Variable exponent perturbation of a parabolic equation with p ( x ) -Laplacian

Variable exponent perturbation of a parabolic equation with p ( x ) _-Laplacian

Aldo T. Lourêdo

, Manuel Milla Miranda

and Marcondes R. Clark

1Universidade Estadual da Paraíba, R. Baraúnas, 351 - C. Grande - PB, CEP 58429-500, Brasil

2Universidade Federal do Piauí, B. Ininga - Teresina - PI, CEP 64049-550, Brasil

Received 26 March 2019, appeared 14 August 2019 Communicated by Vilmos Komornik

Abstract. This paper is concerned with the study of the global existence and the decay of solutions of an evolution problem driven by an anisotropic operator and a nonlinear perturbation, both of them having a variable exponent. Because the nonlinear pertur- bation leads to difficulties in obtaining a priori estimates in the energy method, we had to significantly modify the Tartar method. As a result, we could prove the existence of global solutions at least for small initial data. The decay of the energy is derived by using a differential inequality and applying a non-standard approach.

Keywords: variable exponent, energy not defined, existence of solutions.

2010 Mathematics Subject Classification: 35K92, 35K58, 35K55, 35L60, 35L70.

1 Introduction

Let Ω be an open bounded set ofRⁿ with boundary Γ of class C². Consider p,σ ∈ L^∞(_Ω). The objective of this paper is to analyze the global existence and the decay of solutions of the following parabolic problem:

u⁰−

∑

∂

∂x_i

∂u

∂x_i

p(x)−2

∂u

∂x_i

!

+|u|^σ(x)=0 inΩ×(0,∞), (1.1a) u=0 onΓ×(0,∞) (1.1b) u(x, 0) =u⁰(x) _{inΩ. (1.1c)} The p(x)-Laplacian operatorAgiven byAu =− _∑ⁿ_i=1∂x^∂

i

_∂x^∂u

i

p(x)−2∂u

∂x_i

, arises in some phys- ical problems. For example, in the theory of elasticity and in mechanics of fluids, more pre- cisely, in fluids of electrorheological type (see [8,19,20]), whose equation of motion is given by

u⁰+divS(u) + (u· ∇u) +∇π = f,

BCorresponding author. Email: aldolouredo@gmail.com

whereu: R³⁺¹ →_R³ is the velocity of the fluid at a point in space-time ,∇ = (∂₁,∂₂,∂₃)the gradient operator,π:R³⁺¹ →_Rthe pressure, f :R³⁺¹→_R³represents external forces andS is the stress tensorS:W_loc^1,1→_R^3×3. This operator has the form

S(u)(x) =µ(x)(1+|D(u(x))|^p(x2)−2)D(u(x))

whereD(u) = ¹₂(∇u+ (∇u)^T)is the symmetric part of the gradient ofu. Note that ifp(x)≡2, then this equation reduces to the usual Navier–Stokes equation.

To obtain the existence of global solutions of (1.1a)–(1.1c) we cannot apply the energy method because the termR

Ω|u(x)|^σ(x)u(x)dxdoes not have a definite sign. To overcome this difficulty we apply a new method which has its motivation in the work of Tartar [24] (see also [17]). With this approach and results on monotone operators (see [6,7,25]) we succeed in obtaining a global solution of (1.1a) with small initial data. This is the main contribution of the paper. The decay of the energy is derived by using differential inequalities and applying a new approach.

Problem (1.1a) is an example of an evolution problem driven by an anisotropic operator with variable exponents and a nonlinear perturbation, which has also a variable exponent.

Recent contributions to the study of anisotropic problems can be found, for instance, in [14,20]

and the references contained therein. Parabolic problems with variable exponents can be seen in [3,4,10–12,18]. In [2], Antontsev analized the wave equation with p(x,t)-Laplacian. In [5], those authors considered the energy decay for a class of plate equations with memory and a lower order perturbation of p-Laplacian type. We can find elliptic problems with operators having variable exponents in [1,22] and the references contained therein. Because the energy method works very well, the proof of the existence of a solution in those papers is based on the Galerkin method.

The paper is organized as following. In Section 2, we introduce notation and state the results in form of theorems, whose proofs are given in Section3.

2 Notations and main results

The scalar product and norm ofL²(_Ω)are denoted by(u,v)and|u|, respectively. Consider a functionq∈ L^∞(_Ω)with ess inf_x∈_Ωq(x) =q⁻≥1. The space

L^q(x)(_Ω) =

u:uis a measurable real-valued function, Z

Ω|u(x)|^q(x)dx<_∞

, equipped with the Luxemburg’s norm

kuk_Lq(·)(_Ω) =inf

λ>0 : Z

Ω

u(x) λ

q(x)

dx≤1

is a Banach space. With the notationq⁺=ess inf_x∈_Ωq(x)and the fact 1≤q⁻≤q(x)≤q⁺a.e.

inx∈ _Ω, we have

kuk^q+

L^q(·)(_Ω) ≤

Z

Ω|u(x)|^q(x)dx ≤ kuk^q−

L^q(·)(_Ω) ifkuk_Lq(·)(_Ω) ≤1; (2.1a) kuk^q−

L^q(·)(_Ω) ≤

Z

Ω|u(_x)|^q(x)dx ≤ kuk^q+

L^q(·)(_Ω) ifkuk_Lq(·)(_Ω) >_{1. (2.1b)}

Assume that

p∈C(_Ω), pis Lipschitzian and p(x)≥2, ∀x∈ _Ω; (2.2a)

σ ∈C(_Ω), σ(x)>1, ∀x∈_Ω. (2.2b)

Introduce the notations p⁻=min

x∈_Ω p(x), p⁺=max

x∈_Ω p(x), σ⁻=min

x∈_Ωσ(x), σ⁺=max

x∈_Ω σ(x). Thus

2≤ p⁻≤ p(x)≤ p⁺ and 1<σ⁻≤ σ(x)≤σ⁺. (2.3) The space

W^1,p(x)(_Ω) =

u∈ L^p(x)(_Ω): ∂u

∂x_i ∈ L^p(x)(_Ω),i=1, 2, . . . ,n

, provided with the norm

kuk_W1,p(·)(_Ω)= kuk_Lp(·)(_Ω)+k∇uk_Lp(·)(_Ω), u∈W^1,p(·)(_Ω).

is a reflexive Banach space. The closure ofC₀^∞(_Ω)inW^1,p(x)(_Ω)is denoted byW₀^1,p(x)(_Ω). This reflexive Banach space is equipped with the norm

kuk

W₀^1,p(·)(_Ω) =

∑

∂u

∂x_i L^p(·)(_Ω)

. The dual space ofW^1,p(x)(_Ω)is denoted byW^−1,p0(x)(_Ω)_{, where}

1

p(x)+ ¹

p⁰(x) =1, ∀x∈_Ω.

Let denote byAthe operator

A:W₀^1,p(x)(_Ω)→W^−1,p0(x)(_Ω) defined by

hAu,vi=

∑

Z

Ω

∂u

∂x_i

p(x)−2

∂u

∂x_i

∂v

∂x_idx.

It is known that Ais monotone and hemicontinuous (see Diening [9]).

Proposition 2.1. The operator A takes bounded subsets of W₀^1,p(x)(_Ω) into bounded subsets of W₀^−1,p0(x)(_Ω).

Proof. In fact, it holds that

|hAu,vi| ≤

∑

Z

Ω

∂u

∂x_i

p(x)−1

∂v

∂x_i

dx.

In order to facilitate the notations, we denote the spaceW₀^1,p(x)(_Ω)_byX.

Note that ^∂u

∂xi

∈ L

p(·)

p(·)−1(_Ω)since _∂x^∂u

i ∈ L^p(·)(_Ω). So by the Hölder inequality for the spaces L^p(·)(_Ω)(cf. [13, p. 341]), we obtain the estimate

Z

Ω

∂u

∂x_i

p(x)−1

∂v

∂x_i

dx≤2

∂u

∂x_i

p(x)−1 L

p(·) p(·)−1(_Ω)

∂v

∂x_i L^p(·)(_Ω)

≤2

∂u

∂x_i

p(x)−1 L

p(·) p(·)−1(_Ω)

kvk_X. Therefore, from the above two inequalities we have

|hAu,vi| ≤2





∑

∂u

∂x_i

p(x)−1 L

p(·) p(·)−1(_Ω)



kvk_X. (2.4)

Let us defineαandβas follows:

p(x) p(x)−1

−

=_α,

p(x) p(x)−1

+

= _β.

Ifl_i =

∂u

∂xi

p(x)−1 L

p(·) p(·)−1(_Ω)

≤1, by (2.1a), we obtain

l_i^β ≤

Z

Ω

∂u

∂x_i

p(x)

dx≤

∂u

∂x_i

p⁻ L^p(·)(_Ω)

+

∂u

∂x_i

p⁺

L^p(·)(_Ω)

≤ kuk^p_X⁻+kuk_X^p+. Thus,

l_i ≤kuk^p_X⁻+kuk_X^p+

1 β. In similar way, ifl_i >1 we find

l_i ≤kuk_X^p−+kuk_X^p+

1 α. These last two inequalities imply

∑

l_i ≤n

kuk^p_X⁻+kuk_X^p+

1

β +n

kuk^p_X⁻+kuk_X^p+

1 α. Now, this inequality and (2.4) provide

kAuk_W−1,p0(x)(_Ω) ≤2n

kuk^p_X⁻+kuk_X^p+

1

β +2n

kuk_X^p−+kuk_X^p+

1 α

which proves the proposition.

We also assume that

(p⁺−p⁻)n< p⁺p⁻, (2.5a) p⁺< σ⁻+1≤σ(x) +1≤σ⁺+1< ^np(x)

n−p(x)^, ∀x∈_Ω (2.5b)

if p(x)<n, for allx∈ Ω; and that

σsatisfies hypothesis (2.2b) (2.6)

if p(x)≥nfor allx ∈_Ω.

Note that by (2.5b) we have

p⁺< ^np(x)

n−p(x)^, ∀x∈ _Ω.

Under the hypotheses (2.2a), (2.5a) and (2.5b), we obtain

W₀^1,p(x)(_Ω),→ L^σ++1(_Ω),→L^σ(x)+1(_Ω),→ L^σ−+1(_Ω),→L²(_Ω) (2.7) where,→ denotes continuous embedding. Note that

the embeddding ofW₀^1,p(x)(_Ω)in L^σ++1(_Ω) is compact. (2.8) See Diening et al. [9], Fan and Zhao [13], R˘adulescu et al. [20] and Kováˇcik and Rákosník [23] for detailed proofs of all these results on spaces with variable exponents that we have used in the present paper.

By (2.7) there exists a positive constantKsuch that kuk_Lσ(·)+1(_Ω)≤Kkuk

W₀^1,p(·)(_Ω), ∀u∈W₀^1,p(·)(_Ω)_{. (2.9)} Consider positive constantsa₀,a₁,b₀andb₁ satisfying

a₀

∑

|ξ_i|^p+

! ¹

p+

≤

∑

|ξ_i| ≤a₁

∑

|ξ_i|^p+

! ¹

p+

, ∀ξ = (ξ₁, . . . ,ξ_n)∈_Rⁿ; (2.10a)

b₀

∑

|ξ_i|^p−

! ¹

p−

≤

∑

|ξ_i| ≤b₁

∑

|ξ_i|^p−

! ¹

p−

, ∀ξ = (ξ₁, . . . ,ξ_n)∈_Rⁿ. (2.10b) Further, set the notations

d= ¹

2p⁺a₁^p+, M=K^σ−+1+K^σ++1 (2.11a) λ0=min

( 1,

dp⁺ M(σ⁻+1)

¹

σ−+1−p+)

. (2.11b)

Under the above considerations we have the following result.

Theorem 2.2. Assume that hypotheses(2.2a),(2.5a)and(2.6)hold. If u⁰ ∈W₀^1,p(·)(_Ω)satisfies ku⁰k

W₀^1,p(·)(_Ω)< λ₀, (2.12) and

1 b₀^p−

+ ¹ a^p₀⁺

!

ku⁰k^p−

W₀^1,p(·)(_Ω)+Mku⁰k^σ−+1

W₀^1,p(·)(_Ω) <dλ^p₀⁺. (2.13) Then there exists a function u ∈L^∞(0,∞;W₀^1,p(·)(_Ω)),with u⁰ ∈ L²(0,∞;L²(_Ω))that satisfies

u⁰−

∑

∂

∂x_i

∂u

∂x_i

p(·)−2

∂u

∂x_i

!

+|u|^σ(·) =0 in L²_loc(0,∞;W^−1,p0(·)(_Ω)), (2.14)

u(₀) =u⁰ inΩ. (2.15)

Remark 2.3. We note that in the particular case p(x) = c, c ≥ 2, hypothesis (2.5a) is always holds. Thus by applying the same method used to prove Theorem 2.2, we obtain global solutions to Problem (1.1a)–(1.1c) under only the hypotheses (2.5a) and

σ⁺+1< ^np

n−p if p< n no restriction onσ if p≥ n.

In order to state the result of the decay of solutions, we introduce some notations.

By (2.7), there exists a constantL>0 such that

|v| ≤Lkvk

W₀^1,p(·)(_Ω), ∀v∈W₀^1,p(·)(_Ω), (2.16) where| · |=| · |_L2(_Ω). Set the notation

η= ¹ a₁^p+L^p+.

Letube the solution given by Theorem2.2. Define the energyE(t)by

E(t) =|u(t)|², ∀t≥0. (2.17) Byu∈ L^∞(0,∞;W₀^1,p(·)(_Ω))andu⁰ ∈ L²(0,∞;L²(_Ω)), we have thatE∈ C([0,∞);L²(_Ω)). Theorem 2.4. Let u be the solution given by Theorem2.2. Then

(i) if p⁺=_2,that is, p(x) =_2,∀x∈ _Ω,we have

E(t)≤E(0)e^−ηt, ∀t ≥0; (2.18) (ii) if p⁺>_2,we set ^p₂⁺ =₁+_γ,γ>_0.In this case we have

E(t)≤E(0)(1+E(0)^γηγt)^−1γ, ∀t ≥0. (2.19)

3 Proof of the results

Proof of Theorem2.2. Consider a Schauder basis {w₁,w2,w3, . . .} of W₀^1,p(x)(_Ω). Let um be an approximate solution of Problem (1.1a)–(1.1c), more precisely,

u_m(x,t) =

∑

g_jm(t)w_j(x),

(u⁰_m(t),v) +

∑

Z

Ω

∂um(t)

∂x_i

p(·)−2

∂um(t)

∂x_i

∂v

∂x_idx +

Z

Ω|u_m(t)|^σ(·)vdx=0, for all v∈V_m = [w₁, . . . ,w_m]; um(0) =u⁰_m, u⁰_m ∈Vm, u⁰_m →u⁰ inW₀^1,p(x)(_Ω).

(3.1)

We denote by[_0,t_m)the maximal interval of existence of the solutionu_m.

By (2.12) and (2.13), we obtain

ku⁰_mk_X <λ₀, ∀m≥m₀; (3.2) 1

b₀^p− + ¹

a₀^p+

!

ku⁰_mk^p_X⁻ +Mku⁰_mk^σ_X⁻⁺¹< dλ₀^p+, ∀m≥m0. (3.3) Fixingmsuch thatm≥ m₀, we have the following estimate:

Lemma 3.1. We havekum(t)k_X <λ0, ∀t∈ [0,∞).

Proof. We argue by contradiction. Assume that there existst₁ ∈(0,tm)such that ku_m(t₁)k_X ≥λ₀.

Consider the set

O={τ∈(0,tm):kum(τ)k_X≥λ0} and

inf

τ∈Oτ=t^∗. We have

ku_m(t^∗)k_X =λ₀ and t^∗ >0.

In fact, the function β(t) = kum(t)k_X is continuous on [0,tm) then kum(t^∗)k_X ≥ λ0. If ku_m(t^∗)k_X > λ₀, the Intermediate Value Theorem and noting that ku_m(0)k_X < λ₀, imply that t^∗ is not the infimum on O, which is a contradiction. Thusku_m(t^∗)k_X =λ₀. Alsot^∗ >0 becausekum(0)k_X <λ0. Note that

kum(t)k_X<λ₀, ∀t ∈[0,t^∗). Considert ∈[0,t^∗)andv=u⁰_m in (3.1), we obtain

|u⁰_m(t)|²+

∑

Z

Ω

∂u_m(t)

∂x_i

p(x)−2

∂u_m(t)

∂x_i

∂u⁰_m(t)

∂x_i dx+

Z

Ω|u_m(t)|^σ(x)u⁰_m(t)dx=0.

It follows

|u⁰_m(t)|²+

∑

d dt

Z

Ω

1 p(x)

∂u_m(t)

∂x_i

p(x)

dx+ ^d dt

Z

Ω

1

σ(x) +1|u_m(t)|^σ(x)u_m(t)dx =0.

Integrating on[0,t], we find Z _t

0

|u⁰_m(s)|²ds+

∑

Z

Ω

1 p(x)

∂u_m(t)

∂x_i

p(x)

dx+

Z

Ω

1

σ(x) +₁|u_m(t)|^σ(x)u_m(t)dx

=

∑

Z

Ω

1 p(x)

∂u⁰_m

∂x_i

p(x)

dx+

Z

Ω

1

σ(x) +1|u⁰_m|^σ(x)u⁰_mdx.

By (2.3), we get Z _t

0

|u⁰_m(s)|²ds+

∑

Z

Ω

1 p⁺

∂u_m(t)

∂x_i

p(x)

dx+

Z

Ω

1

σ⁺+1|u_m(t)|^σ(x)u_m(t)dx

≤

∑

Z

Ω

1 p⁻

∂u⁰_m

∂x_i

p(x)

dx+

Z

Ω

1

σ⁻+1|u⁰_m|^σ(x)+1dx.

(3.4)

As λ₀ ≤ 1 and t ∈ [0,t^∗), we have ^∂um

(t)

∂x_i

L^p(·)(_Ω) < λ₀ ≤ 1, i = 1, 2, . . . ,n. Therefore it follows from (2.1a) that

1 p⁺

∑

∂u_m(t)

∂x_i

p⁺ L^p(·)(_Ω)

≤ ¹ p⁺

∑

Z

Ω

∂u_m(t)

∂x_i

p(x)

dx.

By (2.10a) we obtain

∑

∂um(t)

∂x_i L^p(·)(_Ω)

!p⁺

≤a^p₁⁺

∑

∂um(t)

∂x_i

p⁺ L^p(·)(_Ω)

. These last two inequalities furnish

1 p⁺a₁^p+

ku_m(t)k^p+

W₀^1,p(·)(_Ω)≤ ¹ p⁺

∑

Z

Ω

∂u_m(t)

∂x_i

p(x)

dx. (3.5)

We modify the third term of (3.4). From the inequalities (2.9) and (2.1a) and noting that ku_m(t)k_X<1, becauset∈ [0,t^∗)it follows that

Z

Ω|u_m(t)|^σ(x)u_m(t)dx

≤

Z

Ω|u_m(t)|^σ(x)+1dx≤ ku_m(t)k^σ−+1

L^σ(·)+1(_Ω)+ku_m(t)k^σ++1

L^σ(·)+1(_Ω)

≤K^σ−+1ku_m(t)k^σ_X⁻⁺¹+K^σ++1ku_m(t)k^σ_X⁻⁺¹. Thus, noting that _σ+¹+1 <1,

Z

Ω

1

σ⁺+1|um(t)|^σ(x)um(t)dx

≤Mkum(t)k^σ_X⁻⁺¹ (3.6) where Mwas defined in (2.11a).

We modify the last two terms of (3.4). Note that _p¹− ≤ 1, _σ−¹+1 ≤ 1. By (2.1a), (2.10a) and observing thatku⁰_mk_X<1, we obtain

∑

Z

Ω

∂u⁰_m

∂x_i

p(x)

dx ≤

∑

∂u⁰_m

∂x_i

p⁻ L^p(·)(_Ω)

+

∑

∂u⁰_m

∂x_i

p⁺

L^p(·)(_Ω)

≤ ¹ b₀^p−

ku⁰_mk_X^p−+ ¹ a₀^p+

ku⁰_mk_X^p+. Thus

∑

Z

Ω

1 p⁻

∂u⁰_m

∂x_i

p(x)

dx≤ ¹ b₀^p−

+ ¹ a₀^p+

!

ku⁰_mk^p_X⁻. (3.7) In a similar way, we find

Z

Ω|u⁰_m|^σ(x)+1dx≤ ku⁰_mk^σ−+1

L^σ(·)+1(_Ω)+ku⁰_mk^σ++1

L^σ(·)+1(_Ω)

≤K^σ−+1ku⁰_mk^σ_X⁻⁺¹+K^σ++1ku⁰_mk^σ_X⁺⁺¹. That is

1 σ⁻+1

Z

Ω|u⁰_m|^σ(x)+1dx≤ Mku⁰_mk^σ_X⁻⁺¹. (3.8)

Plugging (3.5)–(3.8) into (3.4), we obtain Z _t

0

|u⁰_m(s)|²ds+dkum(t)k^p_X⁺+dkum(t)k_X^p+−Mkum(t)k^σ_X⁻⁺¹

≤ ¹

b₀^p− + ¹

a₀^p+

!

ku⁰_mk^p_X⁻ +Mku⁰_mk^σ_X⁻⁺¹= I(u⁰_m) (3.9) whered, M, a₀ andb₀were defined, respectively, in (2.11a), (2.10a) and (2.10b).

We now compare the third and four term of the last expression. Consider the function θ(λ) =dλ^p+−Mλ^σ−+1, λ≥0.

By hypothesis (2.5a) we have thatσ⁻+1−p⁺ >0. We find that if 0≤ λ≤

dp⁺ M(σ⁻+1)

¹

σ−+1−p+

=P then

θ(λ)≥0.

In particular if

0≤ λ≤min{1,P}= λ₀ (λ0 defined in (2.11a)), we have

θ(λ)≥0.

Askum(t)k_X<λ0, for allt ∈[0,t^∗), we deduce that

θ(ku_m(t)k_X) =dku_m(t)k_X^p+−Mku_m(t)k^σ_X⁻⁺¹≥0, t∈ [0,t^∗). (3.10) Thus by (3.9) we get

Z _t

0

|u⁰_m(s)|²ds+dku_m(t)k^p_X⁺ ≤ I(u⁰_m)_, t∈ [_0,t^∗)_. By (3.3) and (3.4), we obtain

I(u⁰_m)<dλ₀^p+. Therefore,

dku_m(t)k_X^p+ < I(u⁰_m)<r <dλ₀^p+, for somer ∈_R.

Taking the limit t→t^∗,t <t^∗, in the above inequality, we obtain dku_m(t^∗)k_X^p+ ≤r <dλ₀^p+

which is a contradiction becauseku_m(t^∗)k_X =λ₀. Thus the lemma is proved.

Returning to the Proof of Theorem 2.2

By Lemma3.1, (3.9) and properties of operatorA, we obtain that there exists a subsequence of (u_m), still denoted by(u_m), and a functionusuch that

um →uweak star inL^∞(0,∞;W₀^1,p(·)(_Ω)); (3.11a) u⁰_m →u⁰ weak in L²(_0,∞;L²(_Ω))_{; (3.11b)} A(u_m)→χweak star in L^∞(0,∞;W^−1,p0(·)(_Ω)). (3.11c)

The next step is to prove thatχ=Au and for that we need to show that Z _T

0

Z

Ω|um|^σ(x)umdxdt→

Z _T

0

Z

Ω|u|^σ(x)udxdt (3.12) for anyT >0. Introduce the notations:

Q_T =_Ω×(_0,T)_;

F_m ={(x,t)∈Q_T;|u_m(x,t)| ≤1}; G_m ={(x,t)∈Q_T;|u_m(x,t)|>1}.

By compactness (2.8) (see [16] or Corollary 6 in [21]) and convergences (3.11a) and (3.11b), we find

um →uinC([_0,T]_;L^σ++1(_Ω)) therefore,

u_m →uin L^σ++1(Q_T) (3.13)

and

u_m(x,t)→u(x,t)a.e. inQ_T. Hence

|um(x,t)|^σ(x) → |u(x,t)|^σ(x) a.e. inQT. (3.14) By (3.13), we have

Z

Q_T

[|u_m(x,t)|^σ(x)]^{σ+ +σ+1}dxdt=

Z

Fm

[|u_m(x,t)|^σ(x)]^{σ+ +σ+1}dxdt+

Z

Gm

[|u_m(x,t)|^σ(x)]^{σ+ +σ+1}dxdt

≤T(measΩ) +

Z

QT

|u_m(x,t)|^σ++1dxdt≤C, ∀m∈_N, that is,

Z

Q_T

[|u_m(x,t)|^σ(x)]^{σ+ +σ+1}dxdt≤C, ∀m∈_N. (3.15) From (3.14), (3.15) and Lions’ Lemma (see [15] or [16]) it follows that

|u_m|^σ(x)→ |u|^σ(x) weak inL^{σ+ +σ+1}(Q_T). (3.16) This result and convergence (3.13) imply convergence (3.12).

By the theory of monotone operators and the convergences (3.12) and (3.16), we deduce (see Lions [15])

χ= Au. (3.17)

Also by applying the diagonalization process to the sequence of(u_m), we find from (3.16)

|um|^σ(x) → |u|^σ(x) weak in L^{σ+ +σ+1}(QT), ∀T>0. (3.18) Convergences (3.11a), (3.18) and equality (3.17) allows us to pass to the limit in the ap- proximate equation (3.1) and so it holds that

Z _∞

0

(u⁰,ϕ)dt+

Z _∞

0

hAu,ϕidt+

Z _∞

0

Z

Ω|u|^σϕdxdt=0

∀ϕ∈ L²_loc(0,∞;W₀^1,p(x)(·)), suppϕcompact in(0,∞).

Taking ϕ ∈ C^∞₀ (_Ω×(0,T))in the last equality, we find equation (2.14). The initial condition (2.15) follows by convergences (3.11a) and (3.11b). This concludes the proof of Theorem2.2.

Proof of Theorem2.4. Multiply both sides of (2.14) byuand integrate onΩ. We obtain 1

2 d

dt|u(t)|²+

∑

Z

Ω

∂u

∂x_i

p(x)

dx+

Z

Ω|u|^σ(x)udx=0. (3.19) By Lemma 3.1 we have ku(t)k_X ≤ λ₀ < 1 then

_∂x^∂u

i

X < 1, i = 1, 2, . . . ,n. Therefore, from (2.1a) it follows that

∑

∂u

∂x_i

p⁺ L^p(·)(_Ω)

≤

∑

Z

Ω

∂u

∂x_i

p(x)

dx.

On the other side, by (2.10a) we obtain ku(t)k_X^p+ =

∑

∂u

∂x_i L^p(·)(_Ω)

!p⁺

≤ a₁^p+

∑

∂u

∂x_i

p⁺ L^p(·)(_Ω)

. These two preceding inequalities furnish

1 a₁^p+

ku(t)k^p_X⁺ ≤

∑

Z

Ω

∂u

∂x_i

p(x)

dx. (3.20)

Also by (2.1a) and (2.1b), we obtain

Z

Ω|u(t)|^σ(x)u(t)dx

≤ ku(t)k^σ−+1

L^σ(·)+1(_Ω)+ku(t)k^σ++1

L^σ(·)+1(_Ω)

and by (2.9), ku(t)k^σ−+1

L^σ(·)+1(_Ω)+ku(t)k^σ++1

L^σ(·)+1(_Ω)≤K^σ−+1ku(t)k^σ−+1

L^σ(·)+1(_Ω)+K^σ++1ku(t)k^σ++1

L^σ(·)+1(_Ω). Asku(tk_X ≤1, we find

ku(t)k^σ−+1

L^σ(·)+1(_Ω)+ku(t)k^σ++1

L^σ(·)+1(_Ω)≤K^σ−+1ku(t)k^σ_X⁻⁺¹+K^σ++1ku(t)k^σ_X⁺⁺¹. Asku(t)k_X ≤1, we find

ku(t)k^σ−+1

L^σ(·)+1(_Ω)+ku(t)k^σ++1

L^σ(·)+1(_Ω) ≤ Mku(t)k^σ_X⁻¹⁺¹ where M was defined in (2.11a). Then three preceding inequalities provide

Z

Ω|u(t)|^σ(x)u(t)dx

≤ Mku(t)k^σ_X⁻⁺¹. (3.21) Plugging inequalities (3.20) and (3.21) in (3.19), we obtain

d

dt|u(_t)|²+ ² a₁^p+

ku(_t)k^p_X⁺−2Mku(_t)k^σ_X⁻⁺¹≤0.

Noting that ¹

a₁^p+ ≥2dbecause p⁺≥1, we derive of the last inequality d

dt|u(t)|²+ ¹ a₁^p+

ku(t)k_X^p++2

dku(t)k_X^p+−Mku(t)k^σ_X⁻⁺¹≤0.

By (2.16) and (3.10), we get d

dt|u(t)|²+ ¹ a₁^p+L^p+

|u(t)|^p+ ≤0 that is

d

dtE(t) +ηE(t)^p

+

2 ≤0. (3.22)

We prove(i). For p(x) =2, for all x∈Ω, that is, p⁺ =2, we have d

dtE(t) +ηE(t)≤0 that implies (2.18).

Before proving (ii), we make the following considerations. If u⁰ = _{0, we take} u ≡ _{0 as} the solution of Problem (1.1a)–(1.1c). Assume u⁰ 6= 0. If there exists t₁ ∈ (0,∞) such that E(t₁) =0, we consider the set

P ={τ∈ (0,∞);E(τ) =0} and

t^∗ = inf

τ∈Pτ.

Then t^∗ > _{0 because} E(₀) > _{0. Also} E(t^∗) = _{0. As} E⁰(t) ≤ _{0 a.e. in} (_0,∞)_{, then} E(t) _is decreasing, thereforeE(t) =0 for allt ≥t^∗. Thus

eitherE(t) =0, for allt≥t^∗ or E(t)>0, for allt >0.

We prove inequality (2.19) for the second case, that is,E(t)>0, for allt∈ [0,∞). The inequal- ity (2.19) fort∈[0,t^∗)is derived in a similar way. Recalling that ^p₂⁺ =1+γ,γ>0. By (3.22), we obtain

(−γ)E⁰(t)

E(t)^1+γ −ηγ≥0, which implies

[E(t)]^−γ0 ≥ηγ.

Thus

E^−γ(t)≥ E^−η(0) +ηγt that is,

E^−γ(t)≥ (1+E^γ(0)ηγt) E^γ(0) ^. This implies inequality (2.19).

Acknowledgement

We thank to the Referee for his/her very important remarks.