Existence and general decay for a viscoelastic equation with logarithmic nonlinearity

https://doi.org/10.4134/JKMS.j210084 pISSN: 0304-9914 / eISSN: 2234-3008

EXISTENCE AND GENERAL DECAY FOR A VISCOELASTIC EQUATION WITH

LOGARITHMIC NONLINEARITY

Tae Gab Ha and Sun-Hye Park

Abstract. In the present work, we investigate a viscoelastic equation involving a logarithmic nonlinear source term. After proving the existence of solutions, we establish a general decay estimate of the solution using energy estimates and theory of convex functions. This result extends and complements some previous results of [9, 21].

1. Introduction

We consider the viscoelastic equation with logarithmic nonlinear source u_tt− ∆u +

Z t 0

h(t − s)∆u(s)ds = |u|^γ−2u ln |u| in Ω × (0, ∞), (1)

u = 0 on ∂Ω × (0, ∞), (2)

u(0) = u₀, u_t(0) = u₁ in Ω, (3)

where Ω ⊂ R^N (N ≥ 1) is a bounded domain having smooth boundary ∂Ω.

One of the main concerns in the study of viscoelastic problems is to establish more general and optimal decay rates of solutions under minimal conditions of h. And many stability results have been established ([5, 10, 18–21, 23]).

For instance, Messaoudi [18] obtained generalized decay rates of solutions to problem (1)-(3) without the logarithmic nonlinear source term when h verifies

(4) h⁰(t) ≤ −%(t)h(t),

Received January 28, 2021; Revised May 31, 2021; Accepted July 5, 2021.

2010 Mathematics Subject Classification. 35L70, 35B40, 35L05, 35B35.

Key words and phrases. General decay, logarithmic source, viscoelastic equation, convex function.

The first author was supported by Basic Science Research Program through the Na- tional Research Foundation of Korea(NRF) funded by the Ministry of Education (NRF- 2019R1I1A3A01051714). The second author was supported by Basic Science Research Pro- gram through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2020R1I1A3066250).

c

2021 Korean Mathematical Society 1433

where % is positive, differentiable, and non-increasing. The authors of [19]

proved a decay rate of general type for a quasilinear equation under the condi- tion

(5) h⁰(t) ≤ −%(t)h^p(t),

here 1 ≤ p < ³₂. And then, a natural question ‘Can the range of parameter p be extended from 1 ≤ p < ³₂ to 1 ≤ p < 2?’ was raised. Mustafa [21]

answered for the question inspired by the ideas of [12] and [13]. He established more generalized and explicit decay results for (1)-(3) without the logarithmic nonlinearity when h fulfills

(6) h⁰(t) ≤ −%(t)H(h(t)),

where H is a convex function meeting some conditions. He claimed that (5) with 1 ≤ p < 2 is only a special case of (6). For the articles associated with (6), we refer [11, 16].

In this article, we are concerned with a general energy decay rate for problem (1)-(3). This type of equations with logarithmic nonlinearity has its physical applications in the fields of hydrodynamics, thermodynamics and filtration the- ory and so on. For more detail applications, we refer to [2, 8]. Several authors considered nonlinear models of parabolic or hyperbolic equations with such nonlinearity [3, 4, 7, 14, 17, 22]. The authors of [4, 22] studied the equation

ut− div(|∇u|^p−2∇u) − ∆ut= |u|^γ−2u ln |u|.

They proved the existence of solutions by using the potential well method and the logarithmic Sobolev inequality. Di et al. [7] discussed the strongly damped wave equation

utt− ∆u − ∆ut= |u|^γ−2u ln |u|.

They used the modified potential well method to overcome difficulty created by the presence of −∆u instead of −div(|∇u|^p−2∇u). Most recent, Ha and Park [9] proved the existence and uniqueness of local solutions for a strongly damped viscoelastic wave equation with logarithmic nonlinearity, and investi- gated a blow-up result. Motivated by these results, we are concerned with a new energy decay result for solutions to problem (1)-(3). As we know, stability for viscoelastic wave equations involving logarithmic nonlinear source term is seldom studied.

2. Preliminaries Let (ϕ, φ) = R

Ωϕ(x)φ(x)dx. || · ||_q denotes the norm of the space L^q(Ω), 1 ≤ q ≤ ∞. For simplicity, we denote || · ||₂ by || · ||. Let B_q be the optimal constant with ||v||_q ≤ Bq||∇v|| for v ∈ H₀¹(Ω), where

2 ≤ q ≤ 2N

N − 2, if N ≥ 3; 2 ≤ q < ∞, if N = 1, 2.

We endow assumptions on γ and h as below:

(A1) The exponent γ verifies

(7) 2 < γ < ∞, if N = 1, 2; 2 < γ < 2(N − 1)

N − 2 , if N ≥ 3.

(A₂) Let h : [0, ∞) → (0, ∞) is a differentiable and nonincreasing function with

(8) 1 −

Z ∞ 0

h(s)ds := l > 0.

(A3) As in [21], h satisfies

(9) h⁰(t) ≤ −%(t)H(h(t)) for all t ≥ 0,

where H : (0, ∞) → (0, ∞) is a C¹-function, which is either linear or strictly increasing and strictly convex C²-function on (0, r], r ≤ h(0), with H(0) = H⁰(0) = 0, and % is positive, nonincreasing, and differentiable.

The conditions (A2) and (A3) imply the existence of t^∗> 0 with

(10) h(t^∗) = r.

Some examples of the kernel function h satisfying (A₂) and (A₃) are provided by Mustafa [21].

Let

(hz)(t) = Z t

0

h(t − s)||z(t) − z(s)||²ds, kβ(t) = βh(t) − h⁰(t),

and

Cβ = Z ∞

0

h²(s) k_β(s)ds.

By the argument of [21], we have:

Lemma 2.1. For any β > 0 and z ∈ L²_loc(0, ∞; L²(Ω)),

||

Z t 0

h(t − s)(z(t) − z(s))ds||²≤ Cβ(k_βz)(t) (11)

and

||

Z t 0

h⁰(t − s)(z(t) − z(s))ds||²

≤ 2Z t 0

kβ(s)ds

(kβz)(t) + 2β²Cβ(kβz)(t).

(12)

To deal with logarithmic source term, we define J (w) = l

2||∇w||²−1 γ

Z

Ω

|w(x)|^γln |w(x)|dx + 1 γ²||w||^γ_γ, (13)

I(w) = l||∇w||²− Z

Ω

|w(x)|^γln |w(x)|dx, (14)

(15) d = inf

w∈H₀¹(Ω)\{0}

sup

ξ>0

J (ξw).

Then, we know ([6, 7, 25])

(16) d = inf

w∈NJ (w), here N = {w ∈ H₀¹(Ω) \ {0} | I(w) = 0}.

In this article, we use the auxiliary lemma below several times.

Lemma 2.2 (Lemma 2.1 in [9]). For each µ > 0, there hold

| ln τ | ≤ τ^−µ

eµ for 0 < τ < 1 and 0 ≤ ln τ ≤τ^µ

eµ for τ ≥ 1.

3. Existence of solutions

Definition. It is said that a function u is a weak solution of problem (1)-(3) if it fulfills

u ∈ C([0, T ]; H₀¹(Ω)) ∩ C¹([0, T ]; L²(Ω)) ∩ C²([0, T ]; H⁻¹(Ω)), hutt, wi + (∇u, ∇w) −

Z t 0

h(t − s)(∇u(s), ∇w)ds = (|u|^γ−2u ln |u|, w) for w ∈ H₀¹(Ω), where h·, ·i is the duality pairing between H₀¹(Ω) and H⁻¹(Ω), and u(0) = u0in H₀¹(Ω), ut(0) = u1in L²(Ω).

Theorem 3.1. Assume that (A1) and (A2) hold. Let (u0, u1) ∈ H₀¹(Ω)×L²(Ω), I(u0) > 0, and

E(u₀, u₁) := 1

2||u1||²+1

2||∇u0||²−1 γ

Z

Ω

|u0(x)|^γln |u₀(x)|dx + 1

γ²||u0||^γ_γ< d.

Then, problem (1)-(3) admits a unique weak solution.

Proof. Existence. Let {vj}_j∈N be a basis of H₀¹(Ω), which is orthonormal in L²(Ω). For a fixed n ∈ N, we set Vn= span{v₁, v₂, . . . , v_n}. Theory of ordinary differential equations provide a unique local solution uⁿ(x, t) =Pn

j=1f_jⁿ(t)v_j(x) on a maximal interval [0, tn), tn∈ (0, T ], for the approximate problem











(uⁿ_tt(t), v) + (∇uⁿ(t), ∇v) −Rt

0h(t − s)(∇uⁿ(s), ∇v)ds

= (|uⁿ(t)|^γ−2uⁿ(t) ln |uⁿ(t)|, v) v ∈ Vn, uⁿ(0) = uⁿ₀ =Pn

j=1(u0, vj)vj → u0 in H₀¹(Ω), uⁿ_t(0) = uⁿ₁ =Pn

j=1(u₁, v_j)v_j → u₁ in L²(Ω).

(17)

Now, we show that tn = T and that the solution is bounded independent of n, t. Replacing v by uⁿ_t(t) in the first equation of (17), one gets

(18) d

dtE(uⁿ(t), uⁿ_t(t)) = 1

2(h⁰∇uⁿ)(t) −h(t)

2 ||∇uⁿ(t)|| ≤ 0,

where

E(uⁿ(t), uⁿ_t(t)) = 1

2||uⁿ_t(t)||²+1 2

 1 −

Z t 0

h(s)ds

||∇uⁿ(t)||²+1

2(h∇uⁿ)(t)

−1 γ

Z

Ω

|uⁿ(x, t)|^γln |uⁿ(x, t)|dx + 1

γ²||uⁿ(t)||^γ_γ. Thus, we have

E(uⁿ(t), uⁿ_t(t)) ≤ E(uⁿ(0), uⁿ_t(0))

= 1

2||uⁿ₁||²+1

2||∇uⁿ₀||²−1 γ

Z

Ω

|uⁿ₀(x)|^γln |uⁿ₀(x)|dx+ 1

γ²||uⁿ₀||^γ_γ. (19)

Since the functions E and I are continuous,

(20) E(uⁿ₀, uⁿ₁) < d and I(uⁿ₀) > 0 for appropriately large n.

This asserts

(21) E(uⁿ(t), uⁿ_t(t)) < d and

(22) I(uⁿ(t)) > 0 for appropriately large n and t ≥ 0.

Indeed, from (19) and (20), it is clear

(23) E(uⁿ(t), uⁿ_t(t)) < d for appropriately large n and t ≥ 0.

In order to show I(uⁿ(t)) > 0 for t ≥ 0, let us suppose that there exists t0> 0 such that I(uⁿ(t0)) = 0. Then,

E(uⁿ(t₀), uⁿ_t(t₀)) ≥ J (uⁿ(t₀)) ≥ inf

w∈NJ (w) = d, which contradicts (23). Thus, (22) is proved.

From (21), we obtain (24) 0 < 1

2||uⁿ_t(t)||²+l(γ − 2)

2γ ||∇uⁿ(t)||²+1

γI(uⁿ(t)) < d.

Therefore, we can choose a subsequence of {uⁿ}, we still denote it by {uⁿ}, with

(25)

 uⁿ → u weak^∗ in L^∞(0, T ; H₀¹(Ω)), uⁿ_t → ut weak^∗ in L^∞(0, T ; L²(Ω)).

Aubin-Lions compactness theorem ensures

(26) uⁿ(x, t) → u(x, t) a.e. (x, t) ∈ Ω × (0, T ) and

|uⁿ(x, t)|^γ−2uⁿ(x, t) ln |uⁿ(x, t)|

→ |u(x, t)|^γ−2u(x, t) ln |u(x, t)| a.e. (x, t) ∈ Ω × (0, T ).

(27)

Next, we show that

(28) |uⁿ|^γ−1ln |uⁿ| is bounded in L^∞(0, T ; L^γ−1γ (Ω)).

For this, we let

Ω1= {x ∈ Ω : |uⁿ(x, t)| < 1} and Ω2= {x ∈ Ω : |uⁿ(x, t)| ≥ 1}.

Thanks to 2 < γ < _{N −2}^2N , we can take µ1 > 0 such that 2 < γ +_γ−1^µ1γ < _{N −2}^2N . So, by the same argument of Eq. (3.7) of [9], we know

Z

Ω

 

|uⁿ(x, t)|^γ−1ln |uⁿ(x, t)|

_γ−1^γ dx

≤ 1

e(γ − 1)

_γ−1^γ

|Ω1| + B

γ(γ−1+µ1) γ−1 γ(γ−1+µ1)

γ−1

 1 eµ1

_γ−1^γ

||∇uⁿ(t)||^{γ(γ−1+µ1)γ−1} ≤ c (29)

for some c > 0. By (27), (28) and Lemma 1.3 in [15], one gets

|uⁿ|^γ−2uⁿln |uⁿ| → |u|^γ−2u ln |u| weak^∗ in L^∞(0, T ; L^γ−1γ (Ω)).

The rest of the proof can be done as the proofs of Lemma 3.1 and Theorem 3.1

of [9]. 

4. New general decay The energy of problem (1)-(3) is defined by E(t) =: E(u(t), u_t(t)) = 1

2||u_t(t)||²+1 2

 1−

Z t 0

h(s)ds

||∇u(t)||²+1

2(h∇u)(t)

−1 γ

Z

Ω

|u(x, t)|^γln |u(x, t)|dx + 1

γ²||u(t)||^γ_γ. (30)

Then, we find E(t) ≥ 1

2||ut(t)||²+1

2(h∇u)(t)+J(u(t))

= 1

2||ut(t)||²+1

2(h∇u)(t)+l(γ − 2)

2γ ||∇u(t)||²+ 1

γ²||u(t)||^γ_γ+1 γI(u(t)) (31)

and

(32) E⁰(t) = 1

2(h⁰∇u)(t) − h(t)

2 ||∇u(t)||²≤ 0.

By the same arguments of (21) and (22), if I(u0) > 0 and E(0) = E(u0, u1) < d, then

(33) I(u(t)) > 0 and 0 < E(t) < d for t ≥ 0.

Following the ideas of [21], we define

L(t) = M E(t) + M₁Φ(t) + M₂Ψ(t), where M, M1, M2> 0 and

Φ(t) = (u_t(t), u(t)), Ψ(t) = − Z t

0

h(t − s)(u(t) − u(s), u_t(t))ds.

Lemma 4.1. For every 0 < η < 1 and β > 0, it holds Φ⁰(t) ≤ ||ut(t)||²− l

2||∇u(t)||²+ Z

Ω

|u(x, t)|^γln |u(x, t)|dx +Cβ

2l(kβ∇u)(t) (34)

and

Ψ⁰(t) ≤ −Z t 0

h(s)ds − η

||ut(t)||² +c(Cβ+ 1)

η (kβ∇u)(t) + ηCE(0)||∇u(t)||², (35)

where

C_E(0)= 1 + B

γ−1−µ3 2(γ−1−µ3)

eµ₃

22γE(0) l(γ − 2)

γ−2−µ3

+

 B

γ−1+µ4 2(γ−1+µ4)

eµ₄

22γE(0) l(γ − 2)

γ−2+µ4

. Proof. Using (1)-(2) and Young’s inequality, one finds

Φ⁰(t) = ||u_t(t)||²− 1 −

Z t 0

h(s)ds

||∇u(t)||²+ Z

Ω

|u(x, t)|^γln |u(x, t)|dx

+ Z t

0

h(t − s)(∇u(s) − ∇u(t), ∇u(t))ds

≤ ||ut(t)||²− l

2||∇u(t)||²+ Z

Ω

|u(x, t)|^γln |u(x, t)|dx

+ 1 2l||

Z t 0

h(t − s)(∇u(s) − ∇u(t))ds||². (36)

The formula (34) is observed applying (11) to the last term of (36). Similarly, Ψ⁰(t) = −

Z t 0

h(s)ds||u_t(t)||²− Z t

0

h⁰(t − s)(u(t) − u(s), u_t(t))ds

+ 1 −

Z t 0

h(s)dsZ t 0

h(t − s)(∇u(t) − ∇u(s), ∇u(t))ds

+ ||

Z t 0

h(t − s)(∇u(t) − ∇u(s))ds||²

− Z t

0

h(t − s) Z

Ω

(u(x, t) − u(x, s))|u(x, t)|^γ−2u(x, t) ln |u(x, t)|dxds

:= − Z t

0

h(s)ds||ut(t)||²+

6

X

i=3

Ki. (37)

By (11) and (12), it holds, for 0 < η < 1 and β > 0,

|K₃| ≤ η||u_t(t)||²+ 1 2η

Z t 0

k_β(s)ds

(k_βu)(t) + 1

2ηβ²C_β(k_βu)(t)

≤ η||ut(t)||²+(β(1 − l) + h(0))B²₂

2η (kβ∇u)(t) + B₂²β²

2η Cβ(kβ∇u)(t),

|K4| ≤ η||∇u(t)||²+

1 −Rt

0h(s)ds²

4η Cβ(kβ∇u)(t),

|K5| ≤ Cβ(kβ∇u)(t) < Cβ

η (kβ∇u)(t), and

|K6| ≤ η Z

Ω

|u(x, t)|^γ−1ln |u(x, t)|²

dx +B₂²Cβ

4η (kβ∇u)(t).

Let

Ω3= {x ∈ Ω : |u(x, t)| < 1} and Ω4= {x ∈ Ω : |u(x, t)| ≥ 1}.

Due to 2 < 2(γ − 1) < _{N −2}^2N , there exist µ3 > 0 and µ4 > 0 such that 2 <

2(γ − 1) − 2µ3<_{N −2}^2N and 2 < 2(γ − 1 + µ4) < _{N −2}^2N , respectively. So, adapting Lemma 2.2, we get

Z

Ω

|u(x, t)|^γ−1ln |u(x, t|² dx

≤ 1 eµ3

²Z

Ω₃

|u(x, t)|^{2(γ−1)−2µ3}dx + 1 eµ4

²Z

Ω₄

|u(x, t)|^{2(γ−1+µ4)}dx

≤ B

γ−1−µ₃ 2(γ−1−µ3)

eµ3

²2γE(0) l(γ − 2)

^γ−2−µ3

||∇u(t)||²

+ B

γ−1+µ4

2(γ−1+µ₄)

eµ4

22γE(0) l(γ − 2)

γ−2+µ4

||∇u(t)||².

Collecting these and (37), the inequality (35) is deduced.  Lemma 4.2. Let the conditions of Theorem 3.1 be fulfilled. Moreover, assume that

E(0) < min (

d,l(γ − 2) 2γ

 leκ 2B_γ+κ^γ+κ

_γ+κ−2² )

,

where κ > 0 with 2 < γ + κ < _{N −2}^2N . Then, there exist ρ > 0, c₁> 0, and c₂> 0 such that

(38) L⁰(t) ≤ −ρE(t) +1

2(h∇u)(t) − 3(1 − l)||∇u(t)||² for t ≥ t^∗. Moreover, E(t) is equivalent to L(t).

Proof. Recalling h⁰= βh − k_β, (33), (1), and (A₂), we get L⁰(t) ≤ M β

2 (h∇u)(t) −M

2 (k_β∇u)(t) − M

2 h(t)||∇u(t)||²

−n M2

Z t 0

h(s)ds − η

− M1

o||ut(t)||²

+ M1

Z

Ω

|u(x, t)|^γln |u(x, t)|dx −nM₁l

2 − M2ηCE(0)

o||∇u(t)||²

−n

−M₁C_β

2l −M₂c(C_β+ 1) η

o

(kβ∇u)(t), and hence

L⁰(t) ≤ − ρE(t) +nM β 2 +ρ

2 o

(h∇u)(t)

−n M₂

Z t 0

h(s)ds − η



− M1−ρ 2

o||ut(t)||²

+ M1−ρ

γ

Z

Ω

|u(x, t)|^γln |u(x, t)|dx + ρ

γ²||u(t)||^γ_γ

−nM₁l

2 − M2ηCE(0)−ρ 2

 1 −

Z t 0

h(s)dso

||∇u(t)||²

−nM

2 −M₁C_β

2l −M₂c(C_β+ 1) η

o(k_β∇u)(t) for any ρ > 0.

(39)

Owing to the assumption (A1), we can pick κ > 0 with 2 < γ + κ < ∞, if N = 1, 2; 2 < γ + κ < 2N

N − 2, if N ≥ 3.

From this, Lemma 2.2, (31), and (32), we find Z

Ω

|u(x, t)|^γln |u(x, t)|dx ≤ 1 eκ

Z

|u(x,t)|≥1

|u(x, t)|^γ+κdx

≤ 1

eκ||u(t)||^γ+κ_γ+κ

≤ B_γ+κ^γ+κ

eκ (||∇u(t)||²)^γ+κ−22 ||∇u(t)||²

≤ B_γ+κ^γ+κ eκ

2γE(t) l(γ − 2)

^γ+κ−2₂

||∇u(t)||²

≤ B_γ+κ^γ+κ eκ

2γE(0) l(γ − 2)

^γ+κ−2₂

||∇u(t)||² and

||u(t)||^γ_γ ≤ B_γ^γ||∇u(t)||^γ ≤ B_γ^γ2γE(0) l(γ − 2)

^γ−2₂

||∇u(t)||².

Substituting these into (39), letting h^∗=Rt^∗

0 h(s)ds, and selecting M1>_γ^ρ, we observe

L⁰(t) ≤ − ρE(t) +nM β 2 +ρ

2 o

(h∇u)(t) −n

M2(h^∗− η) − M1−ρ 2

o||ut(t)||²

−n M1

l

2 −B_γ+κ^γ+κ eκ

2γE(0) l(γ − 2)

^γ+κ−2₂ 

− M2ηC_E(0)

−ρ 2

1 − Z t

0

h(s)ds

−ρB_γ^γ γ²

2γE(0) l(γ − 2)

^γ−2₂ o

||∇u(t)||²

−nM

2 −M1Cβ

2l −M2c(Cβ+ 1) η

o

(kβ∇u)(t) for t ≥ t^∗. Taking η = _4M^l

2, we have L⁰(t) ≤ − ρE(t) +nM β

2 +ρ 2 o

(h∇u)(t) −n

M2h^∗− l

4− M1−ρ 2

o||ut(t)||²

−n M1

l

2 −B_γ+κ^γ+κ eκ

2γE(0) l(γ − 2)

^γ+κ−2₂ 

−lC_E(0) 4

−ρ 2

 1 −

Z t 0

h(s)ds

−ρB_γ^γ γ²

2γE(0) l(γ − 2)

^γ−2₂ o

||∇u(t)||²

−nM

4 −4cM₂²

l +M

4 − Cβ

M1

2l +4cM₂² l

o

(kβ∇u)(t) for t ≥ t^∗. We fix M1>_γ^ρ suitably large again so that

M1

l

2−B_γ+κ^γ+κ eκ

2γE(0) l(γ − 2)

^γ+κ−2₂ 

−lC_E(0)

4 > 4(1 − l) (40)

and pick M2> ₄^l appropriately large such that

(41) M2h^∗− l

4− M1> 1.

Since ^βh_k^2(s)

β(s) < h(s), lim_β→0+βCβ= lim_β→0+R∞ 0

βh²(s)

k_β(s)ds = 0. Thus, there ex- ists 0 < β0< 1 such that

(42) βC_β < 1

8(^M_2l¹ +^4cM_l ²²)

for β < β₀.

Now, we take β = _2M¹ and M > 0 appropriately so that

(43) β = 1

2M < β0 and M

4 −4cM₂² l > 0.

From (42) and (43), we obtain

(44) M

4 − Cβ

M₁

2l +4cM₂² l



> 0.

Considering (40), (41), (43), (44), we arrive at L⁰(t) ≤ − ρE(t) +n1

4 +ρ 2 o

(h∇u)(t) −n 1 − ρ

2

o||ut(t)||²

−n

4(1 − l) −ρ 2

 1 −

Z t 0

h(s)ds

−ρB_γ^γ γ²

2γE(0) l(γ − 2)

^γ−2₂ o

||∇u(t)||². (38) can be obtained selecting ρ > 0 appropriately small. Furthermore, Young inequality, (33), and (31) provide

|L(t) − M E(t)|

≤ M1+ M2

2 ||ut(t)||²+M2B²₂ 2

Z t 0

h(s)ds

(h∇u)(t) +M1B₂²

2 ||∇u(t)||²

= M₁+ M₂

2 ||ut(t)||²+M₂B²₂ 2

Z t 0

h(s)ds

(h∇u)(t) + M₁B₂²γ

l(γ − 2)

J (u(t)) − 1

γI(u(t)) − 1

γ²||u(t)||^γ_γ

≤ maxnM1+ M2

4 ,M2B²₂(1 − l)

4 , M1B₂²γ l(γ − 2)

o E(t).

Again, taking M > maxn

M₁+M₂

4 ,^M2B22₄^(1−l),^M_l(γ−2)^1B22γo

, we complete the proof.

 Theorem 4.3. Let the conditions of Lemma 4.5 and (A3) be satisfied. Then, the energy of problem (1)-(3) verifies

E(t) ≤ C0H˜⁻¹ ω

Z t h⁻¹(r)

%(s)ds

for t ≥ t^∗, where ω > 0, C0> 0, and

(45) H(s) =˜

Z r s

1 τ H⁰(τ )dτ.

Proof. The proof is similar to those of [21, 24]. But we state the proof here for completeness. Since h and % are nonincreasing and continuous on [0, ∞), they are bounded on [0, t^∗]. Thus, for some c₃> 0 and c₄> 0, it holds

c3≤ %(t)H(h(t)) ≤ c4 for t ∈ [0, t^∗].

Moreover, we observe that

h⁰(t) ≤ −%(t)H(h(t)) ≤ −c3≤ − c₃

h(0)h(t) for t ∈ [0, t^∗].

From this, (38), and (32), we get L⁰(t) ≤ −ρE(t) − h(0)

2c₃ (h⁰∇u)(t) +1 2

Z t t^∗

h(s)||∇u(t) − ∇u(t − s)||²ds

≤ −ρE(t) −h(0) c3

E⁰(t) +1 2

Z t t^∗

h(s)||∇u(t) − ∇u(t − s)||²ds, t ≥ t^∗. Letting F (t) = L(t) + ^h(0)_c

3 E(t), we have F⁰(t) ≤ −ρE(t) +1

2 Z t

t^∗

h(s)||∇u(t) − ∇u(t − s)||²ds for t ≥ t^∗. (46)

Case 1: H is linear, that is, H(s) = as for some a > 0.

Using %⁰(t) ≤ 0, (46), (9), and (32), we get (%(t)F (t) + 1

aE(t))⁰

≤ − ρ%(t)E(t) + 1 2

Z t t^∗

%(s)h(s)||∇u(t) − ∇u(t − s)||²ds +1 aE⁰(t)

≤ − ρ%(t)E(t) − 1 2a

Z t t^∗

h⁰(s)||∇u(t) − ∇u(t − s)||²ds +1 aE⁰(t)

≤ − ρ%(t)E(t) for t ≥ t^∗. (47)

Case 2: H is nonlinear.

Let Ξ(t) =Rt 0

R∞

t−sh(τ )dτ

||∇u(s)||²ds. Then it meets (see Lemma 3.4 in [21])

Ξ⁰(t) ≤ −1

2(h∇u)(t) + 3(1 − l)||∇u(t)||². From Lemma 4.2 and this, we get

(L(t) + Ξ(t))⁰ ≤ −ρE(t).

(48)

This and (33) ensure 0 <

Z ∞ 0

E(s)ds ≤ Z t

t^∗

E(s)ds

≤ −1 ρ

Z t t^∗

(L⁰(s) + Ξ⁰(s))ds ≤ L(t^∗) + Ξ(t^∗) ρ < ∞.

(49)

Now, we define

Γ(t) := m Z t

t^∗

||∇u(t) − ∇u(t − s)||²ds and

χ(t) := − Z t

t^∗

h⁰(s)||∇u(t) − ∇u(t − s)||²ds.

From (49), we can select 0 < m < 1 such that

(50) Γ(t) < 1 for t ≥ t^∗.

It is also observed that

(51) χ(t) ≤ −(h⁰∇u)(t) ≤ −2E⁰(t).

Making use of (A₃), the fact H(λy) ≤ λH(y) for 0 ≤ λ ≤ 1 and y ∈ (0, r], and Jensen’s inequality, we infer

χ(t) = − 1 mΓ(t)

Z t t^∗

Γ(t)h⁰(s)m||∇u(t) − ∇u(t − s)||²ds

≥ 1

mΓ(t) Z t

t^∗

Γ(t)%(s)H(h(s))m||∇u(t) − ∇u(t − s)||²ds

≥ %(t) mΓ(t)

Z t t^∗

H

Γ(t)h(s)

m||∇u(t) − ∇u(t − s)||²ds

≥%(t) m H

m Z t

t^∗

h(s)||∇u(t) − ∇u(t − s)||²ds

=%(t) m H

m Z t

t^∗

h(s)||∇u(t) − ∇u(t − s)||²ds , (52)

where H is an extension of H as H is a strictly increasing and strictly convex C²-function on (0, ∞) and the fact mRt

t^∗h(s)||∇u(t) − ∇u(t − s)||²ds < r is used in the last equality. Thus, we see from (52) that

Z t t^∗

h(s)||∇u(t) − ∇u(t − s)||²ds ≤ 1

mH⁻¹mχ(t)

%(t)

 . Adapting this to (46), we find

F⁰(t) ≤ −ρE(t) + 1

2mH⁻¹mχ(t)

%(t)

 for t ≥ t^∗. (53)

On the other hand, for the convex function H, it is known that yz ≤ H^∗(y) + H(z) for y, z ≥ 0

(54) and

H^∗(y) = y(H⁰)⁻¹(y) − H((H⁰)⁻¹(y)) for y ≥ 0, (55)

where H^∗ is the conjugate function of the convex function H (see [1]).

Next, let 0 < θ < r, E (t) = _E(0)^E(t), and c₅> 0. Since H⁰(s) > 0, H⁰⁰(s) > 0, and H(0) = H⁰(0) = 0, we observe from (53), (54), and (55) that

h

H⁰(θE (t))F (t) + c₅E(t)i0

≤ − ρH⁰(θE (t))E(t) + 1

2mH⁰(θE (t))H⁻¹mχ(t)

%(t)

+ c₅E⁰(t)

≤ − ρH⁰(θE (t))E(t) + 1 2mH^∗

H⁰(θE (t)) + χ(t)

2%(t)+ c5E⁰(t)

≤ − ρE(0)H⁰(θE (t))E (t) + θ

2mE(t)H⁰(θE (t)) + χ(t)

2%(t)+ c5E⁰(t), (56)

we used θE (t) < r in the last equality. Using %⁰(t) ≤ 0 and (51), we have h%(t)n

H⁰(θE (t))F (t) + c₅E(t)o

+ E(t)i⁰

≤ − %(t)

ρE(0) − θ 2m



H⁰(θE (t))E (t).

(57)

Choosing θ > 0 sufficiently small such that c6:= ρE(0) −_2m^θ > 0, we arrive at h

%(t)n

H⁰(θE (t))F (t) + c5E(t)o

+ E(t)i0

≤ − c₆%(t)H⁰(θE (t))E (t).

(58) Letting

F (t) =

( %(t)F (t) +¹_aE(t) if H is linear;

%(t)n

H⁰(θE (t))F (t) + c5E(t)o

+ E(t) if H is nonlinear, (59)

we obtain from (47) and (58) that

(60) F⁰(t) ≤ −c₇%(t)H₀(E (t)) for t ≥ t^∗, where c₇= min{^ρE(0)_a , c₆} and

H₀(s) =

 s if H is linear;

sH⁰(θs) if H is nonlinear.

(61)

Due to F (t) ∼ E(t), there exist c8, c9> 0 such that (62) c₈F (t) ≤ E(t) ≤ c₉F (t).

Finally, putting

(63) L(t) = c8F (t)

E(0) , we see that

(64) L(t) ≤ E(t) ≤ 1.

Due to the fact H0is increasing on (0, 1], (63), (60) and (64), we deduce (65) L⁰(t) ≤ −c₁₀%(t)H₀(L(t)) for t ≥ t^∗,

where c10= _E(0)^c7c8. Thus, we get Z t

t^∗

c10%(s)ds ≤ − Z t

t^∗

L⁰(s)

H0(L(s))ds = − Z t

t^∗

L⁰(s) L(s)H⁰(θL(s))ds

=

Z θL(t^∗) θL(t)

1 sH⁰(s)ds

≤ Z r

θL(t)

1

sH⁰(s)ds = ˜H(θL(t)), (66)

here ˜H is the function defined in (45). Since ˜H is strictly decreasing on (0, r], we conclude, for some ω > 0,

L(t) ≤ 1 θ

H˜⁻¹ ω

Z t t^∗

%(s)ds

for t ≥ t^∗.

This completes the proof. 

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Tae Gab Ha

Department of Mathematics and Institute of Pure and Applied Mathematics Jeonbuk National University

Jeonju 54896, Korea

Email address: [email protected]

Sun-Hye Park

Office for Education Accreditation Pusan National University

Busan 46241, Korea

Email address: [email protected]