Global existence and stability of a class of nonlinear evolution equations with hereditary memory and variable density

R E S E A R C H

Open Access

Global existence and stability of a class of

nonlinear evolution equations with

hereditary memory and variable density

Fushan Li

_{and Zhiqiang Jia}

*_{Correspondence:}

[email protected]

1_{School of Mathematical Sciences,} Qufu Normal University, Shandong, P.R. China

Abstract

In this paper, we consider the initial boundary value problem of nonlinear evolution equation with hereditary memory, variable density, and external force term

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

|ut|ρutt–

α

t

–∞

μ

γ

(x,t)∈

Ω

u(x,t) = 0, (x,t)∈

∂Ω

u(x, 0) =u0(x), ut(x, 0) =u1(x), x∈

Ω

Under suitable assumptions, we prove the existence of a global solution by means of the Galerkin method, establish the exponential stability result by using only one simple auxiliary functional, and give the polynomial stability result.

MSC: 35L05; 35L15; 35L70

Keywords: Hereditary memory; Variable density; Global existence; Exponential stability; Polynomial stability

1 Introduction

In this paper, we are concerned with the following problem:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

|ut|ρ_{utt–αu–utt+}t

–∞μ(t–s)u(s)ds–γ ut=f(u),

(x,t)∈Ω×R+_,

u(x,t) = 0, (x,t)∈∂Ω×R+,

u(x, 0) =u0(x), ut(x, 0) =u1(x), x∈Ω,

(1.1)

whereΩ is a bounded domain ofRn_{(n≥1) with smooth boundary∂Ω,ρis a positive}

constant, andγ ≥0. We prove the existence of a global solution by means of the Galerkin method and establish the exponential stability under suitable assumptions by using a sim-pler auxiliary functional than that in [1]. We also show the polynomial stability under suitable conditions.

Partial differential equations in viscoelastic materials have important physical back-ground and important mathematical significance. The viscous effects are described and

characterized by an integral term, and the integral term indicates a dissipative effect. For mathematical analysis on the motions of evolution equations with memory, we refer to [8, 32]. Problem (1.1) is related to the equations

f(ut)utt–u–utt= 0, (1.2)

which have several modeling features. Iff(ut) is a constant, Eq. (1.2) has been used to model extensional vibrations of thin rods (see [27, Ch. 20]) and it differs from D’Alembert’s wave equation because ofutt, which is not a damping term. On the contrary,utt in-creases the energy functional. Iff(ut) is not a constant, Eq. (1.2) shows that the density of

materials depends on the velocityut.

In the past ten years, several authors studied the homogeneous Dirichlet boundary value problem for the following model with memory (starting from the zero moment) and vari-able density:

|ut|ρ

utt–u+ t

0

g(t–τ)u(τ)dτ+F(u,ut,utt) = 0

in a bounded domainsΩ⊂Rn. Cavalcanti et al. [2] considered the model with integral dissipation and strong damping

|ut|ρ

utt–u–utt+ t

0

g(t–s)u(s)ds–γ ut= 0, (x,t)∈Ω×R+.

Assuming that 0 <ρ≤ _n–22 ifn≥3 orρ> 0 ifn= 1, 2 and thatg(t) decays exponentially, they obtained the global existence of a solution forγ≥0 and the uniform exponential de-cay of the energy forγ > 0. Cavalcanti et al. [3] considered this model and proved intrinsic decays for large classes of relaxation kernels described by the inequalityg+H(g)≤0 with convex functionH. Han and Wang [11] considered the equation with integral dissipation and linear damping

|ut|ρ_u

tt–u–utt+

t

0

g(t–s)u(s)ds+ut= 0, (x,t)∈Ω×R+.

They proved the global existence and exponential decay whengis decaying exponentially by introducing two auxiliary functionals. Han and Wang [12] established the general decay of energy for the equation with integral dissipation and nonlinear damping

|ut|ρ_{utt–u–utt+}

t

0

g(t–s)u(s)ds+|ut|mut= 0, (x,t)∈Ω×R+,

by introducing two auxiliary functionals. Messaoudi and Tatar [29,30] considered the equation only with integral dissipation

|ut|ρ

utt–u–utt+ t

0

g(t–s)u(s)ds= 0, (x,t)∈Ω×R+.

integral dissipation

|ut|ρ_{utt–u–utt+}

t

0

g(t–s)u(s)ds=b|u|p–2u, (x,t)∈Ω×R+. (1.3)

By introducing a new functional and using potential well method they showed that there exists an appropriate setS(called a stable set) such that if the initial datum is inS, then the solution continues to live there forever. They also showed that the solution goes to zero with an exponential or polynomial rate depending on the decay rate of the relaxation func-tiong. Liu [26] considered (1.3) and proved that, for certain class of relaxation functions and certain initial data in the stable set, the decay rate of the solution energy is similar to that of the relaxation function. Conversely, for certain initial data in the unstable set, there are solutions that blow up in finite time.

Now, we list some important literature on the nonlinear evolution equation with hered-itary memoryand variable density. Araújo et al. [1] considered the equation with integral dissipation in infinite interval

|ut|ρ

utt–αu–utt+ t

–∞

μ(t–s)u(s)ds–γ ut+f(u) =h, (x,t)∈Ω×R+,

where Ω is a bounded domain ofRn _{(n≥1) with smooth boundary ∂Ω. They}

estab-lished the uniqueness of the solution, exponential decay, and global attractors. However, the existence of a solutionis not givenin detail,two auxiliary functionalsare introduced to prove the exponential decay result, and the polynomial decay resultis not given. Conti et al. [5] established an existence, uniqueness, and continuous dependence result for weak solutions to the nonlinear viscoelastic equation with hereditary memory on a bounded three-dimensional domain

|∂tu|ρ∂ttu–∂ttu+γ(–)θ∂tu–αu+

∞

0

μ(s)u(t–s)ds+f(u) =h

with Dirichlet boundary conditions. In particular, the parameterρbelongs to the interval [0, 4], the value 4 is critical for the Sobolev embeddings, whereasf can reach the critical polynomial order 5. Lately, Conti et al. [4] studied the nonlinear viscoelastic equation

|∂tu|ρ∂ttu–∂ttu–u+

∞

0

μ(s)u(t–s)ds+f(u) =h

and showed that the sole weak dissipation given by the memory term is enough to en-sure the existence and optimal regularity of the global attractorAρforρ< 4 and critical nonlinearityf.

[16,17,19,22–25]. The authors in [10,18,20,21] studied the blowup phenomenon for some evolution equations. Du and Li [6,7] proved the integrability and regularity of the solution to some equations.

In this paper, we study the equation with hereditary memory (u0(x,t),t≤0) and variable

density

|ut|ρ

utt–αu–utt+ t

–∞

μ(t–τ)u(τ)dτ–γ ut=f(u), (1.4)

that is,

|ut|ρ

utt–αu–utt+ _∞

0

μ(τ)u(t–τ)dτ–γ ut=f(u),

which can be rewritten as

|ut|ρ utt–

α– _∞

0

μ(τ)dτ u–utt

– ∞

0

μ(τ)u(x,t) –u(x,t–τ)dτ–γ ut=f(u).

This equation inspires us to define

η:=η(x,t,τ) =u(x,t) –u(x,t–τ), (x,τ)∈Ω×R+,t≥0,

which implies

ηt(x,t,τ) =ut(x,t) –ut(x,t,τ) =ut(x,t) –ητ(x,t–τ), (x,τ)∈Ω×R+,t≥0,

and

t= 0 :η(x, 0,τ) =u0(x, 0) –u0(x, –τ), (x,τ)∈Ω×R+.

Hence Eq. (1.4) can be rewritten as

|ut|ρ_u tt–

α– ∞

0

μ(τ)dτ u–utt–

∞

0

μ(τ)η(τ)dτ–γ ut=f(u).

Without loss of generality, we assume thatα–₀∞μ(τ)dτ = 1. Then

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

|ut|ρ_{utt–u–utt–}∞

0 μ(τ)ηdτ–γ ut=f(u), (x,t)∈Ω×R +_,

ηt(x,t,τ) =ut(x,t) –ητ(x,t,τ),

u(x, 0) =u0(x), ut(x, 0) =u1(x),

η(x,t, 0) = 0, η(x, 0,τ) =η0(x,τ),

u= 0 ∂Ω×R+, η= 0 ∂Ω×R+×R+,

(1.5)

where

The main contribution of this paper are: (a) the equation with hereditary memory, vari-able density, and external force term is representative; (b) the detailed construction pro-cess of the energy functional is given by an integration method; (c) we give a detailed proof of the existence for the solution; (d) the proof of the exponential decay result is simplified by introducing only one auxiliary functional; (e) the polynomial decay result is established. The outline of this paper is as follows. In Sect.2, we present the preliminaries and im-portant our results. In Sects.3–5, we prove the main Theorems2.1–2.3, respectively.

2 Assumptions and the main results

In this paper, we assume that the following conditions (A1)–(A3) hold:

(A1)

0 <ρ≤ 2

n– 2 ifn≥3; ρ> 0 ifn= 1, 2,

which implies that

H₀1(Ω)→L2(ρ+1)_(Ω).

(A2) f :R→Rand satisfies

f(u) –f(v)≤c0

1 +|u|p+|v|p|u–v|, u,v∈R,

wherec0> 0and

0 <p≤ 2

n– 2 ifn≥3; ρ> 0 ifn= 1, 2,

and

f(s)s≤F(s)≤0, ∀s∈R,

whereF(z) =₀zf(σ)dσ. (A3) μsatisfies

μ∈C1R+∩L1R+, 0≤μ(τ) <∞, μ(0) > 0, μ(+∞) = 0

with ∞

0

μ(τ)dτ=:k0> 0,

and there exists a constantk1> 0satisfying

μ(t)≤–k1μq(t), ∀t∈R+, 1≤q<

3 2.

To consider the relative displacementηas a new function, we introduce the weighted

L2_-space

M:=L2_μR+;H₀1(Ω)=

v:R+→H₀1(Ω) ∞

0

μ(τ)∇v(τ)2₂dτ<∞

which is a Hilbert space endowed with inner product

(v,w)_M= ∞

0

μ(τ)

Ω

∇v(τ)· ∇w(τ)dx dτ

and norm

v2 M=

∞

0

μ(τ)∇v(τ)2₂dτ.

We introduce the notation

H=H₀1(Ω)×H₀1(Ω)×M.

Remark2.1 (1) H1

0(Ω)→Lr(Ω)with

r: ⎧ ⎨ ⎩

2≤r≤_n2_–2n, n≥3,

≥2, n= 1, 2,

which implies

ϕr≤Bϕ2, ∀ϕ∈H01(Ω).

(2) From(A2)we can easily getf(0) = 0.

(3) The conditionq<3₂is imposed to ensure that₀∞μ2–q_{(τ)dτ<∞. In fact,}

assumption (A3) implies

μ(t)≤ C1 (1 +t)q–11

, 2 –q

q– 1> 1,

and therefore₀∞μ2–q(τ)dτ<∞.

Give the initial data (u0,u1,η0)∈H, a functionz= (u,ut,η)∈C([0,T],H) is a weak

so-lution of problem (1.5) if it satisfies the initial conditionz(0) = (u0,u1,η0) and

|ut|ρ

utt,w+ (∇u,∇w) + (∇utt,∇w) +γ(∇ut,∇w) + _∞

0

μ(τ)(∇η,∇w)dτ

=f(u),w,

(∂tη+∂τη,v)M= (ut,v)M,

for allw∈H1

0(Ω),v∈M, and a.e.t∈[0,T].

Multiplying both sides of Eq. (1.5) byut, integrating the resulting equation overΩ, and using the Green formula, we have

Ω

|ut|ρuttutdx+

Ω

∇u· ∇utdx+

Ω

∇utt∇utdx+

Ω ∇ut

∞

0

μ(τ)∇η(τ)dτdx

+γ

Ω

|∇ut|2_dx=

Ω

that is,

d dt

1

ρ+ 2ut

ρ+2 ρ+2+

1 2∇u

2 2+

1

2∇ut(t)

2 2–

Ω F(u)dx

+γ∇ut2₂+

Ω ∇ut

∞

0

μ(τ)∇ηdτdx= 0.

A direct computation and application of (1.5) show that

Ω ∇ut

∞

0

μ∇η(τ)dτdx=

Ω

(∇ηt+∇ητ)

∞

0

μ(τ)∇ηdτdx

=

Ω ∇ηt

∞

0

μ(τ)∇ηdτdx+

Ω ∇ητ

∞

0

μ(τ)∇ηdτdx

=1 2

d dt∇η

2

M+ (∇ητ,∇η)M.

This computation inspires us to define an energy functional as follows:

E(t) = 1

ρ+ 2ut

ρ+2 ρ+2+

1 2∇u

2 2+

1 2∇ut

2 2+

1 2∇η

2 M–

Ω

F(u)dx (2.1)

and

E(t) = –γ∇ut₂2– (ητ,η)M.

Using (A3) and (1.5), we have

(ητ,η)M=

1 2

Ω

∞

0

μ(τ) ∂

∂τ|∇η|

2_{dτ dx= –}1

2

Ω

∞

0

μ(τ)|∇η|2dτ dx.

Then

E(t) = –γ∇ut(t) 2 2+

1 2

∞

0

μ(τ)∇η(τ)2₂dτ≤0. (2.2)

Theorem 2.1 Assume that conditions (A1)–(A3) hold and γ ≥0. If the initial data

(u0,u1,η0)∈H,then for any T> 0,problem(1.5)has a weak solution

(u,ut,η)∈C

[0,T],H

satisfying

u∈L∞R+;H₀1(Ω), ut∈L∞R+;H₀1(Ω),

utt∈L2[0,T];H₀1(Ω), η∈L2R+;M.

Theorem 2.2 Assume that conditions(A1)–(A3)hold andγ> 0.If q= 1,then

E(t)≤Ke–νt, t≥0,

Theorem 2.3 Assume that conditions(A1)–(A3)hold andγ> 0.If1 <q<3₂,then

E(t)≤K(1 +t)–2(q1–1)_{, t}≥_0,

where K is a positive constant.

3 Proof of Theorem2.1

We study the equation

|ut|ρ_u

tt–u–utt–

∞

0

μ(τ)ηdτ–γ ut=f(u).

Let{ωj}∞_j=1be an orthogonal basis ofH01withωjsatisfying

⎧ ⎨ ⎩

–ωj=λjωj, x∈Ω,

ωj|∂Ω= 0.

By normalization we haveωj2= 1 and writeVk=span{ω1, . . . ,ωk}. For any given

inte-gerk, we consider the approximate solution

uk(t) =

k

j=1 cj_k(t)ωj

that satisfies ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

(|ukt|ρ_uktt,ω

j) + (∇uk,∇ωj) + (∇uktt,∇ωj) +γ(∇ukt,∇ωj)

+₀∞μ(τ)(∇ηk,∇ωj)dτ= (f(uk),ωj), uk(0) =u0k, uk,t(0) =u1k, j= 1, 2, . . . ,k,

(3.1)

and, ask→ ∞,

u0k= k

j=1

(u0,ωj)ωj→u0 inH01 and u1k= k

j=1

(u1,ωj)ωj→u1 inH01. (3.2)

Here we denote by (·,·) the inner product inL2(Ω). Then (3.1) can be reduced to the second-order ODE system

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

(|k_i=1ci k

(t)ωi|ρk_i=1ci k

(t)ωi,ωj) +λjcj_k(t) +λjcj_k

(t) +γ λjcj_k(t)

+λj

∞ 0 μ(τ)(c

j k(t) –c

j

k(t–τ))dτ= (f(

k j=1c

j

k(t)ωj),ωj), cj_k(0) = (u0,ωj), cj_k

(0) = (u1,ωj), j= 1, 2, . . . ,k,

(3.3)

According to the standard existence theory for ordinary differential equations, we infer that system (3.3) admits a solutioncj_k(t) in [0,tm), wheretm> 0. Then we can obtain an

Multiplying (3.3) bycj_k(t) and summing with respect toj, we conclude that

d dt

1

ρ+ 2ukt

ρ+2 ρ+2+

1 2∇uk 2 2+ 1 2∇ukt 2

2 +γ∇ukt22+

∞

0

μ(τ)(∇ηk,∇ukt)dτ

=f(uk),u_kt. (3.4)

Simple calculations yield

_∞

0

μ(τ)(∇ηk,∇ukt)dτ

= ∞

0

μ(τ)(∇ηk,∇ηkt+∇ηkτ)dτ=

1 2

d dtηk

2

M+ (∇ητ,∇η)M, (3.5)

(ηkτ,ηk)M=

1 2 Ω ∞ 0

μ(τ)∂

∂τ|∇ηk|

2_{dτ dx}

= –1 2

Ω

∞

0

μ(τ)|∇ηk|2dτ dx

= –1 2

∞

0

μ(τ)∇ηk2₂dτ. (3.6)

Combining (3.4) and (3.6), we find

d dt

1

ρ+ 2ukt

ρ+2 ρ+2+

1 2∇uk 2 2+ 1 2∇ukt 2 2+ 1 2ηk

2 M–

Ω

F(uk)dx

= –γ∇ukt2₂+1 2

∞

0

μ(τ)∇kη2₂dτ≤0. (3.7)

Integrating (3.7) over (0,t) and noting (3.2), we obtain

1

ρ+ 2ukt

ρ+2 ρ+2+

1 2∇uk 2 2+ 1 2∇ukt 2 2+ 1 2ηk

2 M–

Ω

F(uk)dx

≤ 1

ρ+ 2ukt(0)

ρ+2 ρ+2+

1

2∇uk(0)

2 2+

1

2∇ukt(0)

2 2

+1 2ηk(0)

2 M–

Ω Fuk(0)

dx

≤K1, (3.8)

whereK1is a constant independent ofk. It follows from (3.8) and the Poincaré inequality

that ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

{uk}is bounded inL∞(0,T;H1 0), {ukt}is bounded inL∞(0,T;H₀1),

{ηkt}is bounded inL2_(0,T;M).

Multiplying (3.1) bycj_k(t) and summing with respect toj, we obtain

Ω

|ukt|ρ|uktt|2dx+ (∇uk,∇uktt) +∇uktt22+

γ

2

d dt∇ukt

2 2

+ ∞

0

μ(τ)(∇ηk,∇uktt)dτ=

f(uk),uktt,

that is,

Ω

|ukt|ρ_|uktt|2_dx+∇uktt2 2+

γ

2

d dt∇ukt

2 2

= –(∇uk,∇uktt) –

∞

0

μ(τ)(∇ηk,∇uktt)dτ+

f(uk),uktt

. (3.10)

The right-hand side of (3.10) can be estimated as follows:

_{–(∇uk,∇uktt)=–}

Ω

∇uk∇ukttdx≤ε∇uktt2₂+ 1 4ε∇uk

2

2 ∀ε> 0, (3.11)

–

∞

0

μ(τ)(∇ηk,∇uktt)dτ

≤ε∇uktt2₂+ 1 4ε

Ω

∞

0

μ(τ)|∇ηk|dτ

2 dx

≤ε∇uktt2₂+ 1 4ε

Ω

∞

0

μ(τ)μ(τ)|∇ηk|dτ

2 dx

≤ε∇uktt2₂+ 1 4ε

∞

0

μ(τ)dτ

Ω

∞

0

μ(τ)|∇ηk|2dτdx

=ε∇uktt2₂+k0 4ε∇ηk

2

M (3.12)

withk0=

∞

0 μ(τ)dτ. Using (A2), the Sobolev embedding theorem, and the Poincaré

in-equality, we have

f(uk),uktt=–

Ω

f(uk)ukttdx

≤c0

Ω

1 +|uk|p|uk||uktt|dx

≤Cεuk2₂+uk2(_2(pp+1)₊₁₎+εuktt2₂

≤C∗∇uk2₂+∇uk₂2(p+1)+εC∇uktt22. (3.13)

By (3.10)–(3.13) we have

Ω

|ukt|ρ_|uktt|2_dx+∇uktt2 2+

γ

2

d dt∇ukt

2 2

≤(2 +C)ε∇uktt2₂+

1 4ε+C

∗ _∇uk2

2+C∗∇uk 2(p+1)

2 +

k0

4ε∇ηk

that is,

Ω

|ukt|ρ|uktt|2dx+1 – (2 +C)ε∇uktt2₂+γ 2

d dt∇ukt

2 2

≤

1 4ε+C

∗ _∇uk2

2+C∗∇uk 2(p+1)

2 +

k0

4ε∇ηk

2

M. (3.14)

From (3.14) and (3.18) we know that

Ω

|ukt|ρ|uktt|2dx+

1 – (2 +C)ε∇uktt2₂+γ 2

d dt∇ukt

2

2≤K2. (3.15)

Integrating (3.15) over (0,t) (0 <t≤T) and noting (3.2) yield t

0

Ω

|uktt|ρ_|uktt|2_{dx dτ+1 – (2 +C)ε} t 0

∇uktt2 2dτ+

γ

2∇ukt

2

2≤CT. (3.16)

Takingεsuitably small in (3.16), we can obtain the second estimate

t

0

∇uktt2

2dτ+∇ukt22≤CT,

which implies that

{uktt}is uniformly bounded inL20,T;H₀1. (3.17)

According to estimates (3.9) and (3.17), we infer that there exists a subsequence in{um} (denoted by the same symbol) such that

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

uk ∗

u weak-star inL∞(0,T;H1 0(Ω)), ukt∗ ut weak-star inL∞(0,T;H1

0(Ω)), uktt

∗

utt weakly inL2(0,T;H01(Ω)),

ηk ∗

η weakly inL2_(0,T;M),

(3.18)

which, combined with the Aubin–Lions compactness lemma, implies

⎧ ⎨ ⎩

uk→u strongly inC([0,T];H1 0(Ω)), ukt→ut strongly inC([0,T];H01(Ω)).

(3.19)

Using (A2) and (3.19), we get

f(uk)→f(u).

From (3.19) we getukt→uta.e. inΩ×(0,T). Hence

|ukt|ρ_u

On the other hand, by the Sobolev embedding theorem and∇ukt2

2≤L1, we have

_|ukt|ρ

ukt_L2_(0,T;L2_(Ω))=

T

0

Ω

|ukt|2(ρ+1)_{dx dt≤B}2(ρ+1)

T

0

∇ukt2(ρ+1) 2 dt

≤B2(ρ+1)L(₁ρ+1)T≤CT. (3.21)

Thus, using (3.20), (3.21), and the Lions lemma, we derive

|ukt|ρ_u

kt|ut|ρut weakly inL2

0,T;L2(Ω). (3.22)

LetD(0,T) be the space ofC∞ functions with compact support in (0,T). Multiplying (3.1) byθ(t)∈D(0,T) and integrating over (0,T), it follows that

T

0

|ukt|ρ uktt,ωj

θ(t)dt+ T

0

(∇uk,∇ωj)θ(t)dt+

T

0

(∇uktt,∇ωj)θ(t)dt

+γ

T

0

(∇ukt,∇ωj)θ(t)dt+

T

0

_∞

0

μ(τ)(∇ηk,∇ωj)θ(t)dt dτ

= T

0

f(uk),ωj

θ(t)dt. (3.23)

Noting that{ωj}∞_j=1is a basis ofH₀1, via convergences (3.18), (3.19), and (3.22), we can get from (3.23) that

|ut|ρ utt,ωj

+ (∇u,∇ωj) + (∇utt,∇ωj) +γ(∇ut,∇ωj) +

∞

0

μ(τ)∇η(τ),∇ωj

dτ

=f(u),ωj

,

and hence, for allω∈H01(Ω),

|ut|ρ_u tt,ω

+ (∇u,∇ω) +∇utt(t),∇ω

+γ(∇ut,∇ω) +

∞

0

μ(τ)∇η(τ),∇ωdτ

=f(u),ω. (3.24)

Using (3.2) and (3.19), we have

u(0) =u0, ut(0) =u1. (3.25)

On the other hand, by Pata and Zucchi [31] we have that

η∈C[0,T],M. (3.26)

Combining (3.20), (3.25), and (3.26), we complete the proof.

4 Proof of Theorem2.2

Define the functionals

Φ(t) = 1

ρ+ 1

Ω

|ut|ρutu dx–

Ω

utu dx (4.1)

and

L(t) =ME(t) +Φ(t), (4.2)

whereM> 0 will be fixed later. We recall thatE(t) is decreasing sinceE(t)≤0.

Lemma 4.1 For M> 0sufficiently large,there exist constantsβ1,β2> 0such that

β1E(t)≤L(t)≤β2E(t), t≥0.

Proof By the Hölder and Cauchy inequalities we have

Φ(t)≤ 1

ρ+ 1

Ω

|ut|ρ+1|u|dx+

_Ω∇ut· ∇u dx

≤ 1

ρ+ 1

Ω

|ut|2(ρ+1)_dx 1 2

Ω |u|2_dx

1 2

+1 2∇ut

2 2+

1

2∇u

2 2

= 1

ρ+ 1ut

ρ+1

2(ρ+1)u2+

1 2∇ut

2 2+

1 2∇u

2 2.

SinceE(t) is decreasing, from the Sobolev embedding theorem we have

1

ρ+ 1ut

ρ+1

2(ρ+1)u2≤

1 2(ρ+ 1)ut

2(ρ+1) 2(ρ+1)+

1 2(ρ+ 1)u

2 2

≤C1Eρ(0)∇ut22+C1∇u22.

Therefore

Φ(t)≤CE(t).

Then takingM>C, we complete the proof.

Lemma 4.2 There exist C2> 0and C3> 0,dependent on the initial data,such that

Φ(t)≤–E(t) –

1

2–ε ∇u(t)

2

2+C2∇ut(t) 2 2

–C3

∞

0

Proof From the definition ofΨ(t) we get

Φ(t) = 1

ρ+ 1

Ω

|ut|ρ

uttu+|ut|ρutut+u2_t

ρ 2 tutu dx– Ω

(uttu+utut)dx

= 1

ρ+ 1

Ω

_ut(t)ρ

uttu+|ut|ρutut+ρ|ut|ρuttudx–

Ω

uttu+|∇ut|2dx

=

Ω

|ut|ρ_{utt–uttu dx+} 1

ρ+ 1ut

ρ+2

ρ+2+∇ut22.

Using (1.5), we easily see that

Ω

|ut|ρ

utt–uttu dx

= –∇u2 2+

∞

0

μ(τ)

Ω

η(τ)u(t)dx dτ+γ

Ω

utu dx+

Ω

f(u)u dx.

We now estimate the second and third terms in the right-hand side as follows. Using the Cauchy inequality withεand the Hölder inequality, we have

∞

0

μ(τ)

Ω

η(τ)u(t)dx dτ

=

Ω ∇u(t)

_∞

0

μ(τ)∇η(τ)dτdx

≤1

2ε∇u

2 2+ 1 2ε Ω ∞ 0

μ(τ)∇η(τ)dτ

2 dx

≤1

2ε∇u

2 2+

k0

2εη

2 M and γ Ω

utu dx≤1

2ε∇u

2 2+

γ2

2ε∇ut

2 2.

Therefore

Φ(t)≤–(1 –ε)∇u2₂+

1 +γ

2

2ε ∇ut

2 2+

1

ρ+ 1ut

ρ+2 ρ+2+

k0

2εη

2 M.

Noting the definitions ofE(t) and (A2), we obtain

Φ(t)≤–E(t) –

1

2–ε ∇u

2 2+ 3 2+ γ2

2ε ∇ut

2 2+

2

ρ+ 1ut

ρ+2 ρ+2 + 1 2 k0

2ε η

2

M. (4.3)

By Sobolev embedding we have

utρ_ρ+2₊₂≤BE(0)ρ2∇_ut2

Using (A3), we get

η2_M≤–1

k1

∞

0

μ(τ)∇η(τ)2₂dτ. (4.5)

Combining (4.3)–(4.5), we finish the proof of Lemma4.2.

Proof of Theorem2.2 By Lemma4.2we have

Φ(t)≤–E(t) +C2∇ut(t) 2 2–C3

∞

0

μ(τ)∇η(τ)2₂dτ.

Note that

E(t) = –γ∇ut(t)2₂+1 2

_∞

0

μ(τ)∇η(τ)2₂dτ

and

L(t) =ME(t) +Φ(t).

Then takingM=max(2C3,Cγ2), we have

L_{(t)≤–E(t).}

Using Lemma4.1, we obtain

L_(t)≤–1

β2 L(t).

By the Gronwall inequality we obtain

L(t)≤L(0)e–β12t.

By Lemma4.1we have

β1E(t)≤L(t)≤β2E(0)e –_β1

2t,

that is,

E(t)≤Ke–νt,

whereν=_β1

2 andK= β2E(0)

β1 .

5 Proof of Theorem2.3

We define

Φ(t) = 1

ρ+ 1

Ω

|ut|ρutu dx–

Ω

utu dx, (5.1)

Ψ(t) =

Ω

ut

∞

0

μ(τ)ηdτ dx– 1

ρ+ 1

Ω |ut|ρ_u

t

∞

0

and set

L(t) =ME(t) +εΨ(t) +χ(t),

whereMandεwill be fixed later.

Lemma 5.1 For M> 0sufficiently large,there exist constantsβ1,β2> 0such that

β1E(t)≤L(t)≤β2E(t), t≥0,

for any0 <ε≤1.

Proof Since

_Ωut

∞

0

μ(τ)ηdτ dx≤

∞

0

μ(τ)

Ω

utηdx

dτ

= _∞

0

μ(τ)

Ω

∇ut· ∇ηdxdτ

≤k0

2∇ut

2 2+

1 2η

2 M,

by the Hölder and Cauchy inequalities we have

1

ρ+ 1 _Ω|ut|ρ

utηdx≤ 1

ρ+ 1

Ω

_ut(t)ρ+1 |η|dx

≤ 1

ρ+ 1

Ω

ut(t)2(ρ+1)dx 1 2

Ω |η|2dx

1 2

= 1

ρ+ 1ut

ρ+1 2(ρ+1)η2.

SinceE(t) is decreasing, from the Sobolev inequality we have

1

ρ+ 1ut

ρ+1

2(ρ+1)η2≤

1 2(ρ+ 1)ut

2(ρ+1) 2(ρ+1)+

1 2(ρ+ 1)η

2 2

≤C1Eρ(0)∇ut22+C1∇η22.

Therefore – 1

ρ+ 1

Ω |ut|ρ

ut

∞

0

μ(τ)ηdτ dx≤k0C1Eρ(0)∇ut22+C1∇η2_M.

Consequently,

Ψ(t)≤CE(t).

Lemma 5.2 Under the conditions of Theorem2.2,the functional

Φ(t) = 1

ρ+ 1

Ω

ut(t)ρut(t)u(t)dx–

Ω

ut(t)u(t)dx

satisfies

Φ(t)≤–1 – (1 +k0)δ1

∇u2 2+

1 + γ

2

4δ1 ∇ut 2 2+

1

ρ+ 1ut

ρ+2 ρ+2

+ 1 4δ1

∞

0

μ2–q(τ)dτ

∞

0

μq(τ)∇η2₂dτ+

Ω

f(u)u dx

for anyδ1> 0.

Proof As in the proof of Lemma4.2, we get

Φ(t) =

Ω

|ut|ρ

utt–uttu dx+ 1

ρ+ 1ut

ρ+2

ρ+2+∇ut22 (5.3)

and

Ω

|ut|ρutt–utt

u dx

= –∇u2₂+ ∞

0

μ(τ)

Ω

η(τ)u(t)dx dτ

+γ

Ω

utu dx+

Ω

f(u)u dx. (5.4)

By the Cauchy inequality the second and third terms can be estimated as follows:

∞

0

μ(τ)

Ω

ηu(t)dx dτ

≤δ1k0∇u22+

1 4δ1

∞

0

μ2–q(τ)dτ

∞

0

μq(τ)∇η2₂dτ, (5.5)

and γ

Ω

utu dx=–γ

Ω

∇ut· ∇u dx≤δ1∇u22+

γ2

4δ1∇ut 2

2, (5.6)

whereδ1> 0.

Combining (5.3)–(5.6), we establish Lemma5.2.

Lemma 5.3 Under the conditions of Theorem2.2,there exist constants C,C,C> 0such that

Ψ(t) =

Ω

ut

∞

0

μ(τ)ηdτ dx– 1

ρ+ 1

Ω |ut|ρ

ut

∞

0

satisfies

Ψ(t)≤(δ2+δ2C)∇u22+

δ2γ2–k0+ 2δ2C

∇ut2 2–

k0

ρ+ 1ut

ρ+2 ρ+2

+C

∞

0

μ2–q(τ)ds

∞

0

μq(τ)∇η2₂dτ– 2C

∞

0

μ(τ)∇η(τ)2₂dτ

for anyδ2> 0.

Proof From definition ofΨ we have

Ψ(t) =

Ω

utt

∞

0

μ(τ)ηdτ dx+

Ω

ut

∞

0

μ(τ)ηtdτ dx

– 1

ρ+ 1

Ω |ut|ρ

utt

∞

0

μ(τ)ηdτ dx

– 1

ρ+ 1

Ω |ut|ρ

ut

∞

0

μ(τ)ηtdτ dx

– 1

ρ+ 1

Ω

u2_t

ρ

2 tut

∞

0

μ(τ)ηdτ dx

=

Ω

–|ut|ρutt+utt ∞

0

μ(τ)ηdτ dx

+

Ω

–|ut|

ρ_ut

ρ+ 1 +ut

∞

0

μ(τ)ηtdτ dx

:=I1+I2.

From (1.5) we see that

I1=

Ω

–u–

∞

0

μ(τ)ηdτ–γ ut–f(u)

∞

0

μ(τ)ηdτ dx.

By the Green formula and the Cauchy and Hölder inequalities we have the following esti-mates:

Ω

–u

∞

0

μ(τ)ηdτ dx

=

Ω ∇u·

∞

0

μ(τ)∇ηdτ dx

≤δ2∇u22+

1 4δ2

∞

0

μ2–q(τ)dτ

∞

0

μq(τ)∇η2₂dτ,

–γ Ω ut ∞ 0

μ(τ)η(τ)dτ dx

=γ

Ω ∇ut·

∞

0

μ(τ)∇ηdτ dx

≤δ2γ2∇ut22+

1 4δ2

∞

0

μ2–q(τ)dτ

∞

0

and

–

Ω

∞

0

μ(τ)ηdτ

∞

0

μ(τ)ηdτ dx=

Ω

∞

0

μ2–2q_μ

q

2_(τ)∇_η(τ)dτ 2

dx

≤

_∞

0

μ2–q(τ)dτ

_∞

0

μq(τ)∇η2₂dτ.

Using (A2) and the Cauchy, Hölder, and Poincaré inequalities, we obtain

–

Ω f(u)

∞

0

μ(τ)ηdτ dx≤δ2C∇u22+

1 4δ2

∞

0

μ2–q(τ)dτ

∞

0

μq(τ)∇η2₂dτ.

Therefore

I1≤(δ2+δ2C)∇u22+δ2γ2∇ut22+C

∞

0

μ2–q(τ)dτ

∞

0

μq(τ)∇η2₂dτ. (5.7)

From (1.5) we easily obtain

∞

0

μ(τ)ηtdτ= –

∞

0

μ(τ)ητdτ+

∞

0

μ(τ)ut(t)dτ=

∞

0

μ(τ)ηdτ+k0ut(t).

Then

I2= –k0∇ut(t) 2 2–

k0

ρ+ 1ut

ρ+2 ρ+2+

∞

0

μ(τ)

Ω

ut(t)ηdx dτ

+ 1

ρ+ 1 _∞

0

μ(τ)

Ω

–ut(t)ρut(t)ηdx dτ.

Using the Green formula and the Cauchy and Hölder inequalities, we have

_∞

0

μ(τ)

Ω

ut(t)ηdx dτ= – _∞

0

μ(τ)

Ω

∇ut(t)· ∇ηdx dτ

≤– ∞

0

μ(τ)∇ut2∇η2dτ

≤δ2C∇ut22–C

∞

0

μ(τ)∇η2₂dτ.

Using the method similar to that in the proof of Lemma4.1, we get

1

ρ+ 1 _∞

0

μ(τ)

Ω

–ut(t)ρut(t)ηdx dτ≤δ2C∇ut22–C

_∞

0

μ(τ)∇η2₂dτ.

Then

I2≤

–k0+ 2δ2C

∇ut2₂– k0

ρ+ 1ut

ρ+2 ρ+2– 2C

∞

0

μ(τ)∇η2₂dτ. (5.8)

Proof of Theorem2.3 Using

L(t) =ME(t) +εΦ(t) +Ψ(t),

E(t) = –γ∇ut(t)2₂+1 2

∞

0

μ(τ)∇η(τ)2₂dτ≤0,

and Lemmas5.2–5.3, we get

L_{(t) =ME(t) +εΦ(t) +Ψ(t)}

≤

–γM+

1 + γ

2

4δ1

ε+γ2δ2–k0+ 2δ2C

∇ut2 2

+–1 – (k0+ 1)δ1

ε+δ2+δ2C

∇u2₂–

k0

ρ+ 1–

ε ρ+ 1

ut ρ+2 ρ+2 + M

2 – 2C

∞

0

μ(τ)∇η2₂dτ

+

ε

4δ1

+C ∞

0

μ2–q(τ)dτ

∞

0

μq(τ)∇η2₂dτ+ε

Ω

f(u)u dx.

TakingM> 0 sufficiently large and suitableε,δ1,δ2> 0 such that

k0

ρ+ 1–

ε

ρ+ 1> 0, –(1 – 2δ1)ε+δ2+δ2C< 0,

–γM+

1 + γ

2

4δ1

ε+γ2δ2–k0+ 2δ2C< 0, M

2 – 2C

_{> 0,}

by using the inequalityμ(τ)≤–k1μq(τ) we have

M

2 – 2C

∞ 0

μ(τ)∇η2₂dτ≤–k1

M

2 – 2C

∞ 0

μq(τ)∇η2₂dτ.

Therefore

L_{(t)≤–γM+1 +} γ2

4δ1

ε+γ2δ2–k0+ 2δ2C

∇ut2 2

+–1 – (k0+ 1)δ1

ε+δ2+δ2C

∇u2₂–

k0

ρ+ 1–

ε ρ+ 1

ut ρ+2 ρ+2 – k1 M

2 – 2C

_– ε

4δ1

+C

∞

0

μ2–q(τ)dτ

∞

0

μq(τ)∇η2₂dτ

+ε

Ω

f(u)u dx.

By Remark2.1and (A2), taking suitableM> 0,ε,δ1,δ2> 0, we get

L_{(t)≤–C∇ut}2

2+∇u22+ut ρ+2 ρ+2+

∞

0

μq(τ)∇η2₂dτ–

Ω

f(u)u dx

≤–C

∇ut22+∇u22+ut ρ+2 ρ+2+

∞

0

Using (A3), we can easily show that

_∞

0 μ1–

θ_{(τ)dτ<∞for anyθ< 2 –q. Then}

Λ:= ∞

0

μ1–θ(τ)

Ω

|∇η|2dx dτ

≤2 ∞

0

μ1–θ(τ)

Ω

_∇u(t)2

+∇u(t–τ)2dx dτ

≤C

∞

0

μ1–θ(τ)dτ<L (5.10)

with positive constantL> 1.

Using the conditions of Theorem2.3, the Hölder inequality, and (5.10), we see that

η2_M= ∞

0

μ(τ)

Ω

|∇η|2dx dτ= ∞

0

μ(τ)∇η2₂dτ

= ∞

0

μ

(q–1)(1–θ)

q–1+θ _(τ)∇_η2

(q–1)

q–1+θ

2 ·μ

θq

q–1+θ_(τ)∇_η2 θ q–1+θ

2 dτ

≤

∞

0

μ1–θ(τ)∇η2₂dτ

q–1

q–1+θ

·

∞

0

μq(τ)∇η2₂dτ

θ q–1+θ

≤L

∞

0

μq(τ)|∇η|2₂dτ

θ q–1+θ

. (5.11)

Therefore we get, forσ> 1,

Eσ_(t)≤CEσ–1₍₀₎

∇ut2₂+ut ρ+2

ρ+2+∇u22–

Ω F(u)dx

+C

∞

0

μ(τ)∇η2₂dx dτ

σ

≤CEσ–1(0)

∇ut2 2+ut

ρ+2

ρ+2+∇u22–

Ω F(u)dx

+C

∞

0

μq(τ)∇η2₂dτ

σ θ q–1+θ

. (5.12)

By choosingθ=1₂ andσ= 2q– 1 (hence _q–1+σ θ_θ= 1) estimate (5.12) gives

Eσ_(t)≤C

∇ut2₂+ut ρ+2

ρ+2+∇u22–

Ω

F(u)dx+ ∞

0

μq(τ)∇η2₂dτ

. (5.13)

A combination of (5.9) and (5.13) then leads to

L_(t)≤–CEσ

(t).

By Lemma5.1we have

L_(t)≤–C 1

βσ 2

A simple integration of (5.14) over (0,t) yields

L(t)≤C1(1 +t)– 1

σ–1_{, t}≥_{0, (5.15)}

that is,

L(t)≤C1(1 +t)– 1

2(q–1)_{, t}≥_0.

6 Conclusions

In this paper, we consider the Dirichlet boundary value problem of nonlinear evolution equation with hereditary memory, variable density, and external force term. We prove the existence of a global solution by means of the Galerkin method, establish the exponen-tial stability by using only one auxiliary functional (this method is simpler than that in [1]), and also show the polynomial stability under suitable conditions. Under suitable hy-potheses on the external force term functionf and integral kernel functionμwithγ ≥0 in the model, we can further consider the local existence and blowup phenomenon of the solution.

Acknowledgements

The authors would like to thank the referee for his/her careful reading and kind suggestions.

Funding

This work was supported by National Natural Science Foundation of China (No. 11201258).

Availability of data and materials

The data used to support the findings of this study are available from the corresponding author upon request.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally to the writing of this paper. The authors read and approved the final manuscript.

Authors’ information

FL currently works at the School of Mathematical Sciences, Qufu Normal University, P.R. China. He does research in Applied Mathematics and Analysis. He and his group are engaged in the research on the well-posedness and longtime dynamics for some nonlinear evolution equations. ZJ has got Master’s degree in Qufu Normal University and is a member of Li’s group.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 18 November 2018 Accepted: 7 February 2019

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