u(r + ε, τ)dτdr

− ∫_{x}^{x+(Δt)^{α}}(r + ε)m_{ε}(u(r + ε, τ)A′(u(r + ε, τ))\frac{∂}{∂r} u(r + ε, τ)dτdr

≤ ∫_{x+(Δt)^{α}}^{x+2(Δt)^{α}} ∫_{t_{1}}^{t_{2}} kr_{ε}m_{ε}(u(r + ε, τ)|\frac{∂}{∂r} V (rτ)|dτdr

+ ∫_{x+(Δt)^{α}}^{x+2(Δt)^{α}} ∫_{t_{1}}^{t_{2}} r_{ε}m_{ε}(u(r + ε, τ))|A′||\frac{∂u}{∂r}|dτdr

≤ (∫_{x+(Δt)^{α}}^{x+2(Δt)^{α}} ∫_{t_{1}}^{t_{2}} kr_{ε}m_{ε}dτdr)^{1/2} (∫_{x+(Δt)^{α}}^{x+2(Δt)^{α}} ∫_{t_{1}}^{t_{2}} kr_{ε}m_{ε}(\frac{∂V}{∂r} )^{2}dτdr)^{1/2}

+(∫_{x+(Δt)^{α}}^{x+2(Δt)^{α}} ∫_{t_{1}}^{t_{2}} r_{ε}m_{ε}dτdr)^{1/2}(∫_{x+(Δt)^{α}}^{x+2(Δt)^{α}} ∫_{t_{1}}^{t_{2}} r_{ε}m_{ε}|A′_{ε}|^{2}(\frac{∂u}{∂r})^{2}dτdr)^{1/2} .

By (2.8), (2.10) and the assumptions on m(u), A(u), we see that

|x_{ε}^{∗}u(x^{∗}, t_{2}) − x_{ε}^{∗}u(x^{∗}, t_{1})| ≤ C(Δt)^{\frac{1−3α}{2}} ,

which implies, by setting α = 1/4 and using the properties of the functions in W_{∗,ε}^{1,2}(I), that

|r_{ε}u(r, t_{2}) − r_{ε}u(r, t_{1})| ≤ C(Δt)^{1/16}.

The proof is complete.

Proof of Theorem 2.1. Using Lemma 2.1 and 2.2, we see that r_{ε}u is uniformly bounded in C^{1/4,1/16}(\overline{Q}_{T})-norm with the bound independent of ε. Similar to [14], we can further establish the estimates on the Hölder norm of Du. Then, using the classical Schauder theory, we may complete the proof of the remaining part in a standard way.

3 Existence

After the discussion of the regularized problem, we can now turn to the investigation of the existence of weak solutions of the problem (1.1)–(1.3). The main existence result is the following

Theorem 3.1 Under the assumptions (H_{1}), (H_{2}), the problem (1.1)–(1.3) admits at least one weak solution.

Proof. Let u_{ε} be the approximate solution of the problem (2.1)–(2.3) constructed in the previous section. Using the estimates in Lemma 2.1 and 2.2, for any β < \frac{1}{2}, and (r_{1}, t_{2}), (r_{2}, t_{1}) ∈ Q_{T}, we have

|r_{1ε}u_{ε}(r_{1}, t_{2}) − r_{2ε}u_{ε}(r_{2}, t_{1})| ≤ C(|r_{1} − r_{2}|^{β} + |t_{1} − t_{2}|^{β/4})

with constant C independent of ε. So, we may extract a subsequence from {r_{ε}u_{ε}}, denoted also by {r_{ε}u_{ε}}, such that

r_{ε}u_{ε}(r, t) → ru(r, t) uniformly in \overline{Q}_{T} ,

and the limiting function ru ∈ C^{1/4,1/16}(\overline{Q}_{T}). By (2.8), we also have r^{α}u ∈ L^{∞}(Q_{T}) with α > 0 and for any t ∈ (0, T), u(·, t) ∈ W_{∗,0}^{1,2}(I) with the norm ∥u(·, t)∥_{∗,0} bounded by a constant independent of t.

Now, let δ > 0 be fixed and set P_{δ} = {(r, t); rm(u(r, t)) > δ}. We choose ε(δ) > 0, such that

r_{ε}m_{ε}(u_{ε}(r, t)) ≥ \frac{δ}{2}, (r, t) ∈ P_{δ}, 0 < ε < ε_{0}(δ).  (3.1)

Then from (2.10)

∫∫_{P_{δ}} (\frac{∂V_{ε}}{∂r} )^{2} drdt ≤ \frac{C}{δ} .  (3.2)

To prove the integral equality in the definition of solutions, it suffices to pass the limit as ε → 0 in

\begin{matrix}− ∫_{0}^{1} r_{ε}u_{ε}(r, T)φ(r, T)dr + ∫_{0}^{1} r_{ε}u_{0ε}φ(r, 0)dr + ∫∫_{Q_{T}} r_{ε}u_{ε} \frac{∂φ}{∂t}drdt \\ + ∫∫_{P} r_{ε}m_{ε}(u_{ε}) [k \frac{∂V_{ε}}{∂r} − A′(u_{ε})\frac{∂u_{ε}}{∂r} ] \frac{∂φ}{∂r}drdt = 0. \end{matrix}

The limits

lim_{ε→0} ∫_{0}^{1} r_{ε}u_{ε}(r, T)φ(r, T) = ∫_{0}^{1} ru(r, T)φ(r, T)dr,

lim_{ε→0} ∫_{0}^{1} r_{ε}u_{0ε}(r)φ(r, 0)dr = ∫_{0}^{1} u_{0}(r)φ(r, 0)dr,

lim_{ε→0} ∫∫_{Q_{T}} r_{ε}u_{ε} \frac{∂φ}{∂t}drdt = ∫∫_{Q_{T}} ru\frac{∂φ}{∂t}drdt

are obvious. It remains to show

lim_{ε→0} ∫∫_{Q_{T}} r_{ε}m_{ε}(u_{ε})k \frac{∂V_{ε}}{∂r} \frac{∂φ}{∂r}drdt = ∫∫_{P} rm(u)k \frac{∂V}{∂r} \frac{∂φ}{∂r}drdt, (3.3)

lim_{ε→0} ∫∫_{Q_{T}} r_{ε}m_{ε}(u_{ε})A′(u_{ε})\frac{∂u_{ε}}{∂r} \frac{∂φ}{∂r}drdt = ∫∫_{P} rm(u)A′(u)\frac{∂u}{∂r} \frac{∂φ}{∂r}drdt. (3.4)

In fact, for any fixed δ > 0,

| ∫∫_{Q_{T}} r_{ε}m_{ε}(u_{ε})k \frac{∂V_{ε}}{∂r} \frac{∂φ}{∂r}drdt − ∫∫_{P} rm(u)k \frac{∂V}{∂r} \frac{∂φ}{∂r}drdt|

≤ | ∫∫_{P_{δ}} r_{ε}m_{ε}(u_{ε})k \frac{∂V_{ε}}{∂r} \frac{∂φ}{∂r}drdt − ∫∫_{P_{δ}} rm(u)k \frac{∂V}{∂r} \frac{∂φ}{∂r}drdt|

+| ∫∫_{Q_{T} \P_{δ}} r_{ε}m_{ε}(u_{ε})k \frac{∂V_{ε}}{∂r} \frac{∂φ}{∂r}drdt| + | ∫∫_{P \P_{δ}} rm(u)k \frac{∂V}{∂r} \frac{∂φ}{∂r}drdt|.

From the estimates (2.10), we have

\begin{matrix}| ∫∫_{Q_{T} \P_{δ}} r_{ε}m_{ε}(u_{ε})k \frac{∂V_{ε}}{∂r} \frac{∂φ}{∂r}drdt| ≤ Cδ sup |\frac{∂φ}{∂r}|, 0 < ε < ε_{0}(δ) \\ | ∫∫_{P \P_{δ}} rm(u)k \frac{∂V}{∂r} \frac{∂φ}{∂r}drdt| ≤ Cδ sup |\frac{∂φ}{∂r}|, \end{matrix}

| ∫∫_{P_{δ}} r_{ε}m_{ε}(u_{ε})k \frac{∂V_{ε}}{∂r} \frac{∂φ}{∂r}drdt − ∫∫_{P_{δ}} rm(u)k \frac{∂V}{∂r} \frac{∂φ}{∂r}drdt|

≤ ∫∫_{P_{δ}} |r_{ε}m_{ε}(u_{ε}) − rm(u)||\frac{∂V_{ε}}{∂r} ||\frac{∂φ}{∂r}|drdt

+| ∫∫_{P_{δ}} rm(u) (\frac{∂V_{ε}}{∂r} − \frac{∂V}{∂r} ) \frac{∂φ}{∂r}drdt|

≤ sup |r_{ε}m_{ε}(u_{ε}) − rm(u)||\frac{∂φ}{∂r}|\frac{C}{\sqrt{δ}} + | ∫∫_{P_{δ}} rm(u) (\frac{∂V_{ε}}{∂r} − \frac{∂V}{∂r} ) \frac{∂φ}{∂r}drdt|

and hence

\overline{lim_{ε→0}}| ∫∫_{Q_{T}} r_{ε}m_{ε}(u_{ε})k \frac{∂V_{ε}}{∂r} \frac{∂φ}{∂r}drdt − ∫∫_{P} rm(u)k \frac{∂V}{∂r} \frac{∂φ}{∂r}drdt| ≤ Cδ sup |\frac{∂φ}{∂r}|.

By the arbitrariness of δ, we see that the limit (3.3) holds. EJQTDE, 2001 No. 2, p.9 Finally, from the uniform convergence of r_{ε}u_{ε} to ru, we immediately obtain

lim_{ε→0} ∫∫_{Q_{T}} r_{ε}m_{ε}(u_{ε})DA(u_{ε})Dφdrdt

= lim_{ε→0} ∫∫_{Q_{T}} r_{ε}DH(u_{ε})Dφdrdt

= lim_{ε→0} ∫_{0}^{T} (ε + 1)H(u_{ε}(1, t))Dφ(1, t)drdt

− lim_{ε→0} ∫_{0}^{T} εH(u_{ε}(0, t))Dφ(0, t)drdt

− lim_{ε→0} ∫_{0}^{T} H(u_{ε})D(r_{ε}Dφ)drdt

= ∫_{0}^{T} H(u(1, t))Dφ(1, t)drdt − ∫∫_{Q_{T}} H(u)D(rDφ)drdt
= ∫∫_{Q_{T}} DH(u)rDφdrdt = ∫∫_{Q_{T}} rm(u)DA(u)Dφdrdt.

The proof is complete.

4 Nonnegativity

Just as mentioned by several authors, it is much interesting to discuss the physical solutions. For the two-dimensional problem (1.1)–(1.3), a very typical example is the modeling of oil films spreading over an solid surface, where the unknown function u denotes the height from the surface of the oil film to the solid surface. Motivated by this idea, we devote this section to the discussion of the nonnegativity of solutions.

Theorem 4.1 The weak solution u obtained in Section 3 satisfy u(x, t) ≥ 0, if u_{0}(x) ≥ 0.

Proof. Suppose the contrary, that is, the set

E = \{(r, t) ∈ \overline{Q}_{T}; u(r, t) < 0\}  (4.1)

is nonempty.

For any fixed δ > 0, choose a C^{∞} function H_{δ}(s) such that H_{δ}(s) = −σ for s ≥ −δ, H_{δ}(s) = −1, for δ ≤ −2δ and that H_{δ}(s) is nondecreasing for −2δ < s < −δ. Also, we extend the function u(r, t) to be defined in the whole plane ℝ^{2} such that the extension ū(r, t) = 0 for t ≥ T +1 and t ≤ −1. Let α(s) be the kernel of mollifier in one-dimension, that is, α(s) ∈ C^{∞}(ℝ), suppα = [−1, 1],

α(s) > 0 in (−1, 1), and ∫_{−1}^{1} α(s)ds = 1. For any fixed k > 0, δ > 0, define

u^{h}(r, t) = ∫_{ℝ} ū(s, r)α_{h}(t − s)ds,

β_{δ}(t) = ∫_{t}^{+∞} α (\frac{s − \frac{T}{2}}{\frac{T}{2} − δ} ) \frac{1}{\frac{T}{2} − δ} ds,

where α_{h}(s) = \frac{1}{h}α(\frac{s}{h}).

The function

φ_{δ}^{h}(r, t) ≡ [β_{δ}(t)H_{δ}(u^{h})]^{h}

is clearly an admissible test function, that is the following integral equality holds

\begin{matrix}− ∫_{0}^{1} ru(r, T)φ_{δ}^{h}(T, r)dr + ∫_{0}^{1} ru_{0}(r)φ_{δ}^{h}(r, 0)dr + ∫∫_{Q_{T}} ru\frac{∂φ_{δ}^{h}}{∂t} drdt \\ + ∫∫_{P} rm(u) [k \frac{∂V}{∂r} − A′(u)\frac{∂u}{∂r}] \frac{∂φ_{δ}^{h}}{∂r} drdt = 0. \end{matrix} (4.2)

To proceed further, we give an analysis on the properties of the test function φ_{δ}^{h}(r, t). The definition of β_{δ}(t) implies that

φ_{δ}^{h}(r, t) = 0, t ≥ T − \frac{δ}{2}, h < \frac{δ}{2}.  (4.3)

Since ū(r, t) is continuous, for fixed δ, there exists η_{1}(δ) > 0, such that

u^{h}(r, t) ≥ −\frac{δ}{2}, t ≤ η_{1}(δ), 0 ≤ r ≤ 1, h < η_{1}(δ),  (4.4)

which together with the definition of β_{δ}(t), H_{δ}(s) imply that

H_{δ}(u^{h}(r, t)) = −δ, t ≤ η_{1}(δ), 0 ≤ r ≤ 1, h < η_{1}(δ)  (4.5)

and hence

φ_{δ}^{h} = −δ, t ≤ \frac{1}{2}η_{1}(δ), 0 ≤ r ≤ 1, h < \frac{1}{2}η_{1}(δ).  (4.6)

We note also that for any functions f(t), g(t) ∈ L^{2}(R),

\begin{aligned}∫_{ℝ} f(t)g^{h}(t)dt & = ∫_{ℝ} f(t)dt ∫_{ℝ} g(s)α_{n}(t − s)ds = ∫_{ℝ} f(t) ∫_{ℝ} g(s)α_{n}(s − t)ds \\ & = ∫_{ℝ} g(s)ds ∫_{ℝ} f(t)α_{n}(s − t)dt = ∫_{ℝ} f^{h}(t)g(t)dt. \end{aligned}

Taking this into account and using (4.3), (4.5), (4.6), we have

∫∫_{Q_{T}} ru\frac{∂}{∂t}φ_{δ}^{h}drdt

= ∫_{−∞}^{+∞} dt ∫_{0}^{1} ru [\frac{∂}{∂t} (β_{δ}(t)H_{δ}(u^{h}))]^{h} dr
= ∫∫_{Q_{T}} (ru)^{h} \frac{∂}{∂t} (β_{δ}(t)H_{δ}(u^{h})) drdt

and hence by integrating by parts

∫∫_{Q_{T}} (ru)^{h} \frac{∂}{∂t} (β_{δ}(t)H_{δ}(u^{h})) drdt

= ∫_{0}^{1}(ru)^{h}(r, T)β_{δ}(T)H_{δ}(u^{h}(r, T))dr − ∫_{0}^{1}(ru)^{h}(r, 0)β_{δ}(0)H_{δ}(u^{h}(r, 0))dr

− ∫∫_{Q_{T}} β_{δ}(t)H_{δ}(u^{h})\frac{∂(ru)^{h}}{∂t} drdt

= δ ∫_{0}^{1}(ru)^{h}(r, 0)dr − ∫∫_{Q_{T}} rβ_{δ}(t)\frac{∂}{∂t}F_{δ}(u^{h})drdt,

where F_{δ}(s) = ∫_{0}^{s} H_{δ}(σ)dσ.

Again by (4.5)

F_{δ}(u^{h}(r, 0)) = ∫_{0}^{u^{h}(r,0)} H_{δ}(σ)dσ
= ∫_{0}^{1} H_{δ}(λu^{h}(r, 0))dλ · u^{h}(r, 0)
= −δu^{h}(r, 0)

and hence

∫∫_{Q_{T}} (ru)^{h} \frac{∂}{∂t}(β_{δ}(t)H_{δ}(u^{h}))drdt

= δ ∫_{0}^{1}(ru)^{h}(r, 0)dr + ∫_{0}^{1} rβ_{δ}(0)F_{δ}(u^{h}(r, 0))dr + ∫∫_{Q_{T}} rF_{δ}(u^{h})β′_{δ}(t)drdt
= − \frac{1}{\frac{T}{2} − δ} ∫∫_{Q_{T}} rF_{δ}(u^{h})α ( \frac{t − \frac{T}{2}}{\frac{T}{2} − δ} ) drdt.
(4.7)

From (4.3), (4.6) it is clear that

− ∫_{0}^{1} ru(r, T)φ_{δ}^{h}(T, r)dr = 0, 0 < h < \frac{1}{2}η_{1}(δ),  (4.8)
∫_{0}^{1} ru_{0}(r)φ_{δ}^{h}(r, 0)dr = −δ ∫_{0}^{1} ru_{0}(r)dr.  (4.9)

Substituting (4.7), (4.8) and (4.9) into (4.2), we have

\begin{aligned}& −\frac{2}{T − 2δ} ∫∫_{Q_{T}} rF_{δ}(u^{h})α ( \frac{t − \frac{T}{2}}{\frac{T}{2} − δ} ) drdt − δ ∫_{0}^{1} ru_{0}(r)dr \\ + & ∫∫_{P} rm(u) [k \frac{∂V}{∂r} − A′(u)\frac{∂u}{∂r}] \frac{∂φ_{δ}^{h}}{∂r} drdt = 0. \end{aligned}  (4.10)

By the uniform continuity of u(r, t) in \overline{Q}_{T}, there exist η_{2}(δ) > 0, such that

u(r, t) ≥ −\frac{δ}{2} ∀ (r, t) ∈ P^{δ},  (4.11)

where P^{δ} = {(r, t); dist((r, t), P) < η_{2}(δ)}. Here we have used the fact that u(r, t) > 0 in P. Thus

H_{δ}(u^{h}(r, t)) = −δ, ∀ (r, t) ∈ P^{δ/2}, 0 < h < \frac{1}{2}η_{2}(δ)

where P^{δ/2} = \{(r, t); dist((r, t), P) < \frac{1}{2}η_{2}(δ)\}.

This and the definition of u^{h}, H_{δ}(s) show that the function φ_{δ}^{h}(r, t) is only a function of t in P, whenever h < \frac{1}{2}η_{2}(δ). Therefore

Dφ_{δ}^{h}(r, t) = 0, ∀ (r, t) ∈ P, 0 < h < \frac{1}{2}η_{2}(δ)  (4.12)

and so (4.10) becomes

−δ ∫_{0}^{1} ru_{0}(r)dr − \frac{2}{T − 2δ} ∫∫_{Q_{T}} rF_{δ}(u^{h})α (\frac{2t − T}{T − 2δ} ) drdt = 0,  (4.13)

where η(δ) = min(η_{1}(δ), η_{2}(δ)). Letting h tend to zero, we have

−δ ∫_{0}^{1} ru_{0}(r)dr − \frac{2}{T − 2δ} ∫∫_{Q_{T}} rF_{δ}(u)α (\frac{2t − T}{T − 2δ} ) drdt = 0.  (4.14)

From the definition of F_{δ}(s), H_{δ}(s), it is easily seen that

F_{δ}(u(r, t)) → −χ_{E}(r, t)u(r, t) (δ → 0)

and so by letting δ tend to zero in (4.14), we have

∫∫_{E} |u(r, t)|α (\frac{2t − T}{T} ) drdt = 0,

which contradicts the fact that α (\frac{2t − T}{T} ) > 0 for 0 < t < T. We have thus

proved the theorem.

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