Global Solutions for a Nonlinear Wave Equation with the p-Laplacian Operator ^{∗}

Hongjun Gao Institute of Applied Physics and Computational Mathematics 100088 Beijing, China

To Fu Ma Departamento de Matemática - Universidade Estadual de Maringá 87020-900 Maringá-PR, Brazil

Abstract

We study the existence and asymptotic behaviour of the global solutions of the nonlinear equation

u_{tt} − Δ_{p}u + (−Δ)^{α}u_{t} + g(u) = f

where 0 < α ≤ 1 and g does not satisfy the sign condition g(u)u ≥ 0.

Key words: Quasilinear hyperbolic equation, asymptotic behaviour, small data.

AMS Subject Classification: 35B40, 35L70.

1. Introduction

The study of global existence and asymptotic behaviour for initial-boundary value problems involving nonlinear operators of the type

u_{tt} − ∑_{i=1}^{n}{σ_{i}(u_{x_{i}})}_{x_{i}} − Δu_{t} = f(t, x) in (0, T) × Ω

goes back to Greenberg, MacCamy & Mizel [3], where they considered the one-dimensional case with smooth data. Later, several papers have appeared in that direction, and some of its important results can be found in, for example, Ang & Pham Ngoc Dinh [1], Biazutti [2], Nakao [8], Webb [10] and Yamada [11]. In all of the above cited papers, the damping term −Δu_{t} played an essential role in order to obtain global solutions. Our objective is to study this kind of equations under a weaker damping given by (−Δ)^{α}u_{t} with 0 < α ≤ 1. This approach was early considered by Medeiros and Milla Miranda [6] to Kirchhoff equations. We also consider the presence of a forcing term g(x, u) that does not satisfy the sign condition g(x, u)u ≥ 0. Our study is based on the pseudo Laplacian operator

∗Work done while the authors were visiting LNCC/CNPq, Brazil.

−Δ_{p}u = − ∑_{i=1}^{n} \frac{∂}{∂x_{i}} (|\frac{∂u}{∂x_{i}} |^{p−2} \frac{∂u}{∂x_{i}} )

which is used as a model for several monotone hemicontinuous operators.

Let Ω be a bounded domain in ℝ^{n} with smooth boundary ∂Ω. We consider the nonlinear initial-boundary value problem

(1.1) \{ \begin{matrix}u_{tt} − Δ_{p}u + (−Δ)^{α}u_{t} + g(x, u) = f(t, x) in (0, T) × Ω, \\ u = 0 on (0, T) × ∂Ω, \\ u(0, x) = u_{0} and u_{t}(0, x) = u_{1} in Ω, \end{matrix}

where 0 < α ≤ 1 and p ≥ 2. We prove that, depending on the growth of g, problem (1.1) has a global weak solution without assuming small initial data. In addition, we show the exponential decay of solutions when p = 2 and algebraic decay when p > 2. The global solutions are constructed by means of the Galerkin approximations and the asymptotic behaviour is obtained by using a difference inequality due to M. Nakao [7]. Here we only use standard notations. We often write u(t) instead u(t, x) and u′(t) instead u_{t}(t, x). The norm in L^{p}(Ω) is denoted by ∥ · ∥_{p} and in W_{0}^{1,p}(Ω) we use the norm

∥u∥_{1,p}^{p} = ∑_{j=1}^{n} ∥u_{x_{j}}∥_{p}^{p}.

For the reader’s convenience, we recall some of the basic properties of the operators used here. The degenerate operator −Δ_{p} is bounded, monotone and hemicontinuous from W_{0}^{1,p}(Ω) to W^{−1,q}(Ω), where p^{−1} + q^{−1} = 1. The powers for the Laplacian operator is defined by

(−Δ)^{α}u = ∑_{j=1}^{∞} λ_{j}^{α}(u, φ_{j})φ_{j},

where 0 < λ_{1} < λ_{2} ≤ λ_{3} ≤ ⋯ and φ_{1}, φ_{2}, φ_{3}, ⋯ are, respectively, the sequence of the eigenvalues and eigenfunctions of −Δ in H_{0}^{1}(Ω). Then

∥u∥_{D((−Δ)^{α})} = ∥(−Δ)^{α}u∥_{2} ∀ u ∈ D((−Δ)^{α})

and D((−Δ)^{α}) ⊂ D((−Δ)^{β}) compactly if α > β ≥ 0. In particular, for p ≥ 2 and 0 < α ≤ 1, W_{0}^{1,p}(Ω) ↪ D((−Δ)^{α/2}) ↪ L^{2}(Ω).

2. Existence of Global Solutions

Let g : Ω × ℝ → ℝ be a continuous function satisfying the growth condition

|g(x, u)| ≤ a|u|^{σ−1} + b ∀(x, u) ∈ Ω × ℝ,  (2.1)

where a, b are positive constants, 1 < σ < pn/(n−p) if n > p and 1 < σ < ∞ if n ≤ p.

Theorem 2.1 Let us assume condition (2.1) with σ < p. Then given u_{0} ∈ W_{0}^{1,p}(Ω), u_{1} ∈ L^{2}(Ω) and f ∈ L^{2}(0, T; L^{2}(Ω)), there exists a function u : (0, T) × Ω → ℝ such that

u ∈ L^{∞}(0, T; W_{0}^{1,p}(Ω)),  (2.2)
u′ ∈ L^{∞}(0, T; L^{2}(Ω)) ∩ L^{2}(0, T; D((−Δ)^{\frac{α}{2}} )),  (2.3)
u(0) = u_{0} and u′(0) = u_{1} a.e. in Ω,  (2.4)
u_{tt} − Δ_{p}u + (−Δ)^{α}u_{t} + g(x, u) = f in L^{2}(0, T; W^{−1,q}(Ω)),  (2.5)

where p^{−1} + q^{−1} = 1.

Next we consider an existence result when σ ≥ p. In this case, the global solution is obtained with small initial data. For each m ∈ ℕ we put

γ_{m} = \frac{1}{2}∥u_{1m}∥_{2}^{2} + \frac{3}{2p}∥u_{0m}∥_{1,p}^{p} + \frac{aC_{σ}}{σ} ∥u_{0m}∥_{1,p}^{σ} + \frac{2( ^{p} \sqrt{2}bC_{1})^{q}}{q}

where C_{k} > 0 is the Sobolev constant for the inequality ∥u∥_{k} ≤ C_{k}∥u∥_{1,p}, when W_{0}^{1,p}(Ω) ↪ L^{k}(Ω). We also define the polynomial Q by

Q(z) = \frac{1}{2p}z^{p} − \frac{aC_{σ}}{σ} z^{σ},

which is increasing in [0, z_{0}], where

z_{0} = (2aC_{σ})^{\frac{−1}{σ−p}}

is its unique local maximum. We will assume that

∥u_{0}∥_{1,p} < z_{0}  (2.6)

and

γ + \frac{1}{4λ_{1}^{α}} ∫_{0}^{T} ∥f(t)∥_{2}^{2}dt < Q(z_{0}),  (2.7)

where γ = lim_{m→∞} γ_{m}.

Theorem 2.2 Suppose that condition (2.1) holds with σ > p. Suppose in addition that inital data satisfy (2.6) and (2.7). Then there exists a function u : (0, T) × Ω → ℝ satisfying (2.2)-(2.5).

Proof of Theorem 2.1: Let r be an integer for which H_{0}^{r}(Ω) ↪ W_{0}^{1,p}(Ω) is continuous. Then the eigenfunctions of −Δ^{r}w_{j} = α_{j}w_{j} in H_{0}^{r}(Ω) yields a “Galerkin” basis for both H_{0}^{r}(Ω) and L^{2}(Ω). For each m ∈ ℕ, let us put V_{m} = Span{w_{1}, w_{2}, ⋯, w_{m}}. We search for a function

u_{m}(t) = ∑_{j=1}^{m} k_{jm}(t)w_{j}

such that for any v ∈ V_{m}, u_{m}(t) satisfies the approximate equation

∫_{Ω}{u′′_{m}(t) − Δ_{p}u_{m}(t) + (−Δ)^{α}u′_{m}(t) + g(x, u_{m}(t)) − f(t, x)}v dx = 0 (2.8)

with the initial conditions

u_{m}(0) = u_{0m} and u′_{m}(0) = u_{1m},

where u_{0m} and u_{1m} are chosen in V_{m} so that

u_{0m} → u_{0} in W_{0}^{1,p}(Ω) and u_{1m} → u_{1} in L^{2}(Ω).  (2.9)

Putting v = w_{j}, j = 1, ⋯, m, we observe that (2.8) is a system of ODEs in the variable t and has a local solution u_{m}(t) in a interval [0, t_{m}). In the next step we obtain the a priori estimates for the solution u_{m}(t) so that it can be extended to the whole interval [0, T].

A Priori Estimates: We replace v by u′_{m}(t) in the approximate equation (2.8) and after integration we have

\frac{1}{2}∥u′_{m}(t)∥_{2}^{2} + \frac{1}{p}∥u_{m}(t)∥_{1,p}^{p} + ∫_{0}^{t} ∥(−Δ)^{\frac{α}{2}} u′_{m}(s)∥_{2}^{2} ds + ∫_{Ω} G(x, u_{m}(t))dx

≤ ∫_{0}^{t} ∥f(s)∥_{2}∥u′_{m}(s)∥_{2} ds + \frac{1}{2}∥u_{1m}∥_{2}^{2} + \frac{1}{p}∥u_{0m}∥_{1,p}^{p} + ∫_{Ω} G(x, u_{0})dx,

where G(x, u) = ∫_{0}^{u} g(x, s) ds. Now from growth condition (2.1) and the Sobolev embedding, we have that

∫_{Ω} |G(x, u_{m}(t))|dx ≤ \frac{a}{σ} C_{σ}∥u_{m}(t)∥_{1,p}^{σ} + bC_{1}∥u_{m}(t)∥_{1,p}.  (2.10)

But since p > σ, there exists a constant \overline{C} > 0 such that

∫_{Ω} |G(x, u_{m})|dx ≤ \frac{1}{2p}∥u_{m}(t)∥_{1,p}^{p} + \overline{C},

and then we have

\frac{1}{2}∥u′_{m}(t)∥_{2}^{2} + \frac{1}{2p}∥u_{m}(t)∥_{1,p}^{p} + ∫_{0}^{t} ∥(−Δ)^{\frac{α}{2}} u′_{m}(s)∥_{2}^{2} ds

≤ ∫_{0}^{t} ∥f(s)∥_{2}∥u′_{m}(s)∥_{2} ds + \frac{1}{2}∥u_{1m}∥_{2}^{2} + \frac{3}{2p}∥u_{0m}∥_{1,p}^{p} + 2\overline{C}.

Using the convergence (2.9) and the Gronwall’s lemma, there exists a constant C > 0 independent of t, m such that

∥u′_{m}(t)∥_{2}^{2} + ∥u_{m}(t)∥_{1,p}^{p} + ∫_{0}^{t} ∥(−Δ)^{\frac{α}{2}} u′_{m}(s)∥_{2}^{2} ds ≤ C.  (2.11)

With this estimate we can extend the approximate solutions u_{m}(t) to the interval [0, T] and we have that

(u_{m}) is bounded in L^{∞}(0, T; W_{0}^{1,p}(Ω)),  (2.12)
(u′_{m}) is bounded in L^{∞}(0, T; L^{2}(Ω)),  (2.13)
(u′_{m}) is bounded in L^{2}(0, T; D((−Δ)^{\frac{α}{2}} )),  (2.14)

and

(−Δ_{p}u_{m}) is bounded in L^{∞}(0, T; W^{−1,q}(Ω)).  (2.15)

Now we are going to obtain an estimate for (u′′_{m}). Since our Galerkin basis was taken in the Hilbert space H^{r}(Ω) ⊂ W_{0}^{1,p}(Ω), we can use the standard projection arguments as described in Lions [4]. Then from the approximate equation and the estimates (2.12)-(2.15) we get

(u′′_{m}) is bounded in L^{2}(0, T; H^{−r}(Ω)).  (2.16)

Passage to the Limit: From (2.12)-(2.14), going to a subsequence if necessary, there exists u such that

u_{m} ⇀ u weakly star in L^{∞}(0, T; W_{0}^{1,p}(Ω)),  (2.17)
u′_{m} ⇀ u′ weakly star in L^{∞}(0, T; L^{2}(Ω)),  (2.18)
u′_{m} ⇀ u′ weakly in L^{2}(0, T; D((−Δ)^{\frac{α}{2}} ),  (2.19)

and in view of (2.15) there exists χ such that

−Δ_{p}u_{m} ⇀ χ weakly star in L^{∞}(0, T; W^{−1,q}(Ω)).  (2.20)

By applying the Lions-Aubin compactness lemma we get from (2.12)-(2.13)

u_{m} → u strongly in L^{2}(0, T; L^{2}(Ω)),  (2.21)

and since D((−Δ)^{\frac{α}{2}} ) ↪ L^{2}(Ω) compactly, we get from (2.14) and (2.16)

u′_{m} → u′ strongly in L^{2}(0, T; L^{2}(Ω)).  (2.22)

Using the growth condition (2.1) and (2.21) we see that

∫_{0}^{T} ∫_{Ω} |g(x, u_{m}(t, x))|^{\frac{σ}{σ−1}} dxdt

is bounded and

g(x, u_{m}) → g(x, u) a.e. in (0, T) × Ω.

Therefore from Lions [4] (Lemma 1.3) we infer that

g(x, u_{m}) ⇀ g(x, u) weakly in L^{\frac{σ}{σ−1}} (0, T; L^{\frac{σ}{σ−1}} (Ω)).  (2.23)

With these convergence we can pass to the limit in the approximate equation and then

\frac{d}{dt}(u′(t), v) + ⟨χ(t), v⟩ + ((−Δ)^{α}u′(t), v) + (g(x, u(t)), v) = (f(t), v) (2.24)

for all v ∈ W^{1,p}(Ω), in the sense of distributions. This easily implies that (2.2)-(2.4) hold. Finally, since we have the strong convergence (2.22), we can use a standard monotonicity argument as done in Biazutti [2] or Ma & Soriano [5] to show that χ = −Δ_{p}u. This ends the proof. ⊏ ⊐ Proof of Theorem 2.2: We only show how to obtain the estimate (2.11). The remainder of the proof follows as before. We apply an argument made by L. Tartar [9]. From (2.10) we have

∫_{Ω} |G(x, u_{m})|dx ≤ \frac{aC_{σ}}{σ} ∥u_{m}∥_{1,p}^{σ} + \frac{1}{2p}∥u_{m}∥_{1,p}^{p} + \frac{( ^{p} \sqrt{2}bC_{1})^{q}}{q}

and therefore

\frac{1}{2}∥u′_{m}(t)∥_{2}^{2} + Q(∥u_{m}(t)∥_{1,p}) + ∫_{0}^{t} ∥(−Δ)^{\frac{α}{2}} u′_{m}(s)∥_{2}^{2} ds

≤ γ_{m} + ∫_{0}^{t} ∥f(s)∥_{2}∥u′_{m}(s)∥_{2} ds.

Since

∥(−Δ)^{\frac{α}{2}} u′_{m}(s)∥_{2} ≥ λ1^{\frac{α}{2}} ∥u′_{m}(t)∥_{2},  (2.25)

where λ_{1} is the first eigenvalue of −Δ in H_{0}^{1}(Ω), this implies that

\frac{1}{2}∥u′_{m}(t)∥_{2}^{2} + Q(∥u_{m}(t)∥_{1,p}) ≤ γ_{m} + \frac{1}{4λ_{1}^{α}} ∫_{0}^{T} ∥f(t)∥_{2}^{2} dt.  (2.26)

We claim that there exists an integer N such that

∥u_{m}(t)∥_{1,p} < z_{0} ∀ t ∈ [0, t_{m}) m > N  (2.27)

Suppose the claim is proved. Then Q(∥u_{m}(t)∥_{1,p}) ≥ 0 and from (2.7) and (2.26), ∥u′_{m}(t)∥_{2} is bounded and consequently (2.11) follows.

Proof of the Claim: Suppose (2.27) false. Then for each m > N, there exists t ∈ [0, t_{m}) such that ∥u_{m}(t)∥_{1,p} ≥ z_{0}. We note that from (2.6) and (2.9) there exists N_{0} such that

∥u_{m}(0)∥_{1,p} < z_{0} ∀ m > N_{0}.

Then by continuity there exists a first t_{m}^{∗} ∈ (0, t_{m}) such that

∥u_{m}(t_{m}^{∗})∥_{1,p} = z_{0},  (2.28)

from where

Q(∥u_{m}(t)∥_{1,p}) ≥ 0 ∀ t ∈ [0, t_{m}^{∗}].

Now from (2.7) and (2.26), there exist N > N_{0} and β ∈ (0, z_{0}) such that

0 ≤ \frac{1}{2}∥u′_{m}(t)∥_{2}^{2} + Q(∥u_{m}(t)∥_{1,p}) ≤ Q(β) ∀ t ∈ [0, t_{m}^{∗}] ∀ m > N.

Then the monotonicity of Q in [0, z_{0}] implies that

0 ≤ ∥u_{m}(t)∥_{1,p} ≤ β < z_{0} ∀t ∈ [0, t_{m}^{∗}],

and in particular, ∥u_{m}(t_{m}^{∗})∥_{1,p} < z_{0}, which is a contradiction to (2.28). ⊏ ⊐

Remarks: From the above proof we have the following immediate conclusion: The smallness of initial data can be dropped if either condition (2.1) holds with σ = p and coefficient a is sufficiently small, or σ > p and the sign condition g(x, u)u ≥ 0 is satisfied. ⊏ ⊐

3. Asymptotic Behaviour

Theorem 3.1 Let u be a solution of Problem (1.1) given by

(a) either Theorem 2.1 with the additional assumption: there exists ρ > 0 such that

g(x, u)u ≥ ρG(x, u) ≥ 0,  (3.1)

(b) or Theorem 2.2. Then there exists positive constants C and θ such that

∥u′(t)∥_{2}^{2} + ∥u(t)∥_{1,p}^{p} ≤ C exp(−θt) if p = 2,

or

∥u′(t)∥_{2}^{2} + ∥u(t)∥_{1,p}^{p} ≤ C(1 + t)^{\frac{−p}{p−2}} if p > 2.

The proof of Theorem 3.1 is based on the following difference inequality of M. Nakao [7].

Lemma 3.1 (Nakao) Let ϕ : R^{+} → R be a bounded nonnegative function for which there exist constants β > 0 and γ ≥ 0 such that

sup_{t≤s≤t+1}(ϕ(s))^{1+γ} ≤ β(ϕ(t) − ϕ(t + 1)) ∀ t ≥ 0.

Then

(i) If γ = 0, there exist positive constants C and θ such that

ϕ(t) ≤ C exp(−θt) ∀ t ≥ 0.

(ii) If γ > 0, there exists a positive constant C such that

ϕ(t) ≤ C(1 + t)^{\frac{−1}{γ}} ∀ t ≥ 0.

Let us define the approximate energy of the system (1.1) by

E_{m}(t) = \frac{1}{2}∥u′_{m}(t)∥_{2}^{2} + \frac{1}{p}∥u_{m}(t)∥_{1,p}^{p} + ∫_{Ω} G(x, u_{m}(t))dx.  (3.2)

Then the following two lemmas hold.

Lemma 3.2 There exists a constant k > 0 such that

kE_{m}(t) ≤ ∥u_{m}(t)∥_{1,p}^{p} + ∫_{Ω} g(x, u_{m}(t))u_{m}(t)dx.  (3.3)

Proof. In the case (a), condition (3.3) is a direct consequence of (3.1). In the case (b) the result follows from (2.27). In fact, from assumption (2.1) with b = 0 we have

∫_{Ω} |G(x, u_{m})|dx ≤ \frac{aC_{σ}}{σ} ∥u_{m}∥_{1,p}^{σ} and ∫_{Ω} |g(x, u_{m})u_{m}|dx ≤ aC_{σ}∥u_{m}∥_{1,p}^{σ}.

Then given δ > 0,

∥u_{m}∥_{1,p}^{p} + ∫_{Ω} g(x, u_{m})u_{m}dx = \frac{δ}{p}∥u_{m}∥_{1,p}^{p} + δ ∫_{Ω} G(x, u_{m})dx

−δ ∫_{Ω} G(x, u_{m})dx + (1 − \frac{δ}{p})∥u_{m}∥_{1,p}^{p} + ∫_{Ω} g(x, u_{m})u_{m} dx

implies that

∥u_{m}(t)∥_{1,p}^{p} + ∫_{Ω} g(x, u_{m}(t))u_{m}(t)dx ≥ \frac{δ}{p}∥u_{m}(t)∥_{1,p}^{p} + δ ∫_{Ω} G(x, u_{m}(t))dx

+(1 − \frac{δ}{p})∥u_{m}(t)∥_{1,p}^{p} − (1 + \frac{δ}{σ})aC_{σ}∥u_{m}(t)∥_{1,p}^{σ}.

Now, since ∥u_{m}(t)∥_{1,p} ≤ (2aC_{σ})^{\frac{−1}{σ−p}} = z_{0} uniformly in t, m, we have that ∥u_{m}(t)∥_{1,p}^{σ−p} ≤ (2aC_{σ})^{−1}. Then taking δ ≤ (σp)(p + 2σ)^{−1}, we conclude that

(1 − \frac{δ}{p})∥u_{m}(t)∥_{1,p}^{p} − (1 + \frac{δ}{σ})aC_{σ}∥u_{m}(t)∥_{1,p}^{σ}

= [(1 − \frac{δ}{p}) − (1 + \frac{δ}{σ})aC_{σ}∥u_{m}(t)∥_{1,p}^{σ−p}] ∥u_{m}(t)∥_{1,p}^{p} ≥ 0.

This implies that

∥u_{m}(t)∥_{1,p}^{p} + ∫_{Ω} g(x, u_{m}(t))u_{m}(t)dx ≥ \frac{δ}{p}∥u_{m}(t)∥_{1,p}^{p} + δ ∫_{Ω} G(x, u_{m}(t))dx

and therefore (3.3) holds. ⊏ ⊐

Lemma 3.3 For any t > 0,

E_{m}(t) ≥ \frac{1}{2}∥u′_{m}(t)∥_{2}^{2} + \frac{1}{2p}∥u_{m}(t)∥_{1,p}^{p}.

Proof. In the case (a), the Lemma is a consequence of (3.1). In the case (b), we use again the smallness of the approximate solutions.

E_{m}(t) ≥ \frac{1}{2}∥u′_{m}(t)∥_{2}^{2} + \frac{1}{p}∥u_{m}(t)∥_{1,p}^{p} − \frac{aC_{σ}}{σ} ∥u_{m}(t)∥_{1,p}^{σ}
= \frac{1}{2}∥u′_{m}(t)∥_{2}^{2} + \frac{1}{2p}∥u_{m}(t)∥_{1,p}^{p} − Q(∥u_{m}(t)∥_{1,p}).

Since Q(∥u_{m}(t)∥_{1,p}) ≥ 0, the result follows. ⊏ ⊐

Proof of Theorem 3.1: We first obtain uniform estimates for the approximate energy (3.2). Fix an arbitrary t > 0, we get from the approximate problem (2.8) with f = 0 and v = u′_{m}(t)

E_{m}(t) + ∫_{0}^{t} ∥(−Δ)^{\frac{α}{2}} u′_{m}(s)∥_{2}^{2}ds = E_{m}(0),

from where

E′_{m}(t) + ∥(−Δ)^{\frac{α}{2}} u′_{m}(t)∥_{2}^{2} = 0.  (3.4)

Integrating (3.4) from t to t + 1 and putting

D_{m}^{2}(t) = E_{m}(t) − E_{m}(t + 1)

we have in view of (2.25)

D_{m}^{2}(t) = ∫_{t}^{t+1} ∥(−Δ)^{\frac{α}{2}} u′_{m}(s)∥_{2}^{2} ds ≥ λ_{1}^{α} ∫_{t}^{t+1} ∥u′_{m}(s)∥_{2}^{2} ds.  (3.5)

By applying the Mean Value Theorem in (3.5), there exist t_{1} ∈ [t, t + \frac{1}{4}] and t_{2} ∈ [t + \frac{3}{4}, t + 1] such that

∥u′_{m}(t_{i})∥_{2} ≤ \sqrt{2}λ_{1}^{− \frac{α}{2}} D_{m}(t) i = 1, 2.  (3.6)

Now, integrating the approximate equation with v = u_{m}(t) over [t_{1}, t_{2}], we infer from Lemma 3.2 that

k ∫_{t_{1}}^{t_{2}} E_{m}(s) ds ≤ (u′_{m}(t_{1}), u_{m}(t_{1})) − (u′_{m}(t_{2}), u_{m}(t_{2}))

+ ∫_{t_{1}}^{t_{2}} ∥u′_{m}(s)∥_{2}^{2} ds − ∫_{t_{1}}^{t_{2}}((−Δ)^{\frac{α}{2}} u′_{m}(s), (−Δ)^{\frac{α}{2}} u_{m}(s))ds.

Then from Lemma 3.3, Hölder’s inequality and Sobolev embeddings, we have in view of (3.5) and (3.6), there exist positive constants C_{1} and C_{2} such that

∫_{t_{1}}^{t_{2}} E_{m}(s)ds ≤ C_{1}D_{m}^{2}(t) + C_{2}D_{m}(t)Em^{\frac{1}{p}}(t).

By the Mean Value Theorem, there exists t^{∗} ∈ [t_{1}, t_{2}] such that

E_{m}(t^{∗}) ≤ C_{1}D_{m}^{2}(t) + C_{2}D_{m}(t)Em^{\frac{1}{p}}(t).

The monotonicity of E_{m} implies that

E_{m}(t + 1) ≤ C_{1}D_{m}^{2}(t) + C_{2}D_{m}(t)Em^{\frac{1}{p}}(t).

Since E_{m}(t + 1) = E_{m}(t) − D_{m}^{2}(t), we conclude that

E_{m}(t) ≤ (C_{1} + 1)D_{m}^{2}(t) + C_{2}D_{m}(t)Em^{\frac{1}{p}}(t).

Using Young’s inequality, there exist positive constants C_{3} and C_{4} such that

E_{m}(t) ≤ C_{3}D_{m}^{2}(t) + C_{4}Dm^{\frac{p}{p−1}} (t).  (3.7)

If p = 2 then

E_{m}(t) ≤ (C_{3} + C_{4})D_{m}^{2}(t)

and since E_{m} is decreasing, then from Lemma 3.1 there exist positive constants C and θ (independent of m) such that

E_{m}(t) ≤ C exp(−θt) ∀ t > 0.  (3.8)

If p > 2, then relation (3.7) and the boundedness of D_{m}(t) show the existence of C_{5} > 0 such that

E_{m}(t) ≤ C_{5}Dm^{\frac{p}{p−1}} (t),

and then

Em^{\frac{2(p−1)}{p}} (t) ≤ C5^{\frac{2(p−1)}{p}} D_{m}^{2}(t).

Applying Lemma 3.1, with γ = (p − 2)/p, there exists a constant C > 0 (independent of m) such that

E_{m}(t) ≤ C(1 + t)^{\frac{−p}{p−2}} ∀ t ≥ 0.  (3.9)

Finally we pass to the limit (3.8) and (3.9) and the proof is complete in view of Lemma 3.3. ⊏ ⊐

Acknowledgement: The authors thank Professor J. E. Muñoz Rivera for his hospitality during their stay in LNCC/CNPq.

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