Electronic Journal of Qualitative Theory of Differential Equations 2006, No. 5, 1-12; http://www.math.u-szeged.hu/ejqtde/

Renormalized Solutions of a Nonlinear Parabolic Equation with Double Degeneracy^{∗}

Zejia Wang^{1,2}, Yinghua Li^{2}^{†} and Chunpeng Wang^{2}

1 Department of Mathematics, Northeast Normal University, Changchun 130024, P. R. China 2 Department of Mathematics, Jilin University, Changchun 130012, P. R. China

Abstract In this paper, we consider the initial-boundary value problem of a nonlinear parabolic equation with double degeneracy, and establish the existence and uniqueness theorems of renormalized solutions which are stronger than BV solutions.

Keywords: Renormalized solutions, double degeneracy.

2000 MR S. C.: 35K60, 35K65

1 Introduction

This paper is concerned with the following initial-boundary value problem

\frac{∂u}{∂t} + \frac{∂}{∂x}f(u) = \frac{∂}{∂x}[A(\frac{∂}{∂x}B(u))],  (x, t) ∈ Q_{T} ≡ (0, 1) × (0, T),  (1.1)
B(u(0, t)) = B(u(1, t)) = 0,  t ∈ (0, T),  (1.2)
u(x, 0) = u_{0}(x),  x ∈ (0, 1),  (1.3)

where f(s) is an appropriately smooth function and

A(s) = |s|^{p−2}s, B(s) = ∫_{0}^{s} b(σ)dσ, s ∈ ℝ

with p ≥ 2 and b(s) ≥ 0 appropriately smooth.

The equation (1.1) presents two kinds of degeneracy, since it is degenerate not only at

points where b(u) = 0 but also at points where \frac{∂}{∂x}B(u) = 0 if p > 2. Using the method depending on the properties of convex functions, Kalashnikov [10] established the existence of continuous solutions of the Cauchy problem of the equation (1.1) with f ≡ 0 under some convexity assumption on A(s) and B(s). Under such assumption, the equation degenerates only at the zero value of the solutions or their spacial derivatives. The more interesting case is that the equation may present strong degeneracy, namely, the set E = {s ∈ ℝ : b(s) = 0} may have interior points. Generally, the equation may have no classical solutions and even continuous solutions for this case and it is necessary to formulate some suitable weak solutions.

∗Supported by the NNSF of China and Graduate Innovation Lab of Jilin University.

†Corresponding author. Tel.:+86 431 5167051.

For the semilinear case of the equation (1.1) with p = 2, it is Vol’pert and Hudjaev [12] who first introduced BV solutions of the Cauchy problem and proved the existence theorem. Later, Wu and Wang [13] considered the initial-boundary value problem. For the quasilinear case with p > 2, the existence and uniqueness of BV solutions of the problem (1.1)–(1.3) under some natural conditions have been studied by Yin [15], where the BV solutions are defined in the following sense

Definition 1.1 A function u ∈ L^{∞}(Q_{T}) ∩ BV (Q_{T}) is said to be a BV solution of the problem (1.1)–(1.3), if

(i) u satisfies (1.2) and (1.3) in the sense that

B(u^{r}(0, t)) = B(u^{l}(1, t)) = 0,  a.e. t ∈ (0, T),  (1.4)
lim_{t→0^{+}} ū(x, t) = u_{0}(x),  a.e. x ∈ (0, 1),  (1.5)

where u^{r}(·, t) and u^{l}(·, t) denote the right and left approximate limits of u(·, t) respectively and ū denotes the symmetric mean value of u.

(ii) for any k ∈ ℝ and any nonnegative functions φ_{i} ∈ C^{∞}(Q_{T}) (i = 1, 2) with suppφ_{i} ⊂ [0, 1] × (0, T) and

φ_{1}(0, t) = φ_{2}(0, t), φ_{1}(1, t) = φ_{2}(1, t), 0 < t < T,

the following integral inequality holds

∫∫_{Q_{T}} sgn(u − k)[(u − k)\frac{∂φ_{1}}{∂t} + (f(u) − f(k))\frac{∂φ_{1}}{∂x} − A(\frac{∂}{∂x}B(u))\frac{∂φ_{1}}{∂x} ]dxdt

+ sgnk ∫∫_{Q_{T}} [(u − k)\frac{∂φ_{2}}{∂t} + (f(u) − f(k))\frac{∂φ_{2}}{∂x} − A(\frac{∂}{∂x}B(u))\frac{∂φ_{2}}{∂x} ] dxdt ≥ 0. (1.6)

In this paper we discuss the renormalized solutions of the problem (1.1)–(1.3). Such solutions were first introduced by Di Perna and Lions [8] in 1980’s, where the authors studied the existence of solutions of Boltzmann equations. From then on, there have been many results on renormalized solutions of various problems, see [1, 2, 3, 4, 5, 6, 7, 11]. It is shown that such solutions play an important role in prescribing nonsmooth solutions and noncontinuous solutions. The renormalized solutions of the problem (1.1)–(1.3) considered in this paper are defined as follows

Definition 1.2 A function u ∈ L^{∞}(Q_{T}) ∩ BV (Q_{T}) is said to be a renormalized solution of the problem (1.1)–(1.3), if

(i) u satisfies (1.2) and (1.3) in the sense of (1.4) and (1.5);

(ii) for any k ∈ ℝ, any η > 0 and any nonnegative functions φ_{i} ∈ C^{∞}(Q_{T}) (i = 1, 2) with suppφ_{i} ⊂ [0, 1] × (0, T) and

φ_{1}(0, t) = φ_{2}(0, t), φ_{1}(1, t) = φ_{2}(1, t), 0 < t < T,

the following integral inequality holds

∫∫_{Q_{T}} H_{η}(u, k)\frac{∂φ_{1}}{∂t} dxdt + ∫∫_{Q_{T}} F_{η}(u, k)\frac{∂φ_{1}}{∂x} dxdt

− ∫∫_{Q_{T}} A(\frac{∂}{∂x}B(u))H_{η}(u − k)\frac{∂φ_{1}}{∂x} dxdt − ∫∫_{Q_{T}} A(\frac{∂}{∂x}B(u))Hη^{′}(u − k)\frac{∂u}{∂x}φ_{1}dxdt

+ ∫_{k}^{0} H′_{η}(τ − k)(f(τ) − f(k))dτ ∫_{0}^{T} (φ_{1}(1, t) − φ_{1}(0, t))dt

+ H_{η}(k) ∫∫_{Q_{T}} [(u − k)\frac{∂φ_{2}}{∂t} + (f(u) − f(k))\frac{∂φ_{2}}{∂x} − A(\frac{∂}{∂x}B(u))\frac{∂φ_{2}}{∂x} ] dxdt ≥ 0, (1.7)

where

H_{η}(s) = \frac{s}{\sqrt{s^{2} + η}} , Hη^{′}(s) = \frac{η}{(s^{2} + η)^{3/2}} , s ∈ ℝ,

H_{η}(s, k) = ∫_{k}^{s} H_{η}(τ − k)dτ, F_{η}(s, k) = ∫_{k}^{s} H_{η}(τ − k)f′(τ)dτ, s, k ∈ ℝ.

Such solutions are a natural extension of classical solutions, which will be shown at the beginning of the next section. Comparing the two definitions of weak solutions, there are two additional terms in (1.7), i.e. the forth and fifth ones. Moreover, (1.6) follows by letting η → 0^{+} in (1.7) since the forth term of (1.7) is nonnegative and the limit of the fifth term is zero. Therefore, renormalized solutions imply more information than BV solutions and thus it is stronger.

Since renormalized solutions are stronger than BV solutions, the uniqueness of renormalized solutions of the problem (1.1)–(1.3) may be deduced directly from the uniqueness of BV solutions (see [15]). Hence

Theorem 1.1 There exists at most one renormalized solution for the initial-boundary value problem (1.1)–(1.3).

And we will prove the existence of renormalized solutions of the problem (1.1)–(1.3) in this paper, namely

Theorem 1.2 Assume u_{0} ∈ BV ([0, 1]) with u_{0}(0) = u_{0}(1) = 0. Then the initial-boundary value problem (1.1)–(1.3) admits one and only one renormalized solution.

The paper is arranged as follows. The preliminaries are done in §2. We first prove that the classical solution is also a renormalized solution, which shows that the latter is a natural extension of the former. Then we formulate the regularized problem and do some a priori estimates and establish some convergence. Two technical lemmas are introduced at the end of this section. The main result of this paper (Theorem 1.2) is proved in §3 subsequently.

2 Preliminaries

The renormalized solution is a natural extension of the classical solution. In fact, we have

Proposition 2.1 Let u ∈ C^{2}(Q_{T}) ∩ C(\overline{Q}_{T}) be a solution of the equation (1.1) with

u(0, t) = u(1, t) = 0, t ∈ (0, T).  (2.1)

Then for any k ∈ ℝ, any η > 0 and any nonnegative functions φ_{i} ∈ C^{∞}(Q_{T}) (i = 1, 2) with suppφ_{i} ⊂ [0, 1] × (0, T) and

φ_{1}(0, t) = φ_{2}(0, t), φ_{1}(1, t) = φ_{2}(1, t), 0 < t < T,

∫∫_{Q_{T}} H_{η}(u, k)\frac{∂φ_{1}}{∂t} dxdt + ∫∫_{Q_{T}} F_{η}(u, k)\frac{∂φ_{1}}{∂x} dxdt

− ∫∫_{Q_{T}} A(\frac{∂}{∂x}B(u))H_{η}(u − k)\frac{∂φ_{1}}{∂x} dxdt − ∫∫_{Q_{T}} A(\frac{∂}{∂x}B(u))Hη^{′}(u − k)\frac{∂u}{∂x}φ_{1}dxdt

+ ∫_{k}^{0} H′_{η}(τ − k)(f(τ) − f(k))dτ ∫_{0}^{T} (φ_{1}(1, t) − φ_{1}(0, t))dt

+ H_{η}(k) ∫∫_{Q_{T}} [(u − k)\frac{∂φ_{2}}{∂t} + (f(u) − f(k))\frac{∂φ_{2}}{∂x} − A(\frac{∂}{∂x}B(u))\frac{∂φ_{2}}{∂x} ] dxdt = 0. (2.2)

Therefore, if u ∈ C^{2}(Q_{T}) ∩ C(\overline{Q}_{T}) is a solution of the problem (1.1)–(1.3) with (2.1), then u is also a renormalized solution of the problem (1.1)–(1.3) and the inequality (1.7) can be rewritten as the equality.

Proof. Let k ∈ ℝ, η > 0 and φ_{i} ∈ C^{∞}(Q_{T}) (i = 1, 2) be nonnegative functions with suppφ_{i} ⊂ [0, 1] × (0, T) and

φ_{1}(0, t) = φ_{2}(0, t), φ_{1}(1, t) = φ_{2}(1, t), 0 < t < T.  (2.3)

On the one hand, multiply (1.1) with H_{η}(u − k)φ_{1} and then integrate over Q_{T} to get

∫∫_{Q_{T}} φ_{1} \frac{∂}{∂t}H_{η}(u, k)dxdt + ∫∫_{Q_{T}} φ_{1} \frac{∂}{∂x}F_{η}(u, k)dxdt

= ∫∫_{Q_{T}} \frac{∂}{∂x}A(\frac{∂}{∂x}B(u))H_{η}(u − k)φ_{1}dxdt.  (2.4)

From the definition of F_{η}(s, k), (2.1) and the Newton-Leibniz formula,

∫_{0}^{T} φ_{1}F_{η}(u, k)|_{x=0}^{x=1}dt
= ∫_{0}^{T} (∫_{k}^{u(1,t)} H_{η}(τ − k)f′(τ)dτφ_{1}(1, t) − ∫_{k}^{u(0,t)} H_{η}(τ − k)f′(τ)dτφ_{1}(0, t)) dt

= ∫_{k}^{0} H_{η}(τ − k)(f(τ) − f(k))′dτ ∫_{0}^{T} (φ_{1}(1, t) − φ_{1}(0, t))dt
=H_{η}(−k)(f(0) − f(k)) ∫_{0}^{T} (φ_{1}(1, t) − φ_{1}(0, t))dt
− ∫_{k}^{0} H′_{η}(τ − k)(f(τ) − f(k))dτ ∫_{0}^{T} (φ_{1}(1, t) − φ_{1}(0, t))dt.  (2.5)

Then, by using the formula of integrating by parts in (2.4) and from (2.5), we get

∫∫_{Q_{T}} H_{η}(u, k)\frac{∂φ_{1}}{∂t} dxdt + ∫∫_{Q_{T}} F_{η}(u, k)\frac{∂φ_{1}}{∂x} dxdt

+ H_{η}(k)(f(0) − f(k)) ∫_{0}^{T} φ_{1}(1, t) − φ_{1}(0, t))dt
+ ∫_{k}^{0} H′_{η}(τ − k)(f(τ) − f(k))dτ ∫_{0}^{T} (φ_{1}(1, t) − φ_{1}(0, t))dt
= ∫_{0}^{T} H_{η}(k)[A(\frac{∂}{∂x}B(u(1, t)))φ_{1}(1, t) − A(\frac{∂}{∂x}B(u(0, t)))φ_{1}(0, t)]dt
+ ∫∫_{Q_{T}} A(\frac{∂}{∂x}B(u)) (H_{η}(u − k)\frac{∂φ_{1}}{∂x} + H′_{η}(u − k)\frac{∂u}{∂x}φ_{1}) dxdt.  (2.6)

On the other hand, the equation (1.1) leads to

\frac{∂}{∂t}(u − k) + \frac{∂}{∂x}(f(u) − f(k)) = \frac{∂}{∂x}[A(\frac{∂}{∂x}B(u))], (x, t) ∈ Q_{T} .

Multiplying this equation with φ_{2} and then integrating over Q_{T}, we get that by the formula of integrating by parts and (2.1)

∫∫_{Q_{T}} (u − k)\frac{∂φ_{2}}{∂t} dxdt + ∫∫_{Q_{T}} (f(u) − f(k))\frac{∂φ_{2}}{∂x} dxdt

− (f(0) − f(k)) ∫_{0}^{T} (φ_{2}(1, t) − φ_{2}(0, t))dt
= − ∫_{0}^{T} [A(\frac{∂}{∂x}B(u(1, t)))φ_{2}(1, t) − A(\frac{∂}{∂x}B(u(0, t)))φ_{2}(0, t)]dt
+ ∫∫_{Q_{T}} A(\frac{∂}{∂x}B(u))\frac{∂φ_{2}}{∂x} dxdt.  (2.7)

Multiplying (2.7) with H_{η}(k) and then adding what obtained to (2.6), we get (2.2) owing to (2.3). The proof is complete. □

Remark 1 The condition (2.1) is not egregious. In fact, (2.1) and (1.2) are identical if B(s) is strictly increasing, i.e. the set E = {s ∈ ℝ : b(s) = 0} has no interior point. Otherwise, if the set E has interior points, the equation (1.1) is strong degenerate and the equation may have no classical solutions in general, as mentioned in the introduction.

Since the equation (1.1) presents double degeneracy, we regularize the equation to get the existence of renormalized solutions by doing a prior estimates and passing a limit process. We firstly approximate the given initial data u_{0}. For any 0 < ε < 1, choose u_{0,ε} ∈ C_{0}^{∞}(0, 1) satisfying

∫_{0}^{1} |\frac{∂u_{0,ε}}{∂x} |dx ≤ C,

∫_{0}^{1} |\frac{∂}{∂x}A_{ε}(\frac{∂}{∂x}B_{ε}(u_{0,ε}))|dx ≤ C,

u_{0,ε}(x) ⟶ u_{0}(x), uniformly in (0, 1) as ε → 0^{+},

where C > 0 is independent of ε, and

A_{ε}(s) = A(s) + εs, B_{ε}(s) = B(s) + εs, s ∈ ℝ.

Consider the regularized problem

\frac{∂u_{ε}}{∂t} + \frac{∂}{∂x}f(u_{ε}) = \frac{∂}{∂x}[A_{ε}(\frac{∂}{∂x}B_{ε}(u_{ε}))],  (x, t) ∈ Q_{T} ,  (2.8)
u_{ε}(0, t) = u_{ε}(1, t) = 0,  t ∈ (0, T),  (2.9)
u_{ε}(x, 0) = u_{0,ε}(x),  x ∈ (0, 1).  (2.10)

By virtue of the standard theory for uniformly parabolic equations, there exists a unique classical solution u_{ε} ∈ C^{2}(\overline{Q}_{T}) of the above problem. To pass the limit process to the problem (1.1)–(1.3), we need do a priori estimates on u_{ε}. On the one hand, the maximum principle gives

sup_{Q_{T}} |u_{ε}| ≤ C.  (2.11)

Here and hereafter, we denote by C positive constants independent of ε and may be different in different formulae. On the other hand, the following BV estimates and C^{1,1/2} estimates have been proved by Yin [15].

Lemma 2.1 The solutions u_{ε} satisfy

sup_{0<t<T} ∫_{0}^{1} |\frac{∂u_{ε}}{∂x} |dx + sup_{0<t<T} ∫_{0}^{1} |\frac{∂u_{ε}}{∂t} |dx ≤ C,

sup_{0<t<T} ∫_{0}^{1} |\frac{∂}{∂x}A_{ε}(\frac{∂}{∂x}B_{ε}(u_{ε}(x, t)))|dx ≤ C,

sup_{Q_{T}} |\frac{∂}{∂x}B_{ε}(u_{ε})| ≤ C.

Lemma 2.2 For the function

ω_{ε}(x, t) = B_{ε}(u_{ε}(x, t)), (x, t) ∈ Q_{T} ,

we have

\begin{matrix}|ω_{ε}(x_{1}, t) − ω_{ε}(x_{2}, t)| ≤ C|x_{1} − x_{2}|, & x_{1}, x_{2} ∈ (0, 1), t ∈ (0, T), \\ |ω_{ε}(x, t_{1}) − ω_{ε}(x, t_{2})| ≤ C|t_{1} − t_{2}|^{1/2}, & x ∈ (0, 1), t_{1}, t_{2} ∈ (0, T). \end{matrix}

Form the estimate (2.11), Lemma 2.1 and Lemma 2.2, there exist a subsequence of ε ∈ (0, 1), denoted by itself for convenience, and a function u ∈ L^{∞}(Q_{T})∩BV (Q_{T}) with B(u) ∈ C^{1,1/2}(\overline{Q}_{T}) and (1.4) and (1.5), and a function μ ∈ L^{∞}(Q_{T}), such that

u_{ε}(x, t) ⟶ u(x, t),  a.e. in Q_{T} ,  (2.12)
B_{ε}(u_{ε}(x, t)) ⟶ B(u(x, t)),  uniformly in Q_{T} ,  (2.13)
\frac{∂}{∂x}B_{ε}(u_{ε}) ⇀^{∗} \frac{∂}{∂x}B(u),  weakly^{∗} in L^{∞}(Q_{T}),  (2.14)
A_{ε}(\frac{∂}{∂x}B_{ε}(u_{ε})) ⇀^{∗} μ,  weakly^{∗} in L^{∞}(Q_{T}),  (2.15)

as ε → 0^{+}, see more details in Yin [15].

To complete the limit process, we also need the following convergence.

Lemma 2.3 For the solution u_{ε} of the regularized problem (2.8)–(2.10) and the above limit function u, we have

A_{ε}(\frac{∂}{∂x}B_{ε}(u_{ε}(x, t))) ⟶ A(\frac{∂}{∂x}B(u(x, t))),  a.e. in Q_{T} as ε → 0^{+}.  (2.16)

Proof. We first prove that

μ = A(\frac{∂}{∂x}B(u)), a.e. in Q_{T} .  (2.17)

For convenience, we rewrite

A(s) = ∫_{0}^{s} a(σ)dσ, s ∈ ℝ, a(σ) = \frac{1}{p}|σ|^{p}, σ ∈ ℝ.

Multiplying (2.8) with (B_{ε}(u_{ε})−B(u)) and then integrating over Q_{T}, we get that by the formula of integration by parts

∫∫_{Q_{T}} \frac{∂u_{ε}}{∂t} (B_{ε}(u_{ε}) − B(u))dxdt + ∫∫_{Q_{T}} \frac{∂}{∂x}f(u_{ε})(B_{ε}(u_{ε}) − B(u))dxdt

= − ∫∫_{Q_{T}} A_{ε}(\frac{∂}{∂x}B_{ε}(u_{ε}))(\frac{∂}{∂x}B_{ε}(u_{ε}) − \frac{∂}{∂x}B(u))dxdt,

which yields

lim_{ε→0^{+}} ∫∫_{Q_{T}} A_{ε}(\frac{∂}{∂x}B_{ε}(u_{ε}))(\frac{∂}{∂x}B_{ε}(u_{ε}) − \frac{∂}{∂x}B(u))dxdt = 0  (2.18)

from (2.13). On the other hand, by (2.14) and B(u) ∈ C^{1,1/2}(\overline{Q}_{T}),

lim_{ε→0^{+}} ∫∫_{Q_{T}} A_{ε}(\frac{∂}{∂x}B(u))(\frac{∂}{∂x}B_{ε}(u_{ε}) − \frac{∂}{∂x}B(u))dxdt = 0.  (2.19)

Therefore, combining (2.18) with (2.19) gives

0 = lim_{ε→0^{+}} ∫∫_{Q_{T}} (A_{ε}(\frac{∂}{∂x}B_{ε}(u_{ε})) − A_{ε}(\frac{∂}{∂x}B(u)))(\frac{∂}{∂x}B_{ε}(u_{ε}) − \frac{∂}{∂x}B(u))dxdt
= lim_{ε→0^{+}} ∫∫_{Q_{T}} a_{ε}^{∗}(x, t)(\frac{∂}{∂x}B_{ε}(u_{ε}) − \frac{∂}{∂x}B(u))^{2}dxdt,  (2.20)

where

a_{ε}^{∗}(x, t) = ∫_{0}^{1} [a(λ\frac{∂}{∂x}B_{ε}(u_{ε}) − (1 − λ)\frac{∂}{∂x}B(u)) + ε]dλ, (x, t) ∈ Q_{T} .

Since a_{ε}^{∗}(x, t) is positive and uniformly bounded in Q_{T},

∫∫_{Q_{T}} (a_{ε}^{∗})^{2}(\frac{∂}{∂x}B_{ε}(u_{ε}) − \frac{∂}{∂x}B(u))^{2}dxdt ≤ C ∫∫_{Q_{T}} a_{ε}^{∗}(\frac{∂}{∂x}B_{ε}(u_{ε}) − \frac{∂}{∂x}B(u))^{2}dxdt. (2.21)

Then, from the definition of a_{ε}^{∗}, (2.21) and (2.20),

lim_{ε→0^{+}} ∫∫_{Q_{T}} [A_{ε}(\frac{∂}{∂x}B_{ε}(u_{ε})) − A(\frac{∂}{∂x}B(u))]^{2}dxdt

= lim_{ε→0^{+}} ∫∫_{Q_{T}} (a_{ε}^{∗})^{2}(\frac{∂}{∂x}B_{ε}(u_{ε}) − \frac{∂}{∂x}B(u))^{2}dxdt = 0.

This, together with (2.15) yields (2.17), namely

A_{ε}(\frac{∂}{∂x}B_{ε}(u_{ε})) → A(\frac{∂}{∂x}B(u)), in L^{2}(Q_{T}) as ε → 0^{+},

which deduces (2.16). The proof is complete. □

In order to reach Theorem 1.2, we need the following two technical lemmas, which may be found in [9], [14].

Lemma 2.4 Let Ω ⊂ ℝ^{n} be a bounded domain, {u_{m}} be a uniformly bounded sequence in L^{∞}(Ω) and u ∈ L^{∞}(Ω) with

u_{m} ⇀^{∗} u, weakly^{∗} in L^{∞}(Ω) as m → ∞.

Assume that A(s), B(s) are continuous functions, and A(s) is nondecreasing. If for any α ∈ A(ℝ), B(A^{−1}(α)) contains only a single point, and

A(u_{m}(x)) → ω(x), a.e. x ∈ Ω as m → ∞,

then

A(u(x)) = ω(x), lim_{m→∞} B(u_{k}(x)) = B(u(x)), a.e. x ∈ Ω.

Lemma 2.5 Let Ω ⊂ ℝ^{n} be a bounded domain and 1 < q < ∞. Assume {f_{m}} is a sequence in L^{q}(Ω) and f ∈ L^{q}(Ω) with

f_{m} ⇀ f, weakly in L^{q}(Ω) as m → ∞.

Then

\underline{lim}_{m→∞}∥f_{m}∥_{L^{q}(Ω)} ≥ ∥f∥_{L^{q}(Ω)}.

3 Proof of the Main Result

In this section, we will complete the proof of Theorem 1.2 based on the estimates and convergence established in §2.

Proof of Theorem 1.2. For any k ∈ ℝ, any η > 0 and any nonnegative functions φ_{i} ∈ C^{∞}(Q_{T}) (i = 1, 2) with suppφ_{i} ⊂ [0, 1] × (0, T) and

φ_{1}(0, t) = φ_{2}(0, t), φ_{1}(1, t) = φ_{2}(1, t), 0 < t < T,

according to Proposition 2.1,

∫∫_{Q_{T}} H_{η}(u_{ε}, k)\frac{∂φ_{1}}{∂t} dxdt + ∫∫_{Q_{T}} F_{η}(u_{ε}, k)\frac{∂φ_{1}}{∂x} dxdt

− ∫∫_{Q_{T}} A_{ε}(\frac{∂}{∂x}B_{ε}(u_{ε}))H_{η}(u_{ε} − k)\frac{∂φ_{1}}{∂x} dxdt

− ∫∫_{Q_{T}} A_{ε}(\frac{∂}{∂x}B_{ε}(u_{ε}))Hη^{′}(u_{ε} − k)\frac{∂u_{ε}}{∂x} φ_{1}dxdt

+ ∫_{k}^{0} H′_{η}(τ − k)(f(τ) − f(k))dτ ∫_{0}^{T} (φ_{1}(1, t) − φ_{1}(0, t))dt

+ H_{η}(k) ∫∫_{Q_{T}} [(u_{ε} − k)\frac{∂φ_{2}}{∂t} + (f(u_{ε}) − f(k))\frac{∂φ_{2}}{∂x} − A_{ε}(\frac{∂}{∂x}B_{ε}(u_{ε}))\frac{∂φ_{2}}{∂x} ] dxdt = 0. (3.1)

By the definitions of H_{η}(s, k) and F_{η}(s, k), and by using Lemma 2.4 with (2.12), we get

lim_{ε→0^{+}} H_{η}(u_{ε}(x, t), k) = H_{η}(u(x, t), k), lim_{ε→0^{+}} F_{η}(u_{ε}(x, t), k) = F_{η}(u(x, t), k), a.e. in Q_{T} .

Combining this with (2.16) and (2.12), we have

lim_{ε→0^{+}} ( ∫∫_{Q_{T}} H_{η}(u_{ε}, k)\frac{∂φ_{1}}{∂t} dxdt + ∫∫_{Q_{T}} F_{η}(u_{ε}, k)\frac{∂φ_{1}}{∂x} dxdt

− ∫∫_{Q_{T}} A_{ε}(\frac{∂}{∂x}B_{ε}(u_{ε}))H_{η}(u_{ε} − k)\frac{∂φ_{1}}{∂x} dxdt)

= ∫∫_{Q_{T}} H_{η}(u, k)\frac{∂φ_{1}}{∂t} dxdt + ∫∫_{Q_{T}} F_{η}(u, k)\frac{∂φ_{1}}{∂x} dxdt
− ∫∫_{Q_{T}} A(\frac{∂}{∂x}B(u))H_{η}(u − k)\frac{∂φ_{1}}{∂x} dxdt  (3.2)

and

lim_{ε→0^{+}} ∫∫_{Q_{T}} [(u_{ε} − k)\frac{∂φ_{2}}{∂t} + (f(u_{ε}) − f(k))\frac{∂φ_{2}}{∂x} − A_{ε}(\frac{∂}{∂x}B_{ε}(u_{ε}))\frac{∂φ_{2}}{∂x} ] dxdt

= ∫∫_{Q_{T}} [(u − k)\frac{∂φ_{2}}{∂t} + (f(u) − f(k))\frac{∂φ_{2}}{∂x} − A(\frac{∂}{∂x}B(u))\frac{∂φ_{2}}{∂x} ] dxdt.  (3.3)

Note that u satisfies (1.2) and (1.3) in the sense of (1.4) and (1.5). Therefore, owing to (3.2), (3.3) and (3.1), it is shown that u is just a renormalized solution of the problem (1.1)–(1.3) provided

\underline{lim}_{ε→0^{+}} ∫∫_{Q_{T}} A_{ε}(\frac{∂}{∂x}B_{ε}(u_{ε}))Hη^{′}(u_{ε} − k)\frac{∂u_{ε}}{∂x} φ_{1}dxdt

≥ ∫∫_{Q_{T}} A(\frac{∂}{∂x}B(u))Hη^{′}(u − k)\frac{∂u}{∂x}φ_{1}dxdt,  (3.4)

which will be proved below.

Multiplying (2.8) by u_{ε} and then integrating over Q_{T}, we derive that by the formula of integration by parts

\frac{1}{2} ∫∫_{Q_{T}} \frac{∂u_{ε}^{2}}{∂t} dxdt − ∫∫_{Q_{T}} f(u_{ε})\frac{∂u_{ε}}{∂x} dxdt = − ∫∫_{Q_{T}} A_{ε}(\frac{∂}{∂x}B_{ε}(u_{ε}))\frac{∂u_{ε}}{∂x} dxdt.  (3.5)

From (2.11) and Lemma 2.1,

|∫∫_{Q_{T}} \frac{∂u_{ε}^{2}}{∂t} dxdt| = |∫_{0}^{1} (u_{ε}^{2}(x, T) − u_{0,ε}^{2}(x))dx| ≤ C,

|∫∫_{Q_{T}} f(u_{ε})\frac{∂u_{ε}}{∂x} dxdt| ≤ sup_{Q_{T}} |f(u_{ε})| ∫∫_{Q_{T}} |\frac{∂u_{ε}}{∂x} |dxdt ≤ C.

These two estimates and (3.5) yield

∫∫_{Q_{T}} A_{ε}(\frac{∂}{∂x}B_{ε}(u_{ε}))\frac{∂u_{ε}}{∂x} dxdt ≤ C.

From the definitions of A_{ε} and B_{ε}, the above inequality leads to

∫∫_{Q_{T}} |\frac{∂K_{ε}(u_{ε})}{∂x}|^{p}dxdt ≤ C,

where

K_{ε}(s) = ∫_{0}^{s}(b(σ) + ε)^{(p−1)/p}dσ, s ∈ ℝ.

Thus

∫∫_{Q_{T}} |\frac{∂K(u)}{∂x}|^{p}dxdt ≤ C,

namely \frac{∂K(u)}{∂x} ∈ L^{p}(Q_{T}) with

K(s) = ∫_{0}^{s} b^{(p−1)/p}(σ)dσ, s ∈ ℝ.

Moreover, (2.12) and (2.14) imply

\frac{∂K_{ε}(u_{ε})}{∂x} ⇀ \frac{∂K(u)}{∂x}, weakly in L^{p}(Q_{T}) as ε → 0^{+}.  (3.6)

Therefore, for fixed η > 0, \frac{∂K(u)}{∂x}(Hη^{′}(u − k)φ_{1})^{1/p} ∈ L^{p}(Q_{T}), and (3.6) implies

\frac{∂K_{ε}(u_{ε})}{∂x}(Hη^{′}(u_{ε} − k)φ_{1})^{1/p} ⇀ \frac{∂K(u)}{∂x}(Hη^{′}(u − k)φ_{1})^{1/p}, weakly in L^{p}(Q_{T}) as ε → 0^{+}.

By Lemma 2.5,

∥\frac{∂K(u)}{∂x}(Hη^{′}(u − k)φ_{1})^{1/p}∥_{L^{p}(Q)} ≤ \underline{lim}_{ε→0^{+}}∥\frac{∂K_{ε}(u_{ε})}{∂x}(Hη^{′}(u_{ε} − k)φ_{1})^{1/p}∥_{L^{p}(Q)}.

This is just (3.4). The proof of Theorem 1.2 is complete. □

Acknowledgment

We would like to express our thanks to the referee and the editor. Their comments are very important for the improvement of this paper.

References

[1] K. Ammar and P. Wittbold, Existence of renormalized solutions of degenerate ellipticparabolic problems, Proc. Roy. Soc. Edinburgh Sect. A, 133(3)(2003), 477–496.

[2] Ph. Bénilan, L. Boccardo, Th. Gallouët, R. Gariepy, M. Pierre and J. L. Vazquez, An L^{1}- theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22(4)(1995), 241–273.

[3] D. Blanchard and H. Redwane, Solutions rénormalisées d’équations paraboliques à deux non linéarités, C. R. Acad. Sci. Paris, 319(1994), 831–835.

[4] L. Boccardo, D. Giachetti, J. I. Diaz and F. Murat, Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms, J. Differential Equations, 106(2)(1993), 215–237.

[5] F. Bouchut, Renormalized solutions to the Vlasov equation with coefficients of bounded variation, Arch. Ration. Mech. Anal., 1571(2001), 75–90.

[6] J. Carrillo and P. Wittbold, Uniqueness of renormalized solutions of degenerate ellipticparabolic problems, J. Differential Equations, 156(1)(1999), 93–121.

[7] G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Definition and existence of renormalized solutions of elliptic equations with general measure data, C. R. Acad. Sci. Paris Ser. I Math., 325(5)(1997), 481–486.

[8] R. J. Di Perna and P. L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. of Math., 130(2)(1989), 321–366.

[9] L. C. Evans, Weak convergence methods for nonlinear partial differential equations, Conference Board of the Mathematical Sciences, Regional Conference Series in Mathematics Number 74, 1998.

[10] A. S. Kalashnikov, The Cauchy problem for degenerate second-order parabolic equations with nonpower nonlinearities (Russian. English summary), Trudy Sem. Petrovsk., 6(1981), 83–96.

[11] J. M. Rakotoson, Generalized solutions in a new type of sets for problems with measures as data, Differential Integral Equations, 6(1993), 27–36.

[12] A. I. Vol’pert and S. I. Hudjaev, Cauchy problem for second order quasilinear degenerate parabolic equations, Mat. Sb., 78(1969), 398–411.

[13] Z. Q. Wu and J. Y. Wang, Some results on quasilinear degenerate parabolic equations of second order, Proceedings of International Beijing Symposium on Differential Geometry and Partial Differential Equations, 1981.

[14] Z. Q. Wu, J. N. Zhao, J. X. Yin and H. L. Li, Nonlinear Diffusion Equations, World Scientific, 2001.

[15] J. X. Yin, On a class of quasilinear parabolic equations of second order with doubledegeneracy, Journal of Partial Differential Equations, 3(4)(1990), 49–64.

(Received December 30, 2005)

E-mail addresses:

matwzj@jlu.edu.cn
yinghua@jlu.edu.cn