Lobachevskii Journal of Mathematics

http://ljm.ksu.ru
ISSN 1818-9962

Vol. 26, 2007, 79–90 \begin{matrix}c \\ ◯ \end{matrix} Z. Q. Ling and Z. J. Wang

Z. Q. Ling and Z. J. Wang

UNIQUENESS OF SOLUTIONS TO A CLASS OF STRONGLY DEGENERATE PARABOLIC EQUATION

(submitted by M. A. Malakhaltsev)

Abstract. In this paper, by virtue of Holmgren’s approach, we show the uniqueness of the bounded solutions to a class of parabolic equation with two kinds degeneracies at the same time under some necessary conditions on the growth of the convection and sources.

1. Introduction

This paper concerns the uniqueness of the bounded solutions to the initial-boundary value problem of the strongly degenerate parabolic equation

\frac{∂σ(u)}{∂t} = \frac{∂^{2}A(u)}{∂x^{2}} + \frac{∂B(u)}{∂x} + f(u, x, t),  (x, t) ∈ Q_{T} ,  (1.1)
A(u(0, t)) = g_{1}(t), A(u(1, t)) = g_{2}(t),  t ∈ (0, T),  (1.2)
σ(u(x, 0)) = u_{0}(x),  x ∈ (0, 1),  (1.3)

2000 Mathematical Subject Classification. 35K65; 35M10.

Key words and phrases. Holmgren’s approach; Strongly degeneracy.

Supported by the Tianyuan Fund for Mathematics of China (No. 10626024).

where Q_{T} = (0, 1) × (0, T),

σ(u) = ∫_{0}^{u} c(s)ds, A(u) = ∫_{0}^{u} a(s)ds, B(u) = ∫_{0}^{u} b(s)ds,

and c(s) ≥ 0, a(s) ≥ 0, b(s), f(s, x, t), g_{i}(t)(i = 1, 2), u_{0}(x) are suitably smooth functions.

The equation (1.1) can be used to describe a variety of diffusion phenomena appeared widely in nature(see [1]). It is degenerate at the sets E = {s ∈ ℝ; c(s) = 0} or F = {s ∈ ℝ; a(s) = 0}. Generally speaking, the equation (1.1) is a classical parabolic-hyperbolic equation if c(s) ≠ 0 and when a(s) ≠ 0 an elliptical-parabolic equation, whose degeneracy appears respectively in the sets {s; a(s) = 0} and {s; c(s) = 0}. As we know, the equation (1.1) with only one kind of degeneracy, especially for the case A(s) = s and σ(s) = s respectively was studied in many papers, see [1–5] and the references therein. In this paper, we investigate the equation (1.1) with two degeneracies at the same time, the sets E = {s ∈ ℝ; c(s) = 0} and F = {s ∈ ℝ; a(s) = 0} are allowed to have a infinite points.

In [6], the authors considered the equation (1.1) with f(u, x, t) = 0, they proved the uniqueness of the bounded solutions under the assumption that |b(s)|^{2} ≤ h(s)c(s) with h(s) is a given continuous function.

The purpose of the present paper is to generalize the result obtained by the authors in [6] to a more general case, i.e., we establish the uniqueness of the bounded solutions of the problem (1.1)–(1.3) under the assumptions that |b(s)|^{p_{1}} ≤ h_{1}(s)c(s), |f′_{s}(s, x, t)|^{p_{2}} ≤ h_{2}(s)c(s), where 1 < p_{1} ≤ 2, 1 < p_{2} ≤ 2 and h_{i}(s)(i = 1, 2) are given continuous function. It is easy to see that our result also improves the condition in [6].

The main theorem on the uniqueness of the bounded solutions and its proof will be given in the next section.

2. Main Theorem and Its Proof

The bounded solution of the problem (1.1)–(1.3) is defined as follows.

Definition A function u ∈ L^{∞}(Q_{T}) is called a bounded solution of the problem (1.1)–(1.3), if the following integral equality holds

\begin{matrix}∫∫_{Q_{T}} (σ(u)\frac{∂φ}{∂t} + A(u)\frac{∂^{2}φ}{∂x^{2}} − B(u)\frac{∂φ}{∂x} + f(u, x, t)φ)dx dt \\ + ∫_{0}^{T} g_{1}(t)\frac{∂φ}{∂x}(0, t)dt − ∫_{0}^{T} g_{2}(t)\frac{∂φ}{∂x}(1, t)dt + ∫_{0}^{1} u_{0}(x)φ(x, 0)dx = 0 \end{matrix}

for any test function φ ∈ C^{∞}(Q_{T}) with φ(0, t) = φ(1, t) = φ(x, T) = 0.

The following theorem is our main result in this paper.

Theorem 2.1. Assume that the set E = {s; a(s) = 0} has no interior point, there are p_{1}, p_{2} ∈ ℝ : 1 < p_{1} ≤ 2, 1 < p_{2} ≤ 2 and continuous, bounded functions h_{1}(s) ≥ 0, h_{2}(s, x, t) ≥ 0, such that |b(s)|^{p_{1}} ≤ c(s)h_{1}(s), |fs^{′}(s, x, t)|^{p_{2}} ≤ c(s)h_{2}(s, x, t). Then the problem (1.1)–(1.3) admits at most one bounded solution.

The proof of Theorem 2.1 will be completed by the reduction to absurdity. Let u_{1}, u_{2} ∈ L^{∞}(Q_{T}) be two bounded solutions of the problem (1.1)–(1.3). It only needs to show u_{1} = u_{2} a.e. on Q_{T}. We firstly show σ(u_{1}) = σ(u_{2}) a.e. on Q_{T} by the Holmgren’s method which used in [3], [6].

By the definition of bounded solution, we have

∫∫_{Q_{T}} (u_{1} − u_{2})(σ̃ \frac{∂φ}{∂t} + Ã\frac{∂^{2}φ}{∂x^{2}} − B̃ \frac{∂φ}{∂x} + f̃φ)dx dt = 0.

for any function φ ∈ C^{2}(Q_{T}) with φ(0, t) = φ(1, t) = φ(x, T) = 0, where

σ̃ = σ̃(u_{1}, u_{2}) = ∫_{0}^{1} c(θu_{1} + (1 − θ)u_{2})dθ,

Ã = Ã(u_{1}, u_{2}) = ∫_{0}^{1} a(θu_{1} + (1 − θ)u_{2})dθ,

B̃ = B̃(u_{1}, u_{2}) = ∫_{0}^{1} b(θu_{1} + (1 − θ)u_{2})dθ,

f̃ = f̃(u_{1}, u_{2}, x, t) = ∫_{0}^{1} fu^{′}(θu_{1} + (1 − θ)u_{2}, x, t)dθ.

If for any g ∈ C_{0}^{∞}(Q_{T}), the adjoint problem

σ̃ \frac{∂φ}{∂t} + Ã\frac{∂^{2}φ}{∂x^{2}} − B̃ \frac{∂φ}{∂x} + f̃φ = σ̃g,

φ(0, t) = φ(1, t) = 0,

φ(x, T) = 0

admits a solution φ ∈ C^{∞}(\overline{Q}_{T}), then we have

∫∫_{Q_{T}} (σ(u_{1}) − σ(u_{2}))gdx dt = 0.

Then the arbitrariness of g implying that σ(u_{1}) = σ(u_{2}). But we see that the smooth solution of the above problem may not exist since that the coefficients of it are not smooth enough. Thus, we will consider the approximation of the above adjoint problem.

Let σ̃_{ε} > 0 and Ã_{ε} > 0 be a C^{∞} approximation of σ̃ and Ã respectively, such that

\begin{matrix}lim_{ε→0} σ̃_{ε} = σ̃, lim_{ε→0} Ã_{ε} = Ã, a.e. x ∈ Q_{T} , \\ σ̃_{ε} ≤ C, Ã_{ε} ≤ C. \end{matrix}

For sufficiently small η > 0, δ > 0, γ > 0, let

λ_{ηδγB} = \{ \begin{matrix}0, & if (x, t) ∈ G_{δ}, \\ (η + Ã)^{−1/2}(γ + σ̃)^{−1/p_{1}} B̃, & if (x, t) ∈ F_{δ}, \end{matrix}

λ_{ηδγf} = \{ \begin{matrix}0, & if (x, t) ∈ G_{δ}, \\ (η + Ã)^{−1/2}(γ + σ̃)^{−1/p_{2}} f̃, & if (x, t) ∈ F_{δ}, \end{matrix}

where G_{δ} = {(x, t) ∈ Q_{T} , |u_{1}−u_{2}| < δ}, F_{δ} = {(x, t) ∈ Q_{T} , |u_{1}−u_{2}| ≥ δ}. Clearly, the assumptions in Theorem 2.1 imply that

B̃^{p_{1}} = ( ∫_{0}^{1} b(θu_{1} + (1 − θ)u_{2})dθ)^{p_{1}} ≤ Cσ̃,  (2.1)

f̃^{p_{2}} = ( ∫_{0}^{1} fu^{′}(θu_{1} + (1 − θ)u_{2}, x, t)dθ)^{p_{2}} ≤ Cσ̃,  (2.2)

and furthermore,

B̃ ≤ C(σ̃)^{1/p_{1}} ≤ C(γ + σ̃)^{1/p_{1}}, f̃ ≤ C(σ̃)^{1/p_{2}} ≤ C(γ + σ̃)^{1/p_{2}}.

Here and in the sequel, we use C to denote a universal constant independent of δ, η, γ and ε, K(δ) a constant depending only on δ, which may take different values on different occasions.

Since A(s) is strictly increasing and u_{1}, u_{2} ∈ L^{∞}(Q_{T}), there must be constants L(δ) > 0, K(δ) > 0 depending on δ, but independent of η and γ, such that

Ã = \frac{A(u_{1}) − A(u_{2})}{u_{1} − u_{2}} ≥ L(δ), whenever (x, t) ∈ F_{δ},

|λ_{ηδγB}| ≤ K(δ), |λ_{ηδγf}| ≤ K(δ).

Let λ_{ηδγB}^{ε} and λ_{ηδγf}^{ε} be a C^{∞} approximation of λ_{ηδγB} and λ_{ηδγf} respectively, such that

lim_{ε→0} λ_{ηδγB}^{ε} = λ_{ηδγB}, lim_{ε→0} λ_{ηδγf}^{ε} = λ_{ηδγf}, a.e. in Q_{T} ,

|λ_{ηδγB}^{ε}| ≤ K(δ), |λ_{ηδγf}^{ε}| ≤ K(δ).  (2.3)

Denote

\begin{matrix}B̃_{ε} = λ_{ηδγB}^{ε}(η + Ã_{ε})^{1/2}(γ + σ̃_{ε})^{1/p_{1}}, \\ f̃_{ε} = λ_{ηδγf}^{ε}(η + Ã_{ε})^{1/2}(γ + σ̃_{ε})^{1/p_{2}}. \end{matrix}

For any given g ∈ C_{0}^{∞}(Q_{T}), consider the approximate adjoint problem

\frac{∂φ}{∂t} + \frac{η + Ã_{ε}}{γ + σ̃_{ε}} \frac{∂^{2}φ}{∂x^{2}} − \frac{B̃_{ε}}{γ + σ̃_{ε}} \frac{∂φ}{∂x} + \frac{f̃_{ε}}{γ + σ̃_{ε}} φ = g,  (2.4)
φ(0, t) = φ(1, t) = 0,  (2.5)
φ(x, T) = 0.  (2.6)

It is easy to see that the problem (2.4)–(2.6) admits a smooth solution φ by the classical theory of parabolic equations. We give some useful estimates on φ as follows.

Lemma 2.1. The solution φ of the problem (2.4)–(2.6) satisfies

sup_{Q_{T}} |φ(x, t)| ≤ C,  (2.7)
∫∫_{Q_{T}} \frac{η + Ã_{ε}}{γ + σ̃_{ε}} (\frac{∂^{2}φ}{∂x^{2}})^{2}dx dt ≤ K(δ)η^{−1} + K(δ),  (2.8)
∫∫_{Q_{T}} (\frac{∂φ}{∂x})^{2}dx dt ≤ K(δ)η^{−1} + K(δ).  (2.9)

Proof. The inequality (2.7) follows from the maximum principle. To

prove (2.8) and (2.9), multiply (2.4) by \frac{∂^{2}φ}{∂x^{2}} and integrate over Q_{T}. Integrating by parts and using (2.5), (2.6), we have

\begin{matrix}\frac{1}{2} ∫_{0}^{1} (\frac{φ(x, 0)}{∂x})^{2}dx dt + ∫∫_{Q_{T}} \frac{η + Ã_{ε}}{γ + σ̃_{ε}} (\frac{∂^{2}φ}{∂x^{2}})^{2}dx dt \\ − ∫∫_{Q_{T}} \frac{B̃_{ε}}{γ + σ̃_{ε}} \frac{∂φ}{∂x} \frac{∂^{2}φ}{∂x^{2}}dx dt + ∫∫_{Q_{T}} \frac{f̃_{ε}}{γ + σ̃_{ε}} φ\frac{∂^{2}φ}{∂x^{2}}φdx dt \end{matrix}

= ∫∫_{Q_{T}} g \frac{∂^{2}φ}{∂x^{2}}dx dt.  (2.10)

Using Young’s inequality and (2.3), we obtain

\begin{aligned}& ∫∫_{Q_{T}} \frac{B̃_{ε}}{γ + σ̃_{ε}} \frac{∂φ}{∂x} \frac{∂^{2}φ}{∂x^{2}}dx dt \\ = & ∫∫_{Q_{T}} \frac{(η + Ã_{ε})^{1/2}}{(γ + σ̃_{ε})^{1/2}} \frac{∂^{2}φ}{∂x^{2}}λ_{ηδγB}^{ε}(γ + σ̃_{ε})^{1/p_{1}−1/2} \frac{∂φ}{∂x}dx dt \end{aligned}

≤ \frac{1}{4} ∫∫_{Q_{T}} \frac{η + Ã_{ε}}{γ + σ̃_{ε}} (\frac{∂^{2}φ}{∂x^{2}})^{2}dx dt + K(δ) ∫∫_{Q_{T}} (\frac{∂φ}{∂x})^{2}dx dt,  (2.11)

∫∫_{Q_{T}} (\frac{∂φ}{∂x})^{2}dx dt = − ∫∫_{Q_{T}} φ\frac{∂^{2}φ}{∂x^{2}}dx dt

≤ β ∫∫_{Q_{T}} \frac{η + Ã_{ε}}{γ + σ̃_{ε}} (\frac{∂^{2}φ}{∂x^{2}})^{2}dx dt + Cβ^{−1}η^{−1}(β > 0),  (2.12)
∫∫_{Q_{T}} g \frac{∂^{2}φ}{∂x^{2}}dx dt ≤ \frac{1}{8} ∫∫_{Q_{T}} \frac{η + Ã_{ε}}{γ + σ̃_{ε}} (\frac{∂^{2}φ}{∂x^{2}})^{2}dx dt + 8Cη^{−1},  (2.13)

\begin{aligned}& ∫∫_{Q_{T}} \frac{f̃_{ε}}{γ + σ̃_{ε}} φ\frac{∂^{2}φ}{∂x^{2}}dx dt \\ = & ∫∫_{Q_{T}} \frac{(η + Ã_{ε})^{1/2}}{(γ + σ̃_{ε})^{1/2}} \frac{∂^{2}φ}{∂x^{2}}λ_{ηδγf}^{ε}(γ + σ̃_{ε})^{1/p_{2}−1/2}φdx dt \end{aligned}

≤ \frac{1}{8} ∫∫_{Q_{T}} \frac{η + Ã_{ε}}{γ + σ̃_{ε}} (\frac{∂^{2}φ}{∂x^{2}})^{2}dx dt + K(δ).  (2.14)

Let β = 1/(4K(δ)) in (2.12). Then from (2.10)-(2.14) we obtain (2.8) immediately. The inequality (2.9) follows from (2.8) and (2.12)(β = 1). The proof is complete.

Lemma 2.2. The solution φ of the problem (2.4)–(2.6) satisfies

sup_{0<t<T} ∫_{0}^{1} |\frac{∂φ(x, t)}{∂x}|dx ≤ C, ∀t ∈ (0, T).

Proof. For small β > 0, let

sgn_{β}(s) = \{ \begin{matrix}1, & s ≥ β, \\ s/β, & |s| < β, \\ − 1, & s ≤ −β, \end{matrix} I_{β}(s) = ∫_{0}^{s} sgn_{β}(θ)dθ.

Differentiate (2.4) with respect to x, multiply the resulting equality by

sgn_{β}(\frac{∂φ}{∂x}) and integrate over S_{t} = (0, 1) × (t, T). Then integrating by parts and by (2.6) we have

∫_{0}^{1} I_{β}(\frac{∂φ(x, t)}{∂x})dx = − ∫∫_{S_{t}} \frac{η + Ã_{ε}}{γ + σ̃_{ε}} (\frac{∂^{2}φ}{∂x^{2}})^{2}sgnβ^{′}(\frac{∂φ}{∂x})dx dt

+∫∫_{S_{t}} \frac{B̃_{ε}}{γ + σ̃_{ε}} \frac{∂φ}{∂x} \frac{∂^{2}φ}{∂x^{2}}sgnβ^{′}(\frac{∂φ}{∂x})dx dt−∫∫_{S_{t}} \frac{f̃_{ε}}{γ + σ̃_{ε}} φ\frac{∂^{2}φ}{∂x^{2}}sgnβ^{′}(\frac{∂φ}{∂x})dx dt

+ ∫_{t}^{T} (\frac{η + Ã_{ε}}{γ + σ̃_{ε}} \frac{∂^{2}φ}{∂x^{2}} − \frac{B̃_{ε}}{γ + σ̃_{ε}} \frac{∂φ}{∂x} + \frac{f̃_{ε}}{γ + σ̃_{ε}} φ)sgn_{β}(\frac{∂φ}{∂x})|_{x=0}^{x=1}dt

− ∫∫_{S_{t}} \frac{∂g}{∂x}sgn_{β}(\frac{∂φ}{∂x})dx dt. (2.15)

Notice that the first term on the right side is non-positive, the second and third term tends to 0 while β → 0 respectively, the last term is bounded. By the boundary condition (2.5) and g ∈ C_{0}^{∞}(Q_{T}), we see that

\begin{aligned}& ∫_{t}^{T} (\frac{η + Ã_{ε}}{γ + σ̃_{ε}} \frac{∂^{2}φ}{∂x^{2}} − \frac{B̃_{ε}}{γ + σ̃_{ε}} \frac{∂φ}{∂x} + \frac{f̃_{ε}}{γ + σ̃_{ε}} φ)sgn_{β}(\frac{∂φ}{∂x})|_{x=0}^{x=1}dt \\ = & ∫_{t}^{T} (g − \frac{∂φ}{∂t})sgn_{β}(\frac{∂φ}{∂x})|_{x=0}^{x=1}dt = 0. \end{aligned}

Therefore, letting β → 0 in (2.15) gives

sup_{0<t<T} ∫_{0}^{1} |\frac{∂φ(x, t)}{∂x}|dx ≤ C.

Lemma 2.3. The solution φ of the problem (2.4)–(2.6) satisfies

∫∫_{Q_{T}} (\frac{∂φ}{∂t})^{2}dx dt ≤ K(δ)η^{−1}γ^{−1} + K(δ)γ^{−1} + C.  (2.16)

Proof. Multiplying (2.4) by \frac{∂φ}{∂t} and integrating it over Q_{T} yield

∫∫_{Q_{T}} (\frac{∂φ}{∂t})^{2}dx dt + ∫∫_{Q_{T}} \frac{η + Ã_{ε}}{γ + σ̃_{ε}} \frac{∂^{2}φ}{∂x^{2}} \frac{∂φ}{∂t}dx dt − ∫∫_{Q_{T}} \frac{B̃_{ε}}{γ + σ̃_{ε}} \frac{∂φ}{∂x} \frac{∂φ}{∂t}dx dt

+ ∫∫_{Q_{T}} \frac{f̃_{ε}}{γ + σ̃_{ε}} φ\frac{∂φ}{∂t}dx dt = ∫∫_{Q_{T}} g \frac{∂φ}{∂t}dx dt.  (2.17)

Using Young’s inequality, we obtain

\begin{aligned}& ∫∫_{Q_{T}} \frac{η + Ã_{ε}}{γ + σ̃_{ε}} \frac{∂^{2}φ}{∂x^{2}} \frac{∂φ}{∂t}dx dt \\ ≤ & \frac{1}{4} ∫∫_{Q_{T}} (\frac{∂φ}{∂t})^{2}dx dt + \frac{sup(η + Ã_{ε})}{γ} ∫∫_{Q_{T}} \frac{η + Ã_{ε}}{γ + σ̃_{ε}} (\frac{∂^{2}φ}{∂x^{2}})^{2}dx dt \end{aligned}

≤ \frac{1}{4} ∫∫_{Q_{T}} (\frac{∂φ}{∂t})^{2}dx dt + K(δ)η^{−1}γ^{−1} + K(δ)γ^{−1},  (2.18)

∫∫_{Q_{T}} \frac{B̃_{ε}}{γ + σ̃_{ε}} \frac{∂φ}{∂x} \frac{∂φ}{∂t}dx dt

≤\frac{sup(η + Ã_{ε}) sup(γ + σ̃_{ε})^{2/p_{1}−1}}{γ} ∫∫_{Q_{T}} (λ_{ηδγB}^{ε})^{2}(\frac{∂φ}{∂x})^{2}dx dt

+ \frac{1}{4} ∫∫_{Q_{T}} (\frac{∂φ}{∂t})^{2}dx dt

≤ \frac{1}{4} ∫∫_{Q_{T}} (\frac{∂φ}{∂t})^{2}dx dt + K(δ)η^{−1}γ^{−1} + K(δ)γ^{−1},  (2.19)

∫∫_{Q_{T}} \frac{f̃_{ε}}{γ + σ̃_{ε}} φ\frac{∂φ}{∂t}dx dt

≤\frac{sup(η + Ã_{ε}) sup(γ + σ̃_{ε})^{2/p_{2}−1}}{γ} ∫∫_{Q_{T}} (λ_{ηδγf}^{ε})^{2}(\frac{∂φ}{∂x})^{2}dx dt

+ \frac{1}{4} ∫∫_{Q_{T}} (\frac{∂φ}{∂t})^{2}dx dt

≤ \frac{1}{4} ∫∫_{Q_{T}} (\frac{∂φ}{∂t})^{2}dx dt + K(δ)γ^{−1},  (2.20)

| ∫∫_{Q_{T}} g \frac{∂φ}{∂t}dx dt| = | ∫∫_{Q_{T}} φ\frac{∂g}{∂t} dx dt| ≤ C.  (2.21)

By (2.17)–(2.21) we obtain (2.16). The proof is complete.

Lemma 2.4. Under the assumptions in Theorem 2.1, if u_{1} and u_{2} are bounded solutions of the problem (1.1)–(1.3), then σ(u_{1}) = σ(u_{2}) a.e. on Q_{T}.

Proof. For any given g ∈ C_{0}^{∞}(Q_{T}), let φ be a solution of (2.4)–(2.6), then

∫∫_{Q_{T}} (σ(u_{1}) − σ(u_{2}))gdx dt = ∫∫_{Q_{T}} (u_{1} − u_{2})σ̃gdx dt

= ∫∫_{Q_{T}} (u_{1} − u_{2})(σ̃ − σ̃_{ε})gdx dt + ∫∫_{Q_{T}} (u_{1} − u_{2})(γ + σ̃_{ε})gdxdt

− ∫∫_{Q_{T}} (u_{1} − u_{2})γgdx dt = I_{1} + I_{2} + I_{3}. (2.22)

It is not difficult to see that

|I_{1}| = | ∫∫_{Q_{T}} (u_{1} − u_{2})(σ̃ − σ̃_{ε})gdx dt| ≤ C( ∫∫_{Q_{T}} (σ̃ − σ̃_{ε})^{2}dx dt)^{1/2},

(2.23)

|I_{3}| = | ∫∫_{Q_{T}} (u_{1} − u_{2})γgdx dt| ≤ Cγ.  (2.24)

Hence, lim_{ε→0}I_{1} = 0, lim_{γ→0}I_{3} = 0. As indicated above, from the definition of bounded solution, we have

∫∫_{Q_{T}} (u_{1} − u_{2})(σ̃ \frac{∂φ}{∂t} + Ã\frac{∂^{2}φ}{∂x^{2}} − B̃ \frac{∂φ}{∂x} + f̃φ)dx dt = 0.

UNIQUENESS OF SOLUTIONS 87

Combining with the equation (2.4) yields I_{2} = ∫∫_{Q_{T}} (u_{1} − u_{2})(γ + σ̃_{ε})gdx dt

= ∫∫_{Q_{T}} (u_{1} − u_{2})((γ + σ̃_{ε})\frac{∂φ}{∂t} + (Ã_{ε} + η)\frac{∂^{2}φ}{∂x^{2}} − B̃_{ε} \frac{∂φ}{∂x} + f̃_{ε}φ)dx dt

= ∫∫_{Q_{T}} (u_{1} − u_{2})γ \frac{∂φ}{∂t}dx dt + ∫∫_{Q_{T}} (u_{1} − u_{2})(σ̃_{ε} − σ̃)\frac{∂φ}{∂t}dx dt

+ ∫∫_{Q_{T}} (u_{1} − u_{2})(Ã_{ε} − Ã)\frac{∂^{2}φ}{∂x^{2}}dx dt + ∫∫_{Q_{T}} (u_{1} − u_{2})η \frac{∂^{2}φ}{∂x^{2}}dx dt

− ∫∫_{Q_{T}} (u_{1} − u_{2})(B̃_{ε} − B̃)\frac{∂φ}{∂x}dx dt + ∫∫_{Q_{T}} (u_{1} − u_{2})(f̃_{ε} − f̃)φdx dt

= I_{21} + I_{22} + I_{23} + I_{24} + I_{25} + I_{26}.  (2.5)

In the following, we estimate all terms on the right of the above inequality.

|I_{21}| ≤Cγ( ∫∫_{Q_{T}} (\frac{∂φ}{∂t})^{2}dx dt)^{1/2} ≤ Cγ(K(δ)η^{−1}γ^{−1} + K(δ)γ^{−1} + C)^{1/2},
|I_{22}| ≤C( ∫∫_{Q_{T}} (σ̃_{ε} − σ̃)^{2}dx dt)^{1/2}( ∫∫_{Q_{T}} (\frac{∂φ}{∂t})^{2}dx dt)^{1/2}
≤C(K(δ)η^{−1}γ^{−1} + K(δ)γ^{−1} + C)^{1/2}( ∫∫_{Q_{T}} (σ̃_{ε} − σ̃)^{2}dx dt)^{1/2},
|I_{23}| ≤C( ∫∫_{Q_{T}} (Ã_{ε} − Ã)^{2}dx dt)^{1/2}( ∫∫_{Q_{T}} (\frac{∂^{2}φ}{∂x^{2}})^{2}dx dt)^{1/2}
≤Cη^{−1}(K(δ) + K(δ)η)^{1/2}( ∫∫_{Q_{T}} (Ã_{ε} − Ã)^{2}dx dt)^{1/2}.

For any fixed β > 0, F_{β}, G_{δ} are defined just as F_{δ}, G_{δ}. By the Cauchy inequality, we have

|I_{24}| ≤| ∫∫_{F_{β}}(u_{1} − u_{2})η \frac{∂^{2}φ}{∂x^{2}}dx dt| + | ∫∫_{G_{β}}(u_{1} − u_{2})η \frac{∂^{2}φ}{∂x^{2}}dx dt|
≤Cηsup_{F_{β}} \frac{|γ + σ̃_{ε}|^{1/2}}{Ã_{ε}^{1/2}} (∫∫_{F_{β}} \frac{η + Ã_{ε}}{γ + σ̃_{ε}} (\frac{∂^{2}φ}{∂x^{2}})^{2}dx dt)^{1/2}

+ Cηβ( ∫∫_{Q_{T}} (\frac{∂^{2}φ}{∂x^{2}})^{2}dx dt)^{1/2}

≤Cηsup_{F_{β}}|γ + σ̃_{ε}|^{1/2} \frac{(K(δ)η^{−1} + K(δ))^{1/2}}{L(β)^{1/2}} + Cβ(K(δ) + K(δ)η)^{1/2}.

Let β = δK(δ)^{−1/2} and η → 0, δ → 0. We see that I_{24} tends to 0.

|I_{25}| ≤| ∫∫_{G_{δ}}(u_{1} − u_{2})B̃_{ε} \frac{∂φ}{∂x}dx dt| + | ∫∫_{G_{δ}}(u_{1} − u_{2})B̃ \frac{∂φ}{∂x}dx dt|

+ | ∫∫_{F_{δ}}(u_{1} − u_{2})(B̃_{ε} − B̃)\frac{∂φ}{∂x}dx dt|

≤C( ∫∫_{G_{δ}}(λ_{ηδγB}^{ε})^{2}dx dt)^{1/2}( ∫∫_{Q_{T}} (\frac{∂φ}{∂x})^{2}dx dt)^{1/2} + Cδ

+ C( ∫∫_{F_{δ}}(B̃_{ε} − B̃)^{2}dx dt)^{1/2}( ∫∫_{Q_{T}} (\frac{∂φ}{∂x})^{2}dx dt)^{1/2}

≤C(K(δ)η^{−1} + K(δ))^{1/2} (∫∫_{G_{δ}}(λ_{ηδγB}^{ε})^{2}dx dt)^{1/2} + Cδ

+C(K(δ)η^{−1} + K(δ))^{1/2} ∫∫_{F_{δ}}(B̃_{ε} − B̃)^{2}dx dt)^{1/2} .

Since lim_{ε→0}λ_{ηδγB}^{ε} = 0 a.e. on G_{δ}, lim_{ε→0}B̃_{ε} = B̃ a.e. on F_{δ}. Then \overline{lim_{ε→0}}|I_{25}| ≤ Cδ. Similar to the analysis on I_{25}, it yields \overline{lim_{ε→0}}|I_{26}| ≤ Cδ.

From the above inequalities, let ε → 0, γ → 0, η → 0, δ → 0 in turn in (2.25). Then I_{2} → 0. Thus combining with (2.23), (2.24), it is seen from (2.22) that

∫∫_{Q_{T}} (σ(u_{1}) − σ(u_{2}))gdx dt = 0.

Because of the arbitrariness of g, we obtain σ(u_{1}) = σ(u_{2}) a.e. on Q_{T}. The proof is complete.

Proof of Theorem 2.1. By the definition of bounded solutions, we have

∫∫_{Q_{T}} (u_{1} − u_{2})(σ̃ \frac{∂φ}{∂t} + Ã\frac{∂^{2}φ}{∂x^{2}} − B̃ \frac{∂φ}{∂x} + f̃φ)dx dt = 0.

According to Lemma 2.4, it yields

∫∫_{Q_{T}} (u_{1} − u_{2})Ã\frac{∂^{2}φ}{∂x^{2}}dx dt

= ∫∫_{Q_{T}} (u_{1} − u_{2})B̃ \frac{∂φ}{∂x}dx dt − ∫∫_{Q_{T}} (u_{1} − u_{2})f̃φdx dt

= I_{1} − I_{2}.

Using Young’s inequality and Hölder inequality, we have

|I_{1}| ≤ ∫∫_{Q_{T}} |(u_{1} − u_{2})^{1/p_{1}} B̃(u_{1} − u_{2})^{1/q_{1}} \frac{∂φ}{∂x}|dx dt

≤ ( ∫∫_{Q_{T}} |(u_{1} − u_{2})B̃^{p_{1}}|dx dt)^{1/p_{1}}( ∫∫_{Q_{T}} |(u_{1} − u_{2})(\frac{∂φ}{∂x})^{q_{1}}|dx dt)^{1/q_{1}},

|I_{2}| ≤ ( ∫∫_{Q_{T}} |(u_{1} − u_{2})f̃^{p_{2}}|dx dt)^{1/p_{2}}( ∫∫_{Q_{T}} |(u_{1} − u_{2})φ^{q_{2}}|dx dt)^{1/q_{2}},

where q_{1}, q_{2} satisfy 1/p_{1} + 1/q_{1} = 1 and 1/p_{2} + 1/q_{2} = 1. From (2.1) and (2.2), we obtain

|I_{1}| ≤ C( ∫∫_{Q_{T}} |(u_{1} − u_{2})σ̃|dx dt)^{1/p_{1}}

× ( ∫∫_{Q_{T}} |(u_{1} − u_{2})(\frac{∂φ}{∂x})^{q_{1}}|dx dt)^{1/q_{1}} = 0,

|I_{2}| ≤ C( ∫∫_{Q_{T}} |(u_{1} − u_{2})σ̃|dx dt)^{1/p_{2}}( ∫∫_{Q_{T}} |(u_{1} − u_{2})φ^{2}|dx dt)^{1/q_{2}} = 0.

Obviously, we obtain

∫∫_{Q_{T}} (A(u_{1}) − A(u_{2}))\frac{∂^{2}φ}{∂x^{2}}dx dt = ∫∫_{Q_{T}} (u_{1} − u_{2})Ã\frac{∂^{2}φ}{∂x^{2}}dx dt = 0,

Since the function φ is arbitrary, we have A(u_{1}) = A(u_{2}) a.e. on Q_{T}. Furthermore, owing to that A(s) is a strictly increasing function, it has been shown that u_{1} = u_{2} a.e. on Q_{T}. The proof is complete.

Acknowledgments. The authors would like to express their supervisor professor Jingxue Yin in Jilin University for his kindly guidance.

References

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90 Z. LING AND Z. WANG

Department of Mathematics and Computers Science, Yulin Teachers’ College, Yulin 537000, P.R. China E-mail address: matwzj@jlu.edu.cn

Received January 29, 2007