Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 26, 2007, 79–90

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.

________________

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).

1. Introduction

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

\frac{\partial σ (u)}{\partial t} = \frac{\partial^{2} A (u)}{\partial x^{2}} + \frac{\partial B (u)}{\partial x} + f (u, x, t), (x, t) \in Q_{T}, (1.1)

A (u (0, t)) = g_{1} (t), A (u (1, t)) = g_{2} (t), t \in (0, T), (1.2)

σ (u (x, 0)) = u_{0} (x), x \in (0, 1), (1.3)

where $Q_{T} = (0, 1) \times (0, T)$ ,

σ (u) = \int_{0}^{u} c (s) d s, A (u) = \int_{0}^{u} a (s) d s, B (u) = \int_{0}^{u} b (s) d s,

and $c (s) \geq 0, a (s) \geq 0, b (s), f (s, x, t), g_{i} (t) (i = 1, 2), u_{0} (x)$ 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 $E = {s \in ℝ; c (s) = 0}$ or $F = {s \in ℝ; a (s) = 0}$ . Generally speaking, the equation (1.1) is a classical parabolic-hyperbolic equation if $c (s) \neq 0$ and when $a (s) \neq 0$ 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 $E = {s \in ℝ; c (s) = 0}$ and $F = {s \in ℝ; a (s) = 0}$ 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 $∣ b (s) ∣^{2} \leq h (s) c (s)$ 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 $∣ b (s) ∣^{p_{1}} \leq h_{1} (s) c (s)$ , $∣ f_{s}^{'} (s, x, t) ∣^{p_{2}} \leq h_{2} (s) c (s),$ where $1 < p_{1} \leq 2$ , $1 < p_{2} \leq 2$ 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 $u \in L^{\infty} (Q_{T})$ is called a bounded solution of the problem (1.1)–(1.3), if the following integral equality holds

\iint_{Q_{T}} (σ (u) \frac{\partial ϕ}{\partial t} + A (u) \frac{\partial^{2} ϕ}{\partial x^{2}} - B (u) \frac{\partial ϕ}{\partial x} + f (u, x, t) ϕ) d x d t + \int_{0}^{T} g_{1} (t) \frac{\partial ϕ}{\partial x} (0, t) d t - \int_{0}^{T} g_{2} (t) \frac{\partial ϕ}{\partial x} (1, t) d t + \int_{0}^{1} u_{0} (x) ϕ (x, 0) d x = 0

for any test function $ϕ \in C^{\infty} (Q_{T})$ 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 $p_{1}, p_{2} \in ℝ : 1 < p_{1} \leq 2, 1 < p_{2} \leq 2$ and continuous, bounded functions $h_{1} (s) \geq 0, h_{2} (s, x, t) \geq 0$ , such that $∣ b (s) ∣^{p_{1}} \leq c (s) h_{1} (s), ∣ f_{s}^{^{'}} (s, x, t) ∣^{p_{2}} \leq c (s) h_{2} (s, x, t)$ . 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 $u_{1}, u_{2} \in L^{\infty} (Q_{T})$ 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

\iint_{Q_{T}} (u_{1} - u_{2}) (\tilde{σ} \frac{\partial ϕ}{\partial t} + Ã \frac{\partial^{2} ϕ}{\partial x^{2}} - \tilde{B} \frac{\partial ϕ}{\partial x} + \tilde{f} ϕ) d x d t = 0 .

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

\tilde{σ} = \tilde{σ} (u_{1}, u_{2}) = \int_{0}^{1} c (θ u_{1} + (1 - θ) u_{2}) d θ, Ã = Ã (u_{1}, u_{2}) = \int_{0}^{1} a (θ u_{1} + (1 - θ) u_{2}) d θ, \tilde{B} = \tilde{B} (u_{1}, u_{2}) = \int_{0}^{1} b (θ u_{1} + (1 - θ) u_{2}) d θ, \tilde{f} = \tilde{f} (u_{1}, u_{2}, x, t) = \int_{0}^{1} f_{u}^{^{'}} (θ u_{1} + (1 - θ) u_{2}, x, t) d θ .

If for any $g \in C_{0}^{\infty} (Q_{T})$ , the adjoint problem

\tilde{σ} \frac{\partial ϕ}{\partial t} + Ã \frac{\partial^{2} ϕ}{\partial x^{2}} - \tilde{B} \frac{\partial ϕ}{\partial x} + \tilde{f} ϕ = \tilde{σ} g, ϕ (0, t) = ϕ (1, t) = 0, ϕ (x, T) = 0

admits a solution $ϕ \in C^{\infty} ({\bar{Q}}_{T})$ , then we have

\iint_{Q_{T}} (σ (u_{1}) - σ (u_{2})) g d x d t = 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 ${\tilde{σ}}_{ɛ} > 0$ and $Ã_{ɛ} > 0$ be a $C^{\infty}$ approximation of $\tilde{σ}$ and $Ã$ respectively, such that

{lim}_{ɛ \to 0} {\tilde{σ}}_{ɛ} = \tilde{σ}, {lim}_{ɛ \to 0} Ã_{ɛ} = Ã, a . e . x \in Q_{T}, {\tilde{σ}}_{ɛ} \leq C, Ã_{ɛ} \leq C .

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

λ_{η δ γ B} = \{0, i f (x, t) \in G_{δ}, {(η + Ã)}^{- 1 ∕ 2} {(γ + \tilde{σ})}^{- 1 ∕ p_{1}} \tilde{B}, i f (x, t) \in F_{δ}, λ_{η δ γ f} = \{0, i f (x, t) \in G_{δ}, {(η + Ã)}^{- 1 ∕ 2} {(γ + \tilde{σ})}^{- 1 ∕ p_{2}} \tilde{f}, i f (x, t) \in F_{δ},

where $G_{δ} = {(x, t) \in Q_{T}, ∣ u_{1} - u_{2} ∣ < δ}, F_{δ} = {(x, t) \in Q_{T}, ∣ u_{1} - u_{2} ∣ \geq δ}$ . Clearly, the assumptions in Theorem 2.1 imply that

{\tilde{B}}^{p_{1}} = (\int_{0}^{1} b (θ u_{1} + (1 - θ) u_{2}) d θ)^{p_{1}} \leq C \tilde{σ}, (2.1)

{\tilde{f}}^{p_{2}} = (\int_{0}^{1} f_{u}^{^{'}} (θ u_{1} + (1 - θ) u_{2}, x, t) d θ)^{p_{2}} \leq C \tilde{σ}, (2.2)

and furthermore,

\tilde{B} \leq C {(\tilde{σ})}^{1 ∕ p_{1}} \leq C {(γ + \tilde{σ})}^{1 ∕ p_{1}}, \tilde{f} \leq C {(\tilde{σ})}^{1 ∕ p_{2}} \leq C {(γ + \tilde{σ})}^{1 ∕ p_{2}} .

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

Since $A (s)$ is strictly increasing and $u_{1}, u_{2} \in L^{\infty} (Q_{T})$ , 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}} \geq L (δ), w h e n e v e r (x, t) \in F_{δ}, ∣ λ_{η δ γ B} ∣ \leq K (δ), ∣ λ_{η δ γ f} ∣ \leq K (δ) .

Let $λ_{η δ γ B}^{ɛ}$ and $λ_{η δ γ f}^{ɛ}$ be a $C^{\infty}$ approximation of $λ_{η δ γ B}$ and $λ_{η δ γ f}$ respectively, such that

{lim}_{ɛ \to 0} λ_{η δ γ B}^{ɛ} = λ_{η δ γ B}, {lim}_{ɛ \to 0} λ_{η δ γ f}^{ɛ} = λ_{η δ γ f}, a . e . i n Q_{T},

∣ λ_{η δ γ B}^{ɛ} ∣ \leq K (δ), ∣ λ_{η δ γ f}^{ɛ} ∣ \leq K (δ) . (2.3)

Denote

{\tilde{B}}_{ɛ} = λ_{η δ γ B}^{ɛ} {(η + Ã_{ɛ})}^{1 ∕ 2} {(γ + {\tilde{σ}}_{ɛ})}^{1 ∕ p_{1}}, {\tilde{f}}_{ɛ} = λ_{η δ γ f}^{ɛ} {(η + Ã_{ɛ})}^{1 ∕ 2} {(γ + {\tilde{σ}}_{ɛ})}^{1 ∕ p_{2}} .

For any given $g \in C_{0}^{\infty} (Q_{T})$ , consider the approximate adjoint problem

\frac{\partial ϕ}{\partial t} + \frac{η + Ã_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} \frac{\partial^{2} ϕ}{\partial x^{2}} - \frac{{\tilde{B}}_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} \frac{\partial ϕ}{\partial x} + \frac{{\tilde{f}}_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} 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) ∣ \leq C, (2.7)

\iint_{Q_{T}} \frac{η + Ã_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} (\frac{\partial^{2} ϕ}{\partial x^{2}})^{2} d x d t \leq K (δ) η^{- 1} + K (δ), (2.8)

\iint_{Q_{T}} (\frac{\partial ϕ}{\partial x})^{2} d x d t \leq 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 $\frac{\partial^{2} ϕ}{\partial x^{2}}$ and integrate over $Q_{T}$ . Integrating by parts and using (2.5), (2.6), we have

\frac{1}{2} \int_{0}^{1} (\frac{ϕ (x, 0)}{\partial x})^{2} d x d t + \iint_{Q_{T}} \frac{η + Ã_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} (\frac{\partial^{2} ϕ}{\partial x^{2}})^{2} d x d t - \iint_{Q_{T}} \frac{{\tilde{B}}_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} \frac{\partial ϕ}{\partial x} \frac{\partial^{2} ϕ}{\partial x^{2}} d x d t + \iint_{Q_{T}} \frac{{\tilde{f}}_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} ϕ \frac{\partial^{2} ϕ}{\partial x^{2}} ϕ d x d t

= \iint_{Q_{T}} g \frac{\partial^{2} ϕ}{\partial x^{2}} d x d t . (2.10)

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

\iint_{Q_{T}} \frac{{\tilde{B}}_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} \frac{\partial ϕ}{\partial x} \frac{\partial^{2} ϕ}{\partial x^{2}} d x d t = \iint_{Q_{T}} \frac{{(η + Ã_{ɛ})}^{1 ∕ 2}}{{(γ + {\tilde{σ}}_{ɛ})}^{1 ∕ 2}} \frac{\partial^{2} ϕ}{\partial x^{2}} λ_{η δ γ B}^{ɛ} {(γ + {\tilde{σ}}_{ɛ})}^{1 ∕ p_{1} - 1 ∕ 2} \frac{\partial ϕ}{\partial x} d x d t

\leq \frac{1}{4} \iint_{Q_{T}} \frac{η + Ã_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} (\frac{\partial^{2} ϕ}{\partial x^{2}})^{2} d x d t + K (δ) \iint_{Q_{T}} (\frac{\partial ϕ}{\partial x})^{2} d x d t, (2.11)

\iint_{Q_{T}} (\frac{\partial ϕ}{\partial x})^{2} d x d t = - \iint_{Q_{T}} ϕ \frac{\partial^{2} ϕ}{\partial x^{2}} d x d t

\leq β \iint_{Q_{T}} \frac{η + Ã_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} (\frac{\partial^{2} ϕ}{\partial x^{2}})^{2} d x d t + C β^{- 1} η^{- 1} (β > 0), (2.12)

\iint_{Q_{T}} g \frac{\partial^{2} ϕ}{\partial x^{2}} d x d t \leq \frac{1}{8} \iint_{Q_{T}} \frac{η + Ã_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} (\frac{\partial^{2} ϕ}{\partial x^{2}})^{2} d x d t + 8 C η^{- 1}, (2.13)

\iint_{Q_{T}} \frac{{\tilde{f}}_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} ϕ \frac{\partial^{2} ϕ}{\partial x^{2}} d x d t = \iint_{Q_{T}} \frac{{(η + Ã_{ɛ})}^{1 ∕ 2}}{{(γ + {\tilde{σ}}_{ɛ})}^{1 ∕ 2}} \frac{\partial^{2} ϕ}{\partial x^{2}} λ_{η δ γ f}^{ɛ} {(γ + {\tilde{σ}}_{ɛ})}^{1 ∕ p_{2} - 1 ∕ 2} ϕ d x d t

\leq \frac{1}{8} \iint_{Q_{T}} \frac{η + Ã_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} (\frac{\partial^{2} ϕ}{\partial x^{2}})^{2} d x d t + K (δ) . (2.14)

Let $β = 1 ∕ (4 K (δ))$ 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} \int_{0}^{1} | \frac{\partial ϕ (x, t)}{\partial x} | d x \leq C, \forall t \in (0, T) .

Proof. For small $β > 0$ , let

s g n_{β} (s) = \{1, s \geq β, s ∕ β, ∣ s ∣ < β, - 1, s \leq - β, I_{β} (s) = \int_{0}^{s} s g n_{β} (θ) d θ .

Differentiate (2.4) with respect to $x$ , multiply the resulting equality by $s g n_{β} (\frac{\partial ϕ}{\partial x})$ and integrate over $S_{t} = (0, 1) \times (t, T)$ . Then integrating by parts and by (2.6) we have

\begin{matrix} \int_{0}^{1} I_{β} (\frac{\partial ϕ (x, t)}{\partial x}) d x = - \iint_{S_{t}} \frac{η + Ã_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} (\frac{\partial^{2} ϕ}{\partial x^{2}})^{2} s g n_{β}^{^{'}} (\frac{\partial ϕ}{\partial x}) d x d t \\ + \iint_{S_{t}} \frac{{\tilde{B}}_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} \frac{\partial ϕ}{\partial x} \frac{\partial^{2} ϕ}{\partial x^{2}} s g n_{β}^{^{'}} (\frac{\partial ϕ}{\partial x}) d x d t - \iint_{S_{t}} \frac{{\tilde{f}}_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} ϕ \frac{\partial^{2} ϕ}{\partial x^{2}} s g n_{β}^{^{'}} (\frac{\partial ϕ}{\partial x}) d x d t \\ + \int_{t}^{T} (\frac{η + Ã_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} \frac{\partial^{2} ϕ}{\partial x^{2}} - \frac{{\tilde{B}}_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} \frac{\partial ϕ}{\partial x} + \frac{{\tilde{f}}_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} ϕ) s g n_{β} (\frac{\partial ϕ}{\partial x}) |_{x = 0}^{x = 1} d t \\ - \iint_{S_{t}} \frac{\partial g}{\partial x} s g n_{β} (\frac{\partial ϕ}{\partial x}) d x d t . \end{matrix}

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

\int_{t}^{T} (\frac{η + Ã_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} \frac{\partial^{2} ϕ}{\partial x^{2}} - \frac{{\tilde{B}}_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} \frac{\partial ϕ}{\partial x} + \frac{{\tilde{f}}_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} ϕ) s g n_{β} (\frac{\partial ϕ}{\partial x}) |_{x = 0}^{x = 1} d t = \int_{t}^{T} (g - \frac{\partial ϕ}{\partial t}) s g n_{β} (\frac{\partial ϕ}{\partial x}) |_{x = 0}^{x = 1} d t = 0 .

Therefore, letting $β \to 0$ in (2.15) gives

{sup}_{0 < t < T} \int_{0}^{1} | \frac{\partial ϕ (x, t)}{\partial x} | d x \leq C .

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

\iint_{Q_{T}} (\frac{\partial ϕ}{\partial t})^{2} d x d t \leq K (δ) η^{- 1} γ^{- 1} + K (δ) γ^{- 1} + C . (2.16)

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

\iint_{Q_{T}} (\frac{\partial ϕ}{\partial t})^{2} d x d t + \iint_{Q_{T}} \frac{η + Ã_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} \frac{\partial^{2} ϕ}{\partial x^{2}} \frac{\partial ϕ}{\partial t} d x d t - \iint_{Q_{T}} \frac{{\tilde{B}}_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} \frac{\partial ϕ}{\partial x} \frac{\partial ϕ}{\partial t} d x d t

+ \iint_{Q_{T}} \frac{{\tilde{f}}_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} ϕ \frac{\partial ϕ}{\partial t} d x d t = \iint_{Q_{T}} g \frac{\partial ϕ}{\partial t} d x d t . (2.17)

Using Young’s inequality, we obtain

\iint_{Q_{T}} \frac{η + Ã_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} \frac{\partial^{2} ϕ}{\partial x^{2}} \frac{\partial ϕ}{\partial t} d x d t \leq \frac{1}{4} \iint_{Q_{T}} (\frac{\partial ϕ}{\partial t})^{2} d x d t + \frac{sup (η + Ã_{ɛ})}{γ} \iint_{Q_{T}} \frac{η + Ã_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} (\frac{\partial^{2} ϕ}{\partial x^{2}})^{2} d x d t

\leq \frac{1}{4} \iint_{Q_{T}} (\frac{\partial ϕ}{\partial t})^{2} d x d t + K (δ) η^{- 1} γ^{- 1} + K (δ) γ^{- 1}, (2.18)

\iint_{Q_{T}} \frac{{\tilde{B}}_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} \frac{\partial ϕ}{\partial x} \frac{\partial ϕ}{\partial t} d x d t \leq \frac{sup (η + Ã_{ɛ}) sup {(γ + {\tilde{σ}}_{ɛ})}^{2 ∕ p_{1} - 1}}{γ} \iint_{Q_{T}} {(λ_{η δ γ B}^{ɛ})}^{2} (\frac{\partial ϕ}{\partial x})^{2} d x d t + \frac{1}{4} \iint_{Q_{T}} (\frac{\partial ϕ}{\partial t})^{2} d x d t

\leq \frac{1}{4} \iint_{Q_{T}} (\frac{\partial ϕ}{\partial t})^{2} d x d t + K (δ) η^{- 1} γ^{- 1} + K (δ) γ^{- 1}, (2.19)

\iint_{Q_{T}} \frac{{\tilde{f}}_{ɛ}}{γ + {\tilde{σ}}_{ɛ}} ϕ \frac{\partial ϕ}{\partial t} d x d t \leq \frac{sup (η + Ã_{ɛ}) sup {(γ + {\tilde{σ}}_{ɛ})}^{2 ∕ p_{2} - 1}}{γ} \iint_{Q_{T}} {(λ_{η δ γ f}^{ɛ})}^{2} (\frac{\partial ϕ}{\partial x})^{2} d x d t + \frac{1}{4} \iint_{Q_{T}} (\frac{\partial ϕ}{\partial t})^{2} d x d t

\leq \frac{1}{4} \iint_{Q_{T}} (\frac{\partial ϕ}{\partial t})^{2} d x d t + K (δ) γ^{- 1}, (2.20)

| \iint_{Q_{T}} g \frac{\partial ϕ}{\partial t} d x d t | = | \iint_{Q_{T}} ϕ \frac{\partial g}{\partial t} d x d t | \leq 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 $g \in C_{0}^{\infty} (Q_{T})$ , let $ϕ$ be a solution of (2.4)–(2.6), then

\begin{matrix} \iint_{Q_{T}} (σ (u_{1}) - σ (u_{2})) g d x d t = \iint_{Q_{T}} (u_{1} - u_{2}) \tilde{σ} g d x d t \\ = \iint_{Q_{T}} (u_{1} - u_{2}) (\tilde{σ} - {\tilde{σ}}_{ɛ}) g d x d t + \iint_{Q_{T}} (u_{1} - u_{2}) (γ + {\tilde{σ}}_{ɛ}) g d x d t \\ - \iint_{Q_{T}} (u_{1} - u_{2}) γ g d x d t = I_{1} + I_{2} + I_{3} . \end{matrix}

It is not difficult to see that

∣ I_{1} ∣ = | \iint_{Q_{T}} (u_{1} - u_{2}) (\tilde{σ} - {\tilde{σ}}_{ɛ}) g d x d t | \leq C (\iint_{Q_{T}} {(\tilde{σ} - {\tilde{σ}}_{ɛ})}^{2} d x d t)^{1 ∕ 2}, (2.23)

∣ I_{3} ∣ = | \iint_{Q_{T}} (u_{1} - u_{2}) γ g d x d t | \leq C γ . (2.24)

Hence, $lim_{ɛ \to 0} I_{1} = 0, lim_{γ \to 0} I_{3} = 0$ . As indicated above, from the deﬁnition of bounded solution, we have

\iint_{Q_{T}} (u_{1} - u_{2}) (\tilde{σ} \frac{\partial ϕ}{\partial t} + Ã \frac{\partial^{2} ϕ}{\partial x^{2}} - \tilde{B} \frac{\partial ϕ}{\partial x} + \tilde{f} ϕ) d x d t = 0 .

Combining with the equation (2.4) yields

\begin{array}{l} I_{2} = \iint_{Q_{T}} (u_{1} - u_{2}) (γ + {\tilde{σ}}_{ɛ}) g d x d t \\ = \iint_{Q_{T}} (u_{1} - u_{2}) ((γ + {\tilde{σ}}_{ɛ}) \frac{\partial ϕ}{\partial t} + (Ã_{ɛ} + η) \frac{\partial^{2} ϕ}{\partial x^{2}} - {\tilde{B}}_{ɛ} \frac{\partial ϕ}{\partial x} + {\tilde{f}}_{ɛ} ϕ) d x d t \\ = \iint_{Q_{T}} (u_{1} - u_{2}) γ \frac{\partial ϕ}{\partial t} d x d t + \iint_{Q_{T}} (u_{1} - u_{2}) ({\tilde{σ}}_{ɛ} - \tilde{σ}) \frac{\partial ϕ}{\partial t} d x d t \\ + \iint_{Q_{T}} (u_{1} - u_{2}) (Ã_{ɛ} - Ã) \frac{\partial^{2} ϕ}{\partial x^{2}} d x d t + \iint_{Q_{T}} (u_{1} - u_{2}) η \frac{\partial^{2} ϕ}{\partial x^{2}} d x d t \\ - \iint_{Q_{T}} (u_{1} - u_{2}) ({\tilde{B}}_{ɛ} - \tilde{B}) \frac{\partial ϕ}{\partial x} d x d t + \iint_{Q_{T}} (u_{1} - u_{2}) ({\tilde{f}}_{ɛ} - \tilde{f}) ϕ d x d t \\ = I_{21} + I_{22} + I_{23} + I_{24} + I_{25} + I_{26} . & (2.5) \end{array}

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

\begin{array}{l} ∣ I_{21} ∣ \leq & C γ (\iint_{Q_{T}} {(\frac{\partial ϕ}{\partial t})}^{2} d x d t)^{1 ∕ 2} \leq C γ (K (δ) η^{- 1} γ^{- 1} + K (δ) γ^{- 1} + C)^{1 ∕ 2}, \\ ∣ I_{22} ∣ \leq & C (\iint_{Q_{T}} {({\tilde{σ}}_{ɛ} - \tilde{σ})}^{2} d x d t)^{1 ∕ 2} (\iint_{Q_{T}} {(\frac{\partial ϕ}{\partial t})}^{2} d x d t)^{1 ∕ 2} \\ \leq & C (K (δ) η^{- 1} γ^{- 1} + K (δ) γ^{- 1} + C)^{1 ∕ 2} (\iint_{Q_{T}} {({\tilde{σ}}_{ɛ} - \tilde{σ})}^{2} d x d t)^{1 ∕ 2}, \\ ∣ I_{23} ∣ \leq & C (\iint_{Q_{T}} {(Ã_{ɛ} - Ã)}^{2} d x d t)^{1 ∕ 2} (\iint_{Q_{T}} {(\frac{\partial^{2} ϕ}{\partial x^{2}})}^{2} d x d t)^{1 ∕ 2} \\ \leq & C η^{- 1} (K (δ) + K (δ) η)^{1 ∕ 2} (\iint_{Q_{T}} {(Ã_{ɛ} - Ã)}^{2} d x d t)^{1 ∕ 2} . \end{array}

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} ∣ \leq | \iint_{F_{β}} (u_{1} - u_{2}) η \frac{\partial^{2} ϕ}{\partial x^{2}} d x d t | + | \iint_{G_{β}} (u_{1} - u_{2}) η \frac{\partial^{2} ϕ}{\partial x^{2}} d x d t | \leq C η sup_{F_{β}} \frac{∣ γ + {\tilde{σ}}_{ɛ} ∣^{1 ∕ 2}}{{\tilde{A_{ɛ}}}^{1 ∕ 2}} {(\iint_{F_{β}} \frac{η + \tilde{A_{ɛ}}}{γ + {\tilde{σ}}_{ɛ}} (\frac{\partial^{2} ϕ}{\partial x^{2}})^{2} d x d t)}^{1 ∕ 2} + C η β (\iint_{Q_{T}} {(\frac{\partial^{2} ϕ}{\partial x^{2}})}^{2} d x d t)^{1 ∕ 2} \leq C η sup_{F_{β}} ∣ γ + {\tilde{σ}}_{ɛ} ∣^{1 ∕ 2} \frac{(K (δ) η^{- 1} + K (δ))^{1 ∕ 2}}{L {(β)}^{1 ∕ 2}} + C β {(K (δ) + K (δ) η)}^{1 ∕ 2} .

Let $β = δ K {(δ)}^{- 1 ∕ 2}$ and $η \to 0$ , $δ \to 0$ . We see that $I_{24}$ tends to $0$ .

∣ I_{25} ∣ \leq | \iint_{G_{δ}} (u_{1} - u_{2}) {\tilde{B}}_{ɛ} \frac{\partial ϕ}{\partial x} d x d t | + | \iint_{G_{δ}} (u_{1} - u_{2}) \tilde{B} \frac{\partial ϕ}{\partial x} d x d t | + | \iint_{F_{δ}} (u_{1} - u_{2}) ({\tilde{B}}_{ɛ} - \tilde{B}) \frac{\partial ϕ}{\partial x} d x d t | \leq C (\iint_{G_{δ}} {(λ_{η δ γ B}^{ɛ})}^{2} d x d t)^{1 ∕ 2} (\iint_{Q_{T}} {(\frac{\partial ϕ}{\partial x})}^{2} d x d t)^{1 ∕ 2} + C δ + C (\iint_{F_{δ}} {({\tilde{B}}_{ɛ} - \tilde{B})}^{2} d x d t)^{1 ∕ 2} (\iint_{Q_{T}} {(\frac{\partial ϕ}{\partial x})}^{2} d x d t)^{1 ∕ 2} \leq C (K (δ) η^{- 1} + K (δ))^{1 ∕ 2} (\iint_{G_{δ}} {(λ_{η δ γ B}^{ɛ})}^{2} d x d t)^{1 ∕ 2} + C δ {+ C (K (δ) η^{- 1} + K (δ))^{1 ∕ 2} \iint_{F_{δ}} {({\tilde{B}}_{ɛ} - \tilde{B})}^{2} d x d t)}^{1 ∕ 2} .

Since $lim_{ɛ \to 0} λ_{η δ γ B}^{ɛ} = 0$ a.e. on $G_{δ}$ , $lim_{ɛ \to 0} {\tilde{B}}_{ɛ} = \tilde{B}$ a.e. on $F_{δ}$ . Then $\bar{lim_{ɛ \to 0}} ∣ I_{25} ∣ \leq C δ$ . Similar to the analysis on $I_{25}$ , it yields $\bar{lim_{ɛ \to 0}} ∣ I_{26} ∣ \leq C δ$ .

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

\iint_{Q_{T}} (σ (u_{1}) - σ (u_{2})) g d x d t = 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

\iint_{Q_{T}} (u_{1} - u_{2}) (\tilde{σ} \frac{\partial ϕ}{\partial t} + Ã \frac{\partial^{2} ϕ}{\partial x^{2}} - \tilde{B} \frac{\partial ϕ}{\partial x} + \tilde{f} ϕ) d x d t = 0 .

According to Lemma 2.4, it yields

\begin{matrix} \iint_{Q_{T}} (u_{1} - u_{2}) Ã \frac{\partial^{2} ϕ}{\partial x^{2}} d x d t \\ = \iint_{Q_{T}} (u_{1} - u_{2}) \tilde{B} \frac{\partial ϕ}{\partial x} d x d t - \iint_{Q_{T}} (u_{1} - u_{2}) \tilde{f} ϕ d x d t \\ = I_{1} - I_{2} . \end{matrix}

Using Young’s inequality and Hölder inequality, we have

| I_{1} | \leq \iint_{Q_{T}} | {(u_{1} - u_{2})}^{1 ∕ p_{1}} \tilde{B} {(u_{1} - u_{2})}^{1 ∕ q_{1}} \frac{\partial ϕ}{\partial x} | d x d t \leq (\iint_{Q_{T}} ∣ (u_{1} - u_{2}) {\tilde{B}}^{p_{1}} ∣ d x d t)^{1 ∕ p_{1}} (\iint_{Q_{T}} ∣ (u_{1} - u_{2}) {(\frac{\partial ϕ}{\partial x})}^{q_{1}} ∣ d x d t)^{1 ∕ q_{1}}, | I_{2} | \leq (\iint_{Q_{T}} ∣ (u_{1} - u_{2}) {\tilde{f}}^{p_{2}} ∣ d x d t)^{1 ∕ p_{2}} (\iint_{Q_{T}} ∣ (u_{1} - u_{2}) ϕ^{q_{2}} ∣ d x d t)^{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

\begin{matrix} | I_{1} | \leq C (\iint_{Q_{T}} ∣ (u_{1} - u_{2}) \tilde{σ} ∣ d x d t)^{1 ∕ p_{1}} \\ \times (\iint_{Q_{T}} ∣ (u_{1} - u_{2}) {(\frac{\partial ϕ}{\partial x})}^{q_{1}} ∣ d x d t)^{1 ∕ q_{1}} = 0, \end{matrix}

| I_{2} | \leq C (\iint_{Q_{T}} ∣ (u_{1} - u_{2}) \tilde{σ} ∣ d x d t)^{1 ∕ p_{2}} (\iint_{Q_{T}} ∣ (u_{1} - u_{2}) ϕ^{2} ∣ d x d t)^{1 ∕ q_{2}} = 0 .

Obviously, we obtain

\iint_{Q_{T}} (A (u_{1}) - A (u_{2})) \frac{\partial^{2} ϕ}{\partial x^{2}} d x d t = \iint_{Q_{T}} (u_{1} - u_{2}) Ã \frac{\partial^{2} ϕ}{\partial x^{2}} d x d t = 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

[1] Z. Q. Wu, J. N. Zhao, J. X. Yin and H. L. Li, Nonlinear diffusion equations, World Scientiﬁc Publishing Co., Inc., 2001.

[2] B. H. Gilding, A nonlinear degenerate parabolic equation, Annali della Scuola Norm. Sup di Pisa, 4(3)(1977), 393–432.

[3] J. N. Zhao, Uniqueness of solutions of the ﬁrst boundary value problem for quasilinear degenerate parabolic equation, Northeastern Math. J., 1(1)(1985), 153–165.

[4] Z. Q. Wu, Quasilinear degenerate parabolic equations, Adv. Math(China), 16(2)(1987), 121–158.

[5] J. N. Zhao, Continuity of solutions for a class quasilinear degenerate parabolic equations, Northeastern Math. J., 7(3)(1991), 356–365.

[6] Q. Liu, C. P. Wang, Uniqueness of the bounded solution to a strongly degenerate parabolic equations, Nolinear Analysis(2006), doi: 10.1016/j.na.2006.09.053.

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