Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 68, 1-10; http://www.math.u-szeged.hu/ejqtde/

Periodic Solutions of p-Laplacian Systems with a Nonlinear Convection Term

Hamid El Ouardi Equipe Architures des Systemes Université Hassan II Ain Chock ENSEM, BP. 9118, Oasis Casablanca, Morocco, Fax : +212.05.22.23.12.99 e-mail : h.elouardi@ensem.ac.ma / h.elou di@yahoo.fr

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Abstract In this work, we study the existence of periodic solutions for the evolution of p-Laplacian system and we show that these periodic solutions belong to L^{∞}(ω, W^{1,∞}(Ω)) and give a bound of ∥∇u_{i}(t)∥_{∞} under certain geometric conditions on ∂Ω.

Keywords: Periodic solutions; nonlinear parabolic systems; p-Laplacian; existence of solutions; gradients estimates.

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1. Introduction

We consider the following nonlinear system (S) of the form

(S)\{ \begin{matrix}\frac{∂u_{1}}{∂t} − △_{p_{1}}u_{1} + a_{1}(u_{1}).∇u_{1} = f_{1}(t)u_{1}^{α_{1}} + h_{1}(x, u_{2}) in Q, \\ \frac{∂u_{2}}{∂t} − △_{p_{2}}u_{2} + a_{2}(u_{2}).∇u_{2} = f_{2}(t)u_{2}^{α_{2}} + h_{2}(x, u_{1}) in Q, \\ u_{1}(x, t) = u_{2}(x, t) = 0 on S, \\ ((u_{1}(x, t + ω), (u_{2}(x, t + ω)) = (u_{1}(x, t), u_{2}(x, t)) in Q, \end{matrix}

where △_{p}u = div(|∇u|^{p−2} ∇u) is the so-called p−Laplacian operator, Ω is a bounded convex subset in ℝ^{N}, Q := Ω × (0, ω), S := ∂Ω × (0, ω), ∂Ω is the boundary of Ω and ω > 0. Precise conditions on a_{i}, f_{i} and h_{i} will be given later.

The system of form (S) is a class of degenerate parabolic systems and appears in the theory of non-Newtonian fluids perturbed by nonlinear terms and forced by rather irregular period in time excitations, see [1, 5] . Periodic parabolic single equations have been the subject of numbers of extentive works see [3, 5, 11, 12, 14, 15, 16]. In particular, Lui [11], has considered the following single equation:

\frac{∂u}{∂t} − △_{p}u + b(u).∇u = f(t)u^{α} + h(x, t),

with Dirichlet boundary condition and has obtained the existence and gradient estimates of a periodic solution.The basic tools that have been used are a priori estimates, Leray-Shauder fixed point theorem, topological degree theory and Moser method. We first consider the parabolic regularisation of (S) and we employ Moser’s technique as in [4] and we make some devices as in [7, 13] to obtain the existence of periodic solutions and we derive estimates of ∇u_{i}(t).

2. Main results

For convenience, hereafter let E = C_{ω}(\overline{Q}), the set of all functions in C(\overline{Ω}×ℝ) which are periodic in t with period w. ∥.∥_{q} and ∥.∥_{s,q} denote L^{q} = L^{q}(Ω) and W^{s,q} = W^{s,q}(Ω) norms, respectively, 1 ≤ q ≤ ∞. Since , Ω ⊂ ℝ^{N} is a bounded convex domain, we take the equivalent norm in W_{0}^{1,q}(Ω) to be

∥∇u∥_{q} = ( N ; ∑_{j=1} ∫_{Ω} |\frac{∂u}{∂x_{j}} |^{q})^{\frac{1}{q}} for any u ∈ W_{0}^{1,q}(Ω).

In the sequel, the same symbol c will be used to indicate some positive constants, possibly different from each other, appearing in the various hypotheses and computations and depending only on data. When we need to fix the precise value of one constant, we shall use a notation like M_{i}, i = 1, 2....., instead.

We make the following assumptions:

(H1) \{ \begin{matrix}a_{i}(s) = (a_{i1}, a_{i2}, ....., a_{IN}) is an ℝ^{N} − valued function on ℝ, \\ |a_{i}(s)| ≤ k_{i} |s|^{β_{i}} , for some 0 ≤ β_{i} < p_{i} − 2, (i = 1, 2) . \end{matrix}

(H2) { f_{i}(t) ∈ L ^{∞}(0, ω) is periodic in t (i = 1, 2) , with period ω.

(H3) \{ \begin{matrix}h_{i}(x, t) ∈ C_{ω}(\overline{Q})∩L ^{∞}(0, ω; W_{0}^{1,∞}(Ω)), h_{i}(x, 0) = 0 \\ h_{i}(x, t) > 0 , for all (x, t) ∈ Ω × ℝ, (i = 1, 2) . \end{matrix}

(H4) {0 ≤ α_{i} < p_{i} − 1 and N > p_{i} ≥ 2.

We make the following definition of solutions.

Definition 2.1 A function u = (u_{1}, u_{2}) is called a solution of system (S) if

u_{i} ∈ L^{p_{i}}(0, ω; W_{0}^{1,p_{i}}(Ω)) ∩ C_{ω}(\overline{Q})

and u satisfies

∫ ∫_{Q} {−u_{1}φ_{1t} + |∇u_{1}|^{p_{1−2}} ∇u_{1}.∇φ_{1} − A_{1}(u_{1}).∇φ_{1} − f_{1}(t)u_{1}^{α_{1}}φ_{1} − h_{1}(x, u_{2})φ_{1}} dxdt = 0,

∫ ∫_{Q} {−u_{2}φ_{2t} + |∇u_{2}|^{p_{2−2}} ∇u_{2}.∇φ_{2} − A_{2}(u_{2}).∇φ_{2} − f_{2}(t)u_{2}^{α_{2}}φ_{2} − h_{2}(x, u_{1})φ_{2}} dxdt = 0,

for any (φ_{1}, φ_{2}) ∈ C_{0}^{1}(0, ω; C_{0}^{1}(Ω)) with φ_{i}(x, 0) = φ_{i}(x, ω), where A_{i}(u) = ∫_{0}^{u} a_{i}(s)ds is set.

Theorem 1.1.

Let (H1) to (H4) be satisfied. Then there exists a solution (u_{1}, u_{2}) of problem (S) such that for i = 1, 2, we have

u_{i}(t) ∈ L^{∞}(0, ω; W_{0}^{1,p_{i}}(Ω)) ∩ C_{ω}(\overline{Q}),

\frac{∂u_{i}}{∂t} ∈ L^{2}(Q).

Theorem 1.2. Let (H1) to (H4) be satisfied. Then there exists a solution (u_{1}, u_{2}) of problem (S) such that for i = 1, 2, we have

u_{i}(t) ∈ L^{∞}(0, ω; W_{0}^{1,∞}(Ω)),

sup_{t} ∥∇u_{i}(t)∥_{∞} ≤ M_{i} < ∞.

For the proof of the theorems we use the following elementary lemmas. Lemma 2.1. (Gagliardo-Nirenberg, cf[4]) let β ≥ 0, N > p ≥ 1, β + 1 ≤ q, and 1 ≤ r ≤ q ≤ (β + 1)Np/(N − p), then

for v such that |v|^{β} v ∈ W^{1,p}(Ω),

∥v∥_{q} ≤ C^{\frac{1}{β+1}} ∥v∥_{r}^{1−θ} ∥|v|^{β} v∥_{1,p}^{θ/(β+1)} ,

with θ = (β +1)(r^{−1} −q^{−1})/ {N^{−1} − p^{−1} + (β + 1)r^{−1}} , where C is a constant independent of q, r, β, and θ.

Lemma 2.2. (cf [13]) Let y(t) ∈ C^{1}(ℝ) be a nonnegative ω periodic function satisfying the differ-

ential inequality

y′(t) + Ay^{α+1}(t) ≤ By(t) + C, t ∈ ℝ,

with some α, A > 0, B ≥ 0, and C ≥ 0. Then

y(t) ≤ max \{1, (A^{−1}(B + C))^{\frac{1}{α}} \} .

3. Proofs of Theorem 1.1 and Theorem 1.2. In this section we derive a priori estimates of solutions (u_{1}, u_{2}) of problem

(S). We first replace the p-Laplacian by the regularized one

△_{p}^{ε}u = div((|∇u|^{2} + ε)^{\frac{p−2}{2}} ∇u).

To prove theorem 1.1, we consider the following.

By theorem 1 in [3] , we can choose u_{iε}^{0} such that :

\frac{∂u_{iε}^{0}}{∂t} − △_{p_{1}}^{ε}u_{iε}^{0}) + a_{i}(u_{iε}^{0}).∇u_{iε}^{0} = f_{i}(t)(u_{iε}^{n})^{α_{i}} in Q,

u_{iε}^{0} = 0 on S,
u_{iε}^{0}(x, ω) = u_{iε}^{0}(x, t) in Q,

and we construct two sequences of functions (u_{1ε}^{n}) and (u_{2ε}^{n}), such that :

\frac{∂u_{1ε}^{n}}{∂t} − △_{p_{1}}^{ε}u_{1ε}^{n} + a_{1}(u_{1ε}^{n}).∇u_{1ε}^{n} = f_{1}(t)(u_{1ε}^{n})^{α_{1}} + h_{1}(x, u_{2ε}^{n−1}) in Q , (1.1)
u_{1ε}^{n} = 0  on S, (1.2)
u_{1ε}^{n}(x, ω) = u_{1ε}^{n}(x, t)  in Q. (1.3)

\frac{∂u_{2ε}^{n}}{∂t} − △_{p_{2}}^{ε}u_{2ε}^{n} + a_{2}(u_{2ε}^{n}).∇u_{2ε}^{n} = f_{2}(t)(u_{2ε}^{n})^{α_{2}} + h_{2}(x, u_{1ε}^{n−1}) in Q (1.6)
u_{2ε}^{n} = 0  on S, (1.5)
u_{2ε}^{n}(x, ω) = u_{2ε}^{n}(x, t)  in Q. (1.3)

It is clear that for each n = 1, 2, 3, ....., the above systems consist of two nondegenerated and uncoupled initial boundary-value problems. By theorem 1, [3] for fixed n and ε, the problem (1.1)-(1.3) and (1.4)-(1.6) has a solution (u_{1ε}^{n}, u_{2ε}^{n}).

We need lemma 2.3 and lemma 2.4 below to complete the proof of theorem 1.1.

Lemma 2.3. There exists a positive constant M_{i} independent of ε and n, such that

∥u_{iε}^{n}(t)∥_{∞} ≤ M_{i}.  (1.7)

Proof. For n = 0, (1.7) is proved in [11], so suppose (1.7) for (n − 1).

Multiplying (1.1) by |u_{1ε}^{n}|^{k} u_{1ε}^{n}, k integer, and integrating over Ω to obtain :

\frac{1}{k + 2} ∫_{Ω} |u_{1ε}^{n}|^{k+2} dx + ε(k + 1)(\frac{p_{1}}{k + p_{1}} )^{p_{1}} ∥∇ (u_{1ε}^{n})^{(k+p_{1})/p_{1}} ∥_{p_{1}}^{p_{1}}

+ ∫_{Ω} a_{1}(u_{1ε}^{n}).∇u_{1ε}^{n} |u_{1ε}^{n}|^{k} u_{1ε}^{n}dx =

∫_{Ω} f_{1}(t)(u_{1ε}^{n})^{α_{1}+1} |u_{1ε}^{n}|^{k} dx + ∫_{Ω} h_{1}(x, u_{2ε}^{n−1})u_{1ε}^{n} |u_{1ε}^{n}|^{k} dx.  (1.8)

We note that

∫_{Ω} a_{1}(u_{1ε}^{n}).∇u_{1ε}^{n} |u_{1ε}^{n}|^{k} u_{1ε}^{n}dx = ∫_{Ω} N ; ∑_{j=1}a_{1j}(u_{1ε}^{n}) |u_{1ε}^{n}|^{k} u_{1ε}^{n} \frac{∂u_{1ε}^{n}}{∂x_{j}} dx

= N ; ∑_{j=1} ∫_{Ω} (∫_{0}^{u_{1ε}^{n}} a_{1j}(s) |s|^{k} sds)_{x_{j}} dx

= N ; ∑_{j=1} ∫_{∂Ω} (∫_{0}^{u_{1ε}^{n}} a_{1j}(s) |s|^{k} sds) cos(n, x_{j})ds = 0,  (1.9)

∫_{Ω} f_{1}(t)(u_{1ε}^{n})^{α_{1}+1} |u_{1ε}^{n}|^{k} dx + ∫_{Ω} h_{1}(x, u_{2ε}^{n−1})u_{1ε}^{n} |u_{1ε}^{n}|^{k} dx
≤ C [f(t) ∥u_{1ε}^{n}∥_{k+α_{1}+1}^{k+α_{1}+1} + ∥h∥_{k+2} ∥u_{1ε}^{n}∥_{k+2}^{k+1}] .  (1.10)

If 1 ≤ α_{1} < p_{i} − 1, by Hölder’s inequality and Lemma 2.1, we have

∥u_{1ε}^{n}∥_{k+α_{1}+1}^{k+α_{1}+1} ≤ C ∥u_{1ε}^{n}∥_{k+2}^{θ_{1}} ∥u_{1ε}^{n}∥_{q}^{θ_{2}}

≤ C ∥u_{1ε}^{n}∥_{k+2}^{θ_{1}} ∥∇ (u_{1ε}^{n})^{(k+p_{1})/p_{1}} ∥_{p_{i}}^{θ_{2}p_{1}/(k+p_{1})}

≤ \frac{ε}{2M} (k + 1)(\frac{p_{1}}{k + p_{1}} )^{p_{1}} ∥∇ (u_{1ε}^{n})^{(k+p_{1})/p_{1}} ∥_{p_{1}}^{p_{1}}

+C (∥u_{1ε}^{n}∥_{k+2}^{θ_{1}})^{r/θ_{1}} (k + 2)^{σ},  (1.11)

where we set q = (k + p_{1})N/(N − p_{1}), θ_{1} = (k + 2) [q − (p_{1} + α_{1} + 1)] /(q − k − 2) , θ_{2} = q(α_{1} − 1)/(q − k − 2), r < k + 2 and σ > 0 is a constant independent

of k.

If 0 < α_{1} < 1, by Hölder’s inequality and Young’s inequality, we have

∥u_{1ε}^{n}∥_{k+α_{1}+1}^{k+α_{1}+1} ≤ ∥u_{1ε}^{n}∥_{k+2}^{k+2} + C ∥u_{1ε}^{n}∥_{k+2}^{k+1} .  (1.12)

If α_{1} = 0, by Hölder’s inequality , we have

∥u_{1ε}^{n}∥_{k+α_{1}+1}^{k+α_{1}+1} ≤ max (1, |Ω|^{\frac{1}{2}} ) ∥u_{1ε}^{n}∥_{k+2}^{k+1} .  (1.13)

It follows from (1.8) - (1.13) that

\frac{d}{dt} ∥u_{1ε}^{n}∥_{k+2}^{k+2} + c_{1}(k + 2)^{−(p_{1}−2)} ∥∇ (u_{1ε}^{n})^{(k+p_{1})/p_{1}} ∥_{p_{1}}^{p_{1}}

≤ c′ [(k + 2)^{σ+1} ∥u_{1ε}^{n}∥_{k+2}^{k+2} + 1] .  (1.14)

Setting k_{1} = p_{1}, k_{l} = p_{1}k_{l−1} − p_{1} + 2, δ_{l} = \frac{(k_{l}+p_{1}−2)}{θ_{l}} − k_{l} (> p_{1} − 2),

θ_{l} = \frac{N((p_{1}−1)k_{l}−p_{1}+2)}{k_{l}((p_{1}−1)N+p_{1})}. By lemma 2.1, we have

∥u_{1ε}^{n}∥_{k_{l}} ≤ c ∥u_{1ε}^{n}∥_{k_{l−1}}^{1−θ_{l}} ∥∇ (u_{1ε}^{n})^{(k_{l}+p_{1})/p_{1}} ∥_{p_{1}}^{θ_{l}p_{1}/(k_{l}+p_{1}−2)} .  (1.15)

Set k = k_{l} in (1.14) and by (1.15), we have

\frac{d}{dt} ∥u_{1ε}^{n}∥_{k_{l}}^{k_{l}}+c_{1}c^{−(k_{l}+p_{1}−2)θ_{l}^{−1}}k_{l}^{−p_{1}+2}. ∥u_{1ε}^{n}(t)∥_{k_{l−1}}^{(k_{l}+p_{1}−2)(θ_{l}−1)/θ_{l}} ∥u_{1ε}^{n}(t)∥_{k_{l}}^{(k_{l}+p_{1}−2)/θ_{l}}

≤ c [(k_{l})^{σ+1} ∥u_{1ε}^{n}(t)∥_{k_{l}}^{k_{l}} + 1] .  (1.16)

Therefore,

\frac{d}{dt} ∥u_{1ε}^{n}∥_{k_{l}} + c_{1}c^{−(k_{l}+p_{1}−2)θ_{l}^{−1}}k_{l}^{−p_{1}+1} ∥u_{1ε}^{n}(t)∥_{k_{l}}^{(p_{1}−1−δ_{l})} ∥u_{1ε}^{n}(t)∥_{k_{l}}^{δ_{l}+1}

≤ c [(k_{l})^{σ} ∥u_{1ε}^{n}∥_{k_{l}} + 1] .  (1.17)

Let λ_{l} = sup_{t} ∥u_{1ε}^{n}∥_{k_{l}} , by lemma 2.2, we obtain

λ_{l} ≤ max(1, R), R = \{c′c^{(k_{l}+p_{1}−2)θ_{l}^{−1}}k_{l}^{p_{1}−1+σ}.λ_{l−1}^{(p_{1}−1−δ_{l})}\}^{\frac{1}{δl}} .

By [7], we have {λ_{l}} bounded and we set without loss of generality that R > 1, which implies

sup_{t} ∥u_{1ε}^{n}∥_{∞} ≤ c_{0}.  (1.18)

The same holds also for u_{2ε}^{n}.■

Lemma 2.4. u_{iε}^{n} satisfies the following:

∫_{0}^{ω} ∥∇u_{iε}^{n}∥_{p_{i}}^{p_{i}} dt ≤ c_{i} < ∞,  (1.19)

∫_{0}^{ω} ∥\frac{∂u_{iε}^{n}}{∂t} ∥_{2}^{2} dt ≤ c_{i} < ∞.  (1.20)

Proof of lemma 2.4. Multiplying (1.1) by u_{1ε}^{n}, we get :

∫_{0}^{ω} ∫_{Ω} u_{1ε}^{n} \frac{∂u_{1ε}^{n}}{∂t} dxdt + ∫_{0}^{ω} ∫_{Ω} |∇u_{1ε}^{n}|^{p_{1}} dxdt + ∫_{0}^{ω} ∫_{Ω} a_{1}(u_{1ε}^{n}).∇u_{1ε}^{n}u_{1ε}^{n}dxdt

= ∫_{0}^{ω} ∫_{Ω} f(t)(u_{1ε}^{n})^{α_{1}+1}dxdt + ∫_{0}^{ω} ∫_{Ω} h_{1}(t, u_{2ε}^{n−1})dxdt.  (1.21)

By the periodicity, Hölder’s inequality, Poincare’s inequality and the L^{∞} norm bounded of u_{iε}^{n}, we have

∫_{0}^{ω} ∥∇u_{1ε}^{n}∥_{p_{1}}^{p_{1}} dt ≤ ∫_{0}^{ω} ∥f(t)∥_{r^{∗}} ∥u_{1ε}^{n}∥_{p_{1}}^{α_{1}+1} dt + ∫_{0}^{ω} ∥h(., t)∥_{p_{1}} ∥u_{1ε}^{n}∥_{p_{1}}^{p_{1}−1} dt,

≤ (∫_{0}^{ω} ∥f(t)∥_{r^{∗}}^{r^{∗}} dt)^{1/r^{∗}} (∫_{0}^{ω} ∥u_{1ε}^{n}∥_{p_{1}}^{p_{1}} dt)^{(α_{1}+1)/p_{1}}

+ (∫_{0}^{ω} ∥h(., t)∥_{p_{1}}^{p_{1}} dt)^{1/p_{1}} (∫_{0}^{ω} ∥u_{1ε}^{n}∥_{p_{1}}^{p_{1}} dt)^{(p_{1}−1)/p_{1}}

c [(∫_{0}^{ω} ∥∇u_{1ε}^{n}∥_{p_{1}}^{p_{1}} dt)^{(α_{1}+1)/p_{1}} + (∫_{0}^{ω} ∥∇u_{1ε}^{n}∥_{p_{1}}^{p_{1}} dt)^{(p_{1}−1)/p_{1}}] ,  (1.22)

in which r^{∗} = p_{1}/(p_{1} − 1 − α_{1}). Thus, we have

∫_{0}^{ω} ∥∇u_{1}^{n}∥_{p_{1}}^{p_{1}} dt ≤ c < ∞,  (1.23)

Multiplying (1.1) by \frac{∂u_{1ε}^{n}}{∂t} and integrating over [0, ω] × Ω, we get :

∫_{0}^{ω} ∥\frac{∂u_{1ε}^{n}}{∂t} (t)∥_{2}^{2} dt + ∫_{0}^{ω} ∫_{Ω} a_{1}(u_{1ε}^{n}).∇u_{1ε}^{n} \frac{∂u_{1ε}^{n}}{∂t} dxdt

≤ ∫_{0}^{ω} f(t)dt ∫_{Ω}(u_{1ε}^{n})^{α_{1}} \frac{∂u_{1ε}^{n}}{∂t} dx + ∫_{0}^{ω} ∫_{Ω} |h_{1}(t, u_{2ε}^{n−1})\frac{∂u_{1ε}^{n}}{∂t} | dxdt  (1.24)

Hence, by (H1), (1.24) and Young’s inequality, we have

∫_{0}^{ω} ∥\frac{∂u_{1ε}^{n}}{∂t} ∥_{2}^{2} dt ≤ c < ∞.  (1.25)

Whence lemma 2.4. Passing to the limit in ε → 0 and n → +∞ is almost the same in [9].

Proof of theorem 1.2

Multiplying (1.1) by −div(|∇u_{1ε}^{n}|^{k−2} ∇u_{1ε}^{n}) (k > p_{1}), and integrating over Ω, we have

\frac{1}{k} \frac{d}{dt} ∥∇u_{1ε}^{n}(t)∥_{k}^{k} + ∫_{Ω} div [(|∇u_{1ε}^{n}|^{2} + ε)^{\frac{p_{1}−2}{2}} ∇u_{1ε}^{n})] div(|∇u_{1ε}^{n}|^{k−2} ∇u_{1ε}^{n})dx

= ∫_{Ω} a_{1}(u_{1ε}^{n}).∇u_{1ε}^{n} div(|∇u_{1ε}^{n}|^{k−2} ∇u_{1ε}^{n})dx−∫_{Ω} f_{1}(t)(u_{1ε}^{n})^{α_{1}+1} div(|∇u_{1ε}^{n}|^{k−2} ∇u_{1ε}^{n})dx

− ∫_{Ω} h_{1}(x, u_{2ε}^{n−1}) div(|∇u_{1ε}^{n}|^{k−2} ∇u_{1ε}^{n})dx.  (1.26)

By hypothesis and Young’s inequality, we get

∫_{Ω} a_{1}(u_{1ε}^{n}).∇u_{1ε}^{n} div(|∇u_{1ε}^{n}|^{k−2} ∇u_{1ε}^{n})dx

≤ ∫_{Ω} |∇u_{1ε}^{n}|^{k+p_{1}−4} |D^{2}u_{1ε}^{n}|^{2} dx + ∫_{Ω} k^{2} |a_{1}(u_{1ε}^{n})|^{2} |∇u_{1ε}^{n}|^{k−p_{1}−2} dx

≤ ∫_{Ω} |∇u_{1ε}^{n}|^{k+p_{1}−4} |D^{2}u_{1ε}^{n}|^{2} dx + ck^{2}(1 + ∥∇u_{1ε}^{n}(t)∥_{k}^{k}),  (1.27)

− ∫_{Ω} f_{1}(t)(u_{1ε}^{n})^{α_{1}+1} div(|∇u_{1ε}^{n}|^{k−2} ∇u_{1ε}^{n})dx = − ∫_{Ω} ∇f_{1}. |∇u_{1ε}^{n}|^{k−2} ∇u_{1ε}^{n})dx

≤ c (∥∇u_{1ε}^{n}(t)∥_{k}^{k−1} + ∥∇u_{1ε}^{n}(t)∥_{k}^{k}) ,  (1.28)

− ∫_{Ω} h_{1}(x, u_{2ε}^{n−1}) div(|∇u_{1ε}^{n}|^{k−2} ∇u_{1ε}^{n})dx

= − ∫_{Ω} (∇_{x}h_{1}(x, u_{2ε}^{n−1}) + \frac{∂h_{1}(x, s)}{∂s}∇u_{2ε}^{n−1}) . |∇u_{1ε}^{n}|^{k−2} ∇u_{1ε}^{n})dx

≤ c ∥∇u_{1ε}^{n}(t)∥_{k}^{k−1} .  (1.29)

By integration by parts, we have

∫_{Ω} div [(|∇u_{1ε}^{n}|^{2} + ε)^{\frac{p_{i}−2}{2}} ∇u_{1ε}^{n})] div(|∇u_{1ε}^{n}|^{k−2} ∇u_{1ε}^{n})dx ≥ ∫_{Ω} |∇u_{1ε}^{n}|^{k+p_{1}−4} |D^{2}u_{1ε}^{n}|^{2} dx

≥ ∫_{Ω} |∇u_{1ε}^{n}|^{k+p_{1}−4} |D^{2}u_{1ε}^{n}|^{2} dx + \frac{p_{1} − 2}{4} ∫_{Ω} |∇u_{1ε}^{n}|^{k+p_{1}−6} |∇ (|∇u_{1ε}^{n}|^{2})|^{2} dx

−c(N − 1) ∫_{∂Ω} |∇u_{1ε}^{n}|^{k+p_{1}−2} H(x)dS,  (1.30)

where we note that |∇v| = |\frac{∂v}{∂n} | on ∂Ω and H(x) = (N −1)^{−1}((\frac{∂v}{∂n})^{−1} (△v− \frac{∂^{2}v}{∂n^{2}} ) is the mean curvature on ∂Ω. Using that Ω is convex so ∂Ω is of C^{2} and H(x) at x ∈ ∂Ω nonpositive with respect to the ouward normal. We have from (1.26)- (1.30) that

\frac{1}{k} \frac{d}{dt} ∥∇u_{1ε}^{n}(t)∥_{k}^{k} + \frac{c}{k} ∥|∇u_{1ε}^{n}|^{\frac{k+p_{1}−2}{2}} ∥_{1,2}^{2} ≤ ck^{2}(1 + ∥∇u_{1ε}^{n}(t)∥_{k}^{k}) + c ∥∇u_{1ε}^{n}(t)∥_{k}^{k−1}

+\frac{c}{k} ∫_{∂Ω} |∇u_{1ε}^{n}|^{k+p_{1}−2} dx.  (1.31)

By lemma 2.1 and Young’s inequality, we have

∥∇u_{1ε}^{n}(t)∥_{k+p_{1}−2}^{k+p_{1}−2} ≤ \frac{1}{2} ∥|∇u_{1ε}^{n}|^{\frac{k+p_{1}−2}{2}} ∥_{1,2}^{2} + c ∥∇u_{1ε}^{n}∥_{p_{1}}^{p_{1}−1} ∥∇u_{1ε}^{n}∥_{k}^{k−1} . (1.32)

Therefore, (1.31) can be rewritten as

\frac{d}{dt} ∥∇u_{1ε}^{n}(t)∥_{k}^{k} + c ∥|∇u_{1ε}^{n}|^{\frac{k+p_{1}−2}{2}} ∥_{1,2}^{2} ≤ ck^{3}(1 + ∥∇u_{1ε}^{n}(t)∥_{k}^{k}) + ck ∥∇u_{1ε}^{n}∥_{k}^{k−1} .

(1.33)

Then, setting

k_{1} = p_{1} − 2, k_{l} = 2k_{l−1} − p_{1} + 2, θ_{l} = \frac{2N}{N + 2}(1 − \frac{p_{l−1}}{p_{l}}), l = 2, 3, ........

Again by lemma 2.1, we have

∥∇u_{1ε}^{n}(t)∥_{p_{l}} ≤ c^{\frac{2}{p_{l}+p_{1}−2}} ∥∇u_{1ε}^{n}∥_{p_{l−1}}^{1−θ_{l}} ∥|∇u_{1ε}^{n}|^{\frac{p_{l}+p_{1}−2}{2}} ∥_{1,2}^{2θ_{l}/(p_{l}+p_{1}−2)} . (1.34)

Therefore, from (1.33) and (1.34), we obtain

\frac{d}{dt} ∥∇u_{1ε}^{n}(t)∥_{p_{l}}^{p_{l}} + cc^{−2/θ_{l}} ∥|∇u_{1ε}^{n}|∥_{1,2}^{(p_{l}+p_{1}−2)(θ_{l}−1)/θ_{l}} ∥∇u_{1ε}^{n}(t)∥_{p_{l}}^{(p_{l}+p_{1}−2)/θ_{l}})

≤ ck_{l}^{3}(1 + ∥∇u_{1ε}^{n}(t)∥_{k_{l}}^{k_{l}}) + ck_{l} ∥∇u_{1ε}^{n}∥_{k_{l}}^{k_{l}−1} .  (1.35)

Applying Lemma 2.2, by the same argument as in lemma 2.3, we can obtain (1.20).

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(Received March 15, 2009)