Asymptotic Behavior for a Nondissipative and Nonlinear System of the Kirchhoff Viscoelastic Type

Abstract

A wave equation of the Kirchhoff type with several nonlinearities is stabilized by a viscoelastic damping. We consider the case of nonconstant (and unbounded) coefficients. This is a nondissipative case, and as a consequence the nonlinear terms cannot be estimated in the usual manner by the initial energy. We suggest a way to get around this difficulty. It is proved that if the solution enters a certain region, which we determine, then it will be attracted exponentially by the equilibrium.

1. Introduction

We will consider the following wave equation with a viscoelastic damping term: 𝑢_𝑡𝑡+^𝑚_𝑖=1𝑏_𝑖(𝑡)|𝑢|^𝑝𝑖𝑢=1+^𝑘_𝑗=1𝑎_𝑗(𝑡)‖∇𝑢‖^2𝑞𝑗₂−Δ𝑢_𝑡0𝑢=0,ℎon(𝑡−Γ×𝑠)𝐑Δ₊𝑢(𝑠)𝑑𝑠,inΩ×𝐑₊,𝑢(𝑥,0)=𝑢₀,(𝑥),𝑢_𝑡(𝑥,0)=𝑢₁(𝑥),inΩ,(1.1) where Ω is a bounded domain in 𝐑^𝑛 with smooth boundary Γ=𝜕Ω and 𝑝_𝑖,𝑞_𝑗>0,𝑖=1,…,𝑚,𝑗=1,…,𝑘. The functions 𝑢₀(𝑥) and 𝑢₁(𝑥) are given initial data, and the (nonnegative) functions 𝑎_𝑗(𝑡),𝑏_𝑖(𝑡), and ℎ(𝑡) are at least absolutely continuous and will be specified later on. This problem arises in viscoelasticity where it has been shown by experiments that when subject to sudden changes, the viscoelastic response not only does depend on the current state of stress but also on all past states of stress. This gives rise to the integral term called the memory term. One may find a rich literature in this regard (with or without the Kirchhoff terms) treating mainly the stabilization of such systems for different classes of functions ℎ. We refer the reader to [1–25] and the references therein. For problems of the Kirchhoff type, one can consult [26–35] and in particular [36–46] where the equations are supplemented by a nonlinear source. Several questions, such as well-posedness and asymptotic behavior, have been discussed in these references, to cite but a few.

As is clear from the equation in (1.1), we consider here several nonlinearities and the relaxation function is not necessarily decreasing or even nonincreasing. These issues are important but do not constitute the main contribution in the present paper. In case that 𝑎_𝑗(𝑡) and 𝑏_𝑖(𝑡) are not nonincreasing, then we are in a nondissipative situation. This is the case also when the relaxation function oscillates (in case 𝑎_𝑗(𝑡),𝑏_𝑖(𝑡) are nonincreasing). Our argument here is simple and flexible. It relies on a Gronwall-type inequality involving several nonlinearities. We prove that there exists a sufficiently large 𝑇>0 and a constant 𝑈 after which (the modified energy of) global solutions are bounded below by 𝑈 or decay to zero exponentially. We were not able to find conditions directly on the initial data because the Gronwall inequality is applicable only after some large values of time.

For simplicity we shall consider the simpler case 𝑝₁=𝑝,𝑝_𝑖=0,𝑏₁=𝑏,𝑏_𝑖=0,𝑖=2,…,𝑚 and 𝑞₁=𝑞,𝑞_𝑗=0,𝑎₁=𝑎,𝑎_𝑗=0,𝑗=2,…,𝑘.

The local existence and uniqueness may be found in [36, 37].

Theorem 1.1. Assume that (𝑢₀,𝑢₁)∈𝐻¹⁰(Ω)×𝐿²(Ω) and ℎ(𝑡) is a nonnegative summable kernel. If 0<𝑝<2/(𝑛−2) when 𝑛≥3 and 𝑝>0 when 𝑛=1,2, then there exists a unique solution 𝑢 to problem (1.1) such that 𝑢∈𝐶[0,𝑇];𝐻¹⁰(Ω)∩𝐶¹[0,𝑇];𝐿²(Ω)(1.2) for 𝑇 small enough.

The plan of the paper is as follows. In the next section we prepare some materials needed to prove our result. Section 3 is devoted to the statement and proof of our theorem.

2. Preliminaries

In this section we define the different functionals we will work with. We prove an equivalence result between two functionals. Further, some useful lemmas are presented. We define the (classical) energy by 𝐸(𝑡)=12‖‖𝑢^𝑡‖‖²²+‖∇𝑢‖²²+𝑎(𝑡)2(𝑞+1)‖∇𝑢‖^22(𝑞+1)+𝑏(𝑡)𝑝+2‖𝑢‖^𝑝+2𝑝+2,𝑡≥0,(2.1) where ‖⋅‖_𝑝 denotes the norm in 𝐿^𝑝(Ω). Then by  (1.1)  it is easy to see that for 𝑡≥0𝐸^(𝑡)=_Ω∇𝑢_𝑡⋅^𝑡0ℎ(𝑡−𝑠)∇𝑢(𝑠)𝑑𝑠𝑑𝑥+𝑎^(𝑡)2(𝑞+1)‖∇𝑢‖^22(𝑞+1)+𝑏^(𝑡)𝑝+2‖𝑢‖^𝑝+2𝑝+2.(2.2) The first term in the right-hand side of (2.2) may be written as the derivative of some expression; namely, _Ω∇𝑢_𝑡⋅^𝑡0ℎ(𝑡−𝑠)∇𝑢(𝑠)𝑑𝑠𝑑𝑥=12_Ωℎ^□∇𝑢𝑑𝑥−12ℎ(𝑡)‖∇𝑢‖²²−12𝑑𝑑𝑡_Ω(ℎ□∇𝑢)𝑑𝑥−^𝑡0ℎ(𝑠)𝑑𝑠‖∇𝑢‖²²,(2.3) where (ℎ□𝑣)(𝑡)∶=^𝑡0ℎ(𝑡−𝑠)||𝑣(𝑡)−𝑣(𝑠)||²𝑑𝑠.(2.4) Therefore, if we modify 𝐸(𝑡) to ℰ(𝑡)∶=12‖‖𝑢^𝑡+𝑎(‖‖𝑡)²²+1−^𝑡0ℎ(𝑠)𝑑𝑠‖∇𝑢‖²²+_Ω(ℎ□∇𝑢)𝑑𝑥2(𝑞+1)‖∇𝑢‖^22(𝑞+1)+𝑏(𝑡)𝑝+2‖𝑢‖^𝑝+2𝑝+2,(2.5) we obtain for 𝑡≥0ℰ^(𝑡)=12_Ωℎ^□∇𝑢−ℎ(𝑡)||∇𝑢||²𝑑𝑥+𝑎^(𝑡)2(𝑞+1)‖∇𝑢‖^22(𝑞+1)+𝑏^(𝑡)𝑝+2‖𝑢‖^𝑝+2𝑝+2.(2.6) Assuming that 1−^0+∞ℎ(𝑠)𝑑𝑠=∶1−𝜅>0(2.7) makes ℰ(𝑡) a nonnegative functional. The following functionals ΦΦ₁(𝑡)∶=_Ω𝑢_𝑡𝑢𝑑𝑥,₂(𝑡)∶=−_Ω𝑢_𝑡^𝑡0ℎ(𝑡−𝑠)(𝑢(𝑡)−𝑢(𝑠))𝑑𝑠𝑑𝑥(2.8) are standard and will be used here. The next ones have been introduced by the present author in [24] Φ₃(𝑡)∶=^𝑡0𝐻_𝛾(𝑡−𝑠)‖∇𝑢(𝑠)‖²²𝑑𝑠,Φ₄(𝑡)∶=^𝑡0Ψ_𝛾(𝑡−𝑠)‖∇𝑢(𝑠)‖²²𝑑𝑠,(2.9) where 𝐻_𝛾(𝑡)∶=𝛾(𝑡)⁻¹^∞𝑡ℎ(𝑠)𝛾(𝑠)𝑑𝑠,Ψ_𝛾(𝑡)∶=𝛾(𝑡)⁻¹^∞𝑡𝜉(𝑠)𝛾(𝑠)𝑑𝑠,𝑡≥0,(2.10) and 𝛾(𝑡) and 𝜉(𝑡) are two nonnegative functions which will be precised later (see (H2), (H3)). The functional 𝐿(𝑡)∶=ℰ(𝑡)+⁴_𝑖=1𝜆_𝑖Φ_𝑖(𝑡)(2.11) for some 𝜆_𝑖>0,𝑖=1,2,3,4, to be determined is equivalent to ℰ(𝑡)+Φ₃(𝑡)+Φ₄(𝑡).

Proposition 2.1. There exist 𝜌_𝑖>0,𝑖=1,2 such that 𝜌₁ℰ(𝑡)+Φ₃(𝑡)+Φ₄(𝑡)≤𝐿(𝑡)≤𝜌₂ℰ(𝑡)+Φ₃(𝑡)+Φ₄(𝑡)(2.12) for all 𝑡≥0 and small 𝜆_𝑖,𝑖=1,2.

Proof. By the inequalities Φ₁(𝑡)=_Ω𝑢_𝑡𝑢𝑑𝑥≤12‖‖𝑢^𝑡‖‖²²+𝐶^𝑝2‖∇𝑢‖²²Φ₂,(𝑡)≤12‖‖𝑢^𝑡‖‖²²+𝐶^𝑝𝜅2_Ω(ℎ□∇𝑢)𝑑𝑥,(2.13) where 𝐶_𝑝 is the Poincaré constant, we have 𝐿(𝑡)≤121+𝜆¹+𝜆₂‖‖𝑢_𝑡‖‖²²+121−^𝑡0ℎ(𝑠)𝑑𝑠+𝜆₁𝐶_𝑝‖∇𝑢‖²²+𝑏(𝑡)𝑝+2‖𝑢‖^𝑝+2𝑝+2+121+𝜆²𝐶_𝑝𝜅_Ω(ℎ□∇𝑢)𝑑𝑥+𝑎(𝑡)2(𝑞+1)‖∇𝑢‖^22(𝑞+1)+𝜆₃Φ₃(𝑡)+𝜆₄Φ₄(𝑡),𝑡≥0.(2.14) On the other hand, 2𝐿(𝑡)≥1−𝜆₁−𝜆₂‖‖𝑢_𝑡‖‖²²+1−𝜆₂𝐶_𝑝𝜅_Ω(ℎ□∇𝑢)𝑑𝑥+2𝑏(𝑡)𝑝+2‖𝑢‖^𝑝+2𝑝+2+𝑎(𝑡)𝑞+1‖∇𝑢‖^22(𝑞+1)+1−𝜅−𝜆₁𝐶_𝑝‖∇𝑢‖²²+2𝜆₃Φ₃(𝑡)+2𝜆₄Φ₄(𝑡).(2.15) Therefore, 𝜌₁[ℰ(𝑡)+Φ₃(𝑡)+Φ₄(𝑡)]≤𝐿(𝑡)≤𝜌₂[ℰ(𝑡)+Φ₃(𝑡)+Φ₄(𝑡)] for some constant 𝜌_𝑖>0,𝑖=1,2 and small 𝜆_𝑖,𝑖=1,2 such that 𝜆₁<min{1,(1−𝜅)/𝐶_𝑝} and 𝜆₂<min{1/𝐶_𝑝𝜅,1−𝜆₁}.
The identity to follow is easy to justify and is helpful to prove our result.

Lemma 2.2. One has for ℎ∈𝐶(0,∞) and 𝑣∈𝐶((0,∞);𝐿²(Ω))_Ω𝑣(𝑡)^𝑡0ℎ(𝑡−𝑠)𝑣(𝑠)𝑑𝑠𝑑𝑥=12^𝑡0ℎ(𝑠)𝑑𝑠‖𝑣(𝑡)‖²²+12^𝑡0ℎ(𝑡−𝑠)‖𝑣(𝑠)‖²²𝑑𝑠−12_Ω(ℎ□𝑣)𝑑𝑥,𝑡≥0.(2.16)

The next lemma is crucial in estimating (partially) our nonlinear terms. It can be found in [47].

Let 𝐼⊂𝐑, and let 𝑔₁,𝑔₂∶𝐼→𝐑⧵{𝟎}. We write 𝑔₁∝𝑔₂ if 𝑔₂/𝑔₁ is nondecreasing in 𝐼.

Lemma 2.3. Let 𝑎(𝑡) be a positive continuous function in 𝐽∶=[𝛼,𝛽),𝑘_𝑗(𝑡),𝑗=1,…,𝑛 are nonnegative continuous functions, 𝑔_𝑗(𝑢),𝑗=1,…,𝑛 are nondecreasing continuous functions in 𝐑₊, with 𝑔_𝑗(𝑢)>0 for 𝑢>0, and 𝑢(𝑡) is a nonnegative continuous functions in 𝐽. If 𝑔₁∝𝑔₂∝⋯∝𝑔_𝑛 in (0,∞), then the inequality 𝑢(𝑡)≤𝑎(𝑡)+^𝑛_𝑗=1^𝑡𝛼𝑘_𝑗(𝑠)𝑔_𝑗(𝑢(𝑠))𝑑𝑠,𝑡∈𝐽,(2.17) implies that 𝑢(𝑡)≤𝑐_𝑛(𝑡),𝛼≤𝑡<𝛽₀,(2.18) where 𝑐₀(𝑡)∶=sup_{0≤𝑠≤𝑡}𝑎(𝑠), 𝑐_𝑗(𝑡)∶=𝐺^𝑗−1𝐺^𝑗𝑐_𝑗−1𝐺(𝑡)+^𝑡𝛼𝑘_𝑗(𝑠)𝑑𝑠,𝑗=1,…,𝑛,_𝑗(𝑢)∶=^𝑢𝑢_𝑗𝑑𝑥𝑔_𝑗(𝑥),𝑢>0𝑢^𝑗>0,𝑗=1,...,𝑛,(2.19) and 𝛽₀ is chosen so that the functions 𝑐_𝑗(𝑡),𝑗=1,…,𝑛 are defined for 𝛼≤𝑡<𝛽₀.

Lemma 2.4. Assume that 2≤𝑞<+∞ if 𝑛=1,2 or 2≤𝑞<2𝑛/(𝑛−2) if 𝑛≥3. Then there exists a positive constant 𝐶_𝑒=𝐶_𝑒(Ω,𝑞) such that ‖𝑢‖_𝑞≤𝐶_𝑒‖∇𝑢‖₂(2.20) for 𝑢∈𝐻¹⁰(Ω).

3. Asymptotic Behavior

In this section we state and prove our result. To this end we need some notation. For every measurable set 𝒜⊂𝐑⁺, we define the probability measure ℎ by ℎ(𝒜)∶=1𝜅_𝒜ℎ(𝑠)𝑑𝑠.(3.1) The nondecreasingness set and the non-decreasingness rate of ℎ are defined by 𝒬_ℎ∶=𝑠∈ℛ𝐑⁺∶ℎ(𝑠)>0,ℎ^(𝑠)≥0,(3.2)_ℎ∶=ℎ𝒬_ℎ,(3.3) respectively.

The following assumptions on the kernel ℎ(𝑡) will be adopted.(H1)ℎ(𝑡)≥0 for all 𝑡≥0 and 0<𝜅=∫^0+∞ℎ(𝑠)𝑑𝑠<1.(H2)ℎ is absolutely continuous and of bounded variation on (0,∞) and ℎ^(𝑡)≤𝜉(𝑡) for some nonnegative summable function 𝜉(𝑡) (=max{0,ℎ^(𝑡)} where ℎ^(𝑡) exists) and almost all 𝑡>0.(H3) There exists a nondecreasing function 𝛾(𝑡)>0 such that 𝛾^(𝑡)/𝛾(𝑡)=𝜂(𝑡) is a nonincreasing function: ∫^0+∞ℎ(𝑠)𝛾(𝑠)𝑑𝑠<+∞ and ∫^0+∞𝜉(𝑠)𝛾(𝑠)𝑑𝑠<+∞.

Note that a wide class of functions satisfies the assumption (H3). In particular, exponentially and polynomially (or power type) decaying functions are in this class.

Let 𝑡_∗>0 be a number such that ∫^𝑡∗₀ℎ(𝑠)𝑑𝑠=ℎ_∗>0. We denote by ℬ_𝑡 the set ℬ_𝑡∶=ℬ∩[0,𝑡].

Lemma 3.1. One has for 𝑡≥𝑡_∗ and 𝛿_𝑖>0,𝑖=1,…,5Φ^2(𝑡)≤1−ℎ_∗𝛿₁+32_𝒬𝑡ℎ(𝑡−𝑠)𝑑𝑠‖∇𝑢‖²²+𝛿₃−ℎ_∗‖‖𝑢_𝑡‖‖²²+1−ℎ_∗𝜅4𝛿₁+1+1𝛿₂𝜅_Ω_𝒜𝑡+1ℎ(𝑡−𝑠)||∇𝑢(𝑡)−∇𝑢(𝑠)||²𝑑𝑠𝑑𝑥21−ℎ^∗_𝒬𝑡ℎ(𝑡−𝑠)‖∇𝑢(𝑠)‖²²𝑑𝑠−𝐶^𝑝4𝛿₃+1+𝛿₂𝐵𝑉[ℎ]_Ωℎ^□∇𝑢𝑑𝑥+2_𝒬𝑡ℎ(𝑡−𝑠)𝑑𝑠_Ω_𝒬𝑡ℎ(𝑡−𝑠)||∇𝑢(𝑡)−∇𝑢(𝑠)||²𝑑𝑠𝑑𝑥_𝑝+1𝛿₄𝐶_𝑒𝑏²(𝑡)(1−𝜅)^𝑝+1ℰ^𝑝+1(𝑡)+𝐶^𝑝𝜅4𝛿₄_Ω(ℎ□∇𝑢)𝑑𝑥+𝛿₅𝑎(𝑡)‖∇𝑢‖^22(𝑞+1)+2^𝑞−1𝜅𝑎(𝑡)𝛿₅(1−𝜅)^𝑞+𝐶ℰ^𝑞+1(𝑡)_𝑝4𝛿₃_𝒬𝑡𝜉(𝑡−𝑠)𝑑𝑠_Ω_𝒬𝑡𝜉(𝑡−𝑠)||∇𝑢(𝑡)−∇𝑢(𝑠)||²𝑑𝑠𝑑𝑥,(3.4) where 𝐵𝑉[ℎ] is the total variation of ℎ.

Proof. This lemma is proved by a direct differentiation of Φ₂(𝑡) along solutions of (1.1) and estimation of the different terms in the obtained expression of the derivative. Indeed, we have Φ^2(𝑡)=−_Ω𝑢_𝑡𝑡^𝑡0−ℎ(𝑡−𝑠)(𝑢(𝑡)−𝑢(𝑠))𝑑𝑠𝑑𝑥_Ω𝑢_𝑡^𝑡0ℎ^(𝑡−𝑠)(𝑢(𝑡)−𝑢(𝑠))𝑑𝑠+𝑢_𝑡^𝑡0ℎ(𝑠)𝑑𝑠𝑑𝑥(3.5) or Φ^2(𝑡)=−_Ω1−^𝑡0+ℎ(𝑠)𝑑𝑠Δ𝑢−𝑏(𝑡)|𝑢|^𝑝𝑢+𝑎(𝑡)‖∇𝑢‖^22𝑞Δ𝑢_𝑡0−ℎ(𝑡−𝑠)(Δ𝑢(𝑡)−Δ𝑢(𝑠))𝑑𝑠^𝑡0ℎ(𝑡−𝑠)(𝑢(𝑡)−𝑢(𝑠))𝑑𝑠𝑑𝑥_𝑡0ℎ(𝑠)𝑑𝑠‖‖𝑢_𝑡‖‖²²−_Ω𝑢_𝑡^𝑡0ℎ^(𝑡−𝑠)(𝑢(𝑡)−𝑢(𝑠))𝑑𝑠𝑑𝑥,𝑡≥0.(3.6) Therefore, Φ^2(𝑡)=1−^𝑡0ℎ(𝑠)𝑑𝑠+𝑎(𝑡)‖∇𝑢‖^22𝑞×_Ω∇𝑢⋅^𝑡0ℎ(𝑡−𝑠)(∇𝑢(𝑡)−∇𝑢(𝑠))𝑑𝑠𝑑𝑥−^𝑡0ℎ(𝑠)𝑑𝑠‖‖𝑢_𝑡‖‖²²+𝑏(𝑡)_Ω|𝑢|^𝑝𝑢^𝑡0ℎ(𝑡−𝑠)(𝑢(𝑡)−𝑢(𝑠))𝑑𝑠𝑑𝑥−_Ω𝑢_𝑡^𝑡0+ℎ^(𝑡−𝑠)(𝑢(𝑡)−𝑢(𝑠))𝑑𝑠𝑑𝑥_Ω||||^𝑡0ℎ(𝑡−𝑠)(∇𝑢(𝑡)−∇𝑢(𝑠))𝑑𝑠||||²𝑑𝑥,𝑡≥0.(3.7) For all measurable sets 𝒜 and 𝒬 such that 𝒜=𝐑⁺⧵𝒬, it is clear that _Ω∇=𝑢⋅^𝑡0ℎ(𝑡−𝑠)(∇𝑢(𝑡)−∇𝑢(𝑠))𝑑𝑠𝑑𝑥_Ω+∇𝑢⋅_{𝒜∩[0,𝑡]}ℎ(𝑡−𝑠)(∇𝑢(𝑡)−∇𝑢(𝑠))𝑑𝑠𝑑𝑥_Ω=∇𝑢⋅_{𝒬∩[0,𝑡]}ℎ(𝑡−𝑠)(∇𝑢(𝑡)−∇𝑢(𝑠))𝑑𝑠𝑑𝑥_Ω+∇𝑢⋅_{𝒜∩[0,𝑡]}ℎ(𝑡−𝑠)(∇𝑢(𝑡)−∇𝑢(𝑠))𝑑𝑠𝑑𝑥_{𝒬∩[0,𝑡]}ℎ(𝑡−𝑠)𝑑𝑠‖∇𝑢‖²²−_Ω∇𝑢⋅_{𝒬∩[0,𝑡]}ℎ(𝑡−𝑠)∇𝑢(𝑠)𝑑𝑠𝑑𝑥,𝑡≥0.(3.8) For 𝛿₁>0, the first term in the right-hand side of (3.8) satisfies _Ω∇≤𝑢⋅𝛿_𝒜𝑡1ℎ(𝑡−𝑠)(∇𝑢(𝑡)−∇𝑢(𝑠))𝑑𝑠𝑑𝑥‖∇𝑢‖²²+𝜅4𝛿₁_Ω_𝒜𝑡ℎ(𝑡−𝑠)||∇𝑢(𝑡)−∇𝑢(𝑠)||²𝑑𝑠𝑑𝑥,𝑡≥0,(3.9) and the third one fulfills _Ω∇≤𝑢⋅1_𝒬𝑡ℎ(𝑡−𝑠)∇𝑢(𝑠)𝑑𝑠𝑑𝑥2_𝒬𝑡ℎ(𝑡−𝑠)𝑑𝑠‖∇𝑢‖²²+12_𝒬𝑡ℎ(𝑡−𝑠)‖∇𝑢(𝑠)‖²²𝑑𝑠,𝑡≥0.(3.10)
Back to (3.8) we may write _Ω∇≤𝑢⋅𝛿^𝑡0₁ℎ(𝑡−𝑠)(∇𝑢(𝑡)−∇𝑢(𝑠))𝑑𝑠𝑑𝑥‖∇𝑢‖²²+𝜅4𝛿₁_Ω_𝒜𝑡+3ℎ(𝑡−𝑠)||∇𝑢(𝑡)−∇𝑢(𝑠)||²𝑑𝑠𝑑𝑥2‖∇𝑢‖²²_𝒬𝑡ℎ(𝑡−𝑠)𝑑𝑠+12_𝒬𝑡ℎ(𝑡−𝑠)‖∇𝑢(𝑠)‖²²𝑑𝑠,𝑡≥0.(3.11) The last term in the right-hand side of (3.7) will be estimated as follows: _Ω||||^𝑡0≤ℎ(𝑡1−+𝑠)1(∇𝑢(𝑡)−∇𝑢(𝑠))𝑑s||||²𝑑𝑥𝛿₂𝜅_Ω_𝒜𝑡+1+𝛿₂ℎ(𝑡−𝑠)||∇𝑢(𝑡)−∇𝑢(𝑠)||²𝑑𝑠𝑑𝑥_𝒬𝑡ℎ(𝑡−𝑠)𝑑𝑠_Ω_𝒬𝑡ℎ(𝑡−𝑠)||∇𝑢(𝑡)−∇𝑢(𝑠)||²𝑑𝑠𝑑𝑥,𝛿₂>0.(3.12) For the fourth term in (3.7), it holds that −_Ω𝑢_𝑡^𝑡0≤𝛿₃ℎ^(𝑡−𝑠)(𝑢(𝑡)−𝑢(𝑠))𝑑𝑠𝑑𝑥‖‖𝑢_𝑡‖‖²²−𝐶^𝑝4𝛿₃+𝐶𝐵𝑉[ℎ]_Ωℎ^□∇𝑢𝑑𝑥_𝑝4𝛿₃_𝒬𝑡𝜉(𝑡−𝑠)𝑑𝑠_Ω_𝒬𝑡𝜉(𝑡−𝑠)||∇𝑢(𝑡)−∇𝑢(𝑠)||²𝑑𝑠𝑑𝑥,𝛿₃>0,𝑡≥0.(3.13) Moreover, from Lemma 2.4, for 𝑝>0 if 𝑛=1,2 and 0<𝑝<2/(𝑛−2) if 𝑛≥3, we find 𝑏(𝑡)≤_Ω𝛿|𝑢₄|^𝑝𝑢^𝑡0ℎ(𝑡−𝑠)(𝑢(𝑡)−𝑢(𝑠))𝑑𝑠𝑑𝑥𝑏²(𝑡)‖𝑢‖^{2(𝑝+1)2(𝑝+1)}+𝐶^𝑝4𝛿₄^𝑡0≤𝛿₄ℎ(𝑠)𝑑𝑠_Ω(ℎ□∇𝑢)𝑑𝑥𝐶_𝑒𝑏²(𝑡)‖∇𝑢‖^22(𝑝+1)+𝐶^𝑝𝜅4𝛿₄_Ω≤2(ℎ□∇𝑢)𝑑𝑥_𝑝+1𝛿₄𝐶_𝑒𝑏²(𝑡)(1−𝜅)^𝑝+1ℰ^𝑝+1(𝑡)+𝐶^𝑝𝜅4𝛿₄_Ω(ℎ□∇𝑢)𝑑𝑥,𝛿₄>0,𝑡≥0.(3.14) The definition of ℰ(𝑡) in (2.5) allows us to write 𝑎(𝑡)‖∇𝑢‖^22𝑞_Ω≤𝑎(𝑡)‖∇𝑢∇‖𝑢_22𝑞⋅^𝑡0ℎ(𝑡−𝑠)(∇𝑢(𝑡)−∇𝑢(𝑠))𝑑𝑠𝑑𝑥𝛿⁵‖∇𝑢‖²²+𝜅4𝛿₅_Ω≤𝛿₅(ℎ□∇𝑢)𝑑𝑥𝑎(𝑡)‖∇𝑢‖^22(𝑞+1)+2^𝑞−1𝜅𝑎(𝑡)𝛿₅(1−𝜅)^𝑞ℰ^𝑞+1(𝑡),𝛿₅>0,𝑡≥0.(3.15)
Gathering all the relations (3.11)–(3.15) together with (3.7), we obtain for 𝑡≥𝑡_∗Φ^2(𝑡)≤1−ℎ_∗𝛿₁+32_𝒬𝑡ℎ(𝑡−𝑠)𝑑𝑠‖∇𝑢‖²²+𝛿₃−ℎ_∗‖‖𝑢_𝑡‖‖²²+1−ℎ_∗𝜅4𝛿₁+1+1𝛿₂𝜅_Ω_𝒜𝑡+1ℎ(𝑡−𝑠)||∇𝑢(𝑡)−∇𝑢(𝑠)||²𝑑𝑠𝑑𝑥21−ℎ^∗_𝒬𝑡ℎ(𝑡−𝑠)‖∇𝑢(𝑠)‖²²𝑑𝑠−𝐶^𝑝4𝛿₃+1+𝛿₂𝐵𝑉[ℎ]_Ωℎ^□∇𝑢𝑑𝑥+2_𝒬𝑡ℎ(𝑡−𝑠)𝑑𝑠_Ω_𝒬𝑡ℎ(𝑡−𝑠)||∇𝑢(𝑡)−∇𝑢(𝑠)||²𝑑𝑠𝑑𝑥_𝑝+1𝛿₄𝐶_𝑒𝑏²(𝑡)(1−𝜅)^𝑝+1ℰ^𝑝+1(𝑡)+𝐶^𝑝𝜅4𝛿₄_Ω(ℎ□∇𝑢)𝑑𝑥+𝛿₅𝑎(𝑡)‖∇𝑢‖^22(𝑞+1)+2^𝑞−1𝜅𝑎(𝑡)𝛿₅(1−𝜅)^𝑞+𝐶ℰ^𝑞+1(𝑡)_𝑝4𝛿₃_𝒬𝑡𝜉(𝑡−𝑠)𝑑𝑠_Ω_𝒬𝑡𝜉(𝑡−𝑠)||∇𝑢(𝑡)−∇𝑢(𝑠)||²𝑑𝑠𝑑𝑥.(3.16)

In the following theorem we will assume that 𝑝<𝑞 just to fix ideas. The result is also valid for 𝑝>𝑞. It suffices to interchange 𝑝↔𝑞 and 𝐴(𝑡)↔𝐵(𝑡) in the proof following it. The case 𝑝=𝑞 is easier.

We will make use of the following hypotheses for some positive constants 𝐴,𝐵,𝑈, and 𝑉 to be determined.(A)𝑎(𝑡) is a continuously differentiable function such that 𝑎^(𝑡)<𝐴𝑎(𝑡),𝑡≥0.(B)𝑏(𝑡) is a continuously differentiable function such that 𝑏^(𝑡)<𝐵𝑏(𝑡),𝑡≥0.(C)𝑝>0 if 𝑛=1,2 and 0<𝑝<2/(𝑛−2) if 𝑛≥3.(D)[∫^∞0𝑎(𝑠)𝑒^−𝑞𝑠𝑑𝑠]^1/𝑞[∫^∞0𝑏²(𝑠)𝑒^−𝑝𝑠𝑑𝑠]^1/𝑝<𝑈. (E)[∫^∞0𝑎(𝑠)𝑒^{∫−𝑞𝑠0𝜂(𝜏)𝑑𝜏}𝑑𝑠]^1/𝑞[∫^∞0𝑏²(𝑠)𝑒^{∫−𝑝𝑠0𝜂(𝜏)𝑑𝜏}𝑑𝑠]^1/𝑝<𝑉.

Theorem 3.2. Assume that the hypotheses (H1)–(H3), (A)–(C) hold and ℛ_ℎ<1/4. If lim_𝑡→∞𝜂(𝑡)=𝜂≠0, then, for global solutions and small ∫_𝒬𝜉(𝑠)𝑑𝑠, there exist 𝑇₁>0 and 𝑈>0 such that 𝐿(𝑡)>𝑈,𝑡≥𝑇₁ or 𝐸(𝑡)≤𝑀₁𝑒^−𝜈1𝑡,𝑡≥0(3.17) for some positive constants 𝑀₁ and 𝜈₁ as long as (D) holds. If 𝜂=0, then there exist 𝑇₂>0 and 𝑉>0 such that 𝐿(𝑡)>𝑉,𝑡≥𝑇₂ or 𝐸(𝑡)≤𝑀₂𝑒^{−𝜈2∫𝑡0𝜂(𝑠)𝑑𝑠},𝑡≥0(3.18) for some positive constants 𝑀₂ and 𝜈₂ as long as (E) holds.

Proof. A differentiation of Φ₁(𝑡) with respect to 𝑡 along trajectories of (1.1) gives Φ^1(𝑡)∶=‖‖−𝑢_𝑡𝑎‖‖²²(𝑡−)‖‖∇∇𝑢𝑢‖‖²²+_22(𝑞+1)_Ω∇𝑢⋅^𝑡0ℎ(𝑡−𝑠)∇𝑢(𝑠)𝑑𝑠𝑑𝑥−𝑏(𝑡)‖𝑢‖^𝑝+2𝑝+2,(3.19) and Lemma 2.2 implies Φ^1(𝑡)≤‖‖𝑢_𝑡‖‖²²−1−𝜅2‖∇𝑢‖²²+12^𝑡0−1ℎ(𝑡−𝑠)‖∇𝑢(𝑠)‖²²𝑑𝑠2_Ω(ℎ□∇𝑢)𝑑𝑥−𝑎(𝑡)‖∇𝑢‖^22(𝑞+1)−𝑏(𝑡)‖𝑢‖^𝑝+2𝑝+2,𝑡≥0.(3.20) Next, a differentiation of Φ₃(𝑡) and Φ₄(𝑡) yields Φ^3(𝑡)==𝐻𝐻_𝛾𝛾(0)‖∇𝑢‖²²+^𝑡0𝐻^𝛾(𝑡−𝑠)‖∇𝑢(𝑠)‖²²𝑑𝑠(0)‖∇𝑢‖²²−^𝑡0𝛾^(𝑡−𝑠)𝛾(𝑡−𝑠)𝐻^𝛾≤𝐻_𝛾(𝑡−𝑠)‖∇𝑢(𝑠)‖²²𝑑𝑠−^𝑡0ℎ(𝑡−𝑠)‖∇𝑢(𝑠)‖²²𝑑𝑠Φ(0)‖∇𝑢‖²²−𝜂(𝑡)Φ₃(𝑡)−^𝑡0ℎ(𝑡−𝑠)‖∇𝑢(𝑠)‖²²𝑑𝑠,𝑡≥0,_4(𝑡)==ΨΨ_𝛾𝛾(0)‖∇𝑢‖²²+^𝑡0Ψ^′_𝛾(𝑡−𝑠)‖∇𝑢(𝑠)‖²²𝑑𝑠(0)‖∇𝑢‖²²−^𝑡0𝛾^(𝑡−𝑠)𝛾(𝑡−𝑠)Ψ^𝛾≤Ψ_𝛾(𝑡−𝑠)‖∇𝑢(𝑠)‖²²𝑑𝑠−^𝑡0𝜉(𝑡−𝑠)‖∇𝑢(𝑠)‖²²𝑑𝑠(0)‖∇𝑢‖²²−𝜂(𝑡)Φ₄(𝑡)−^𝑡0𝜉(𝑡−𝑠)‖∇𝑢(𝑠)‖²²𝑑𝑠,𝑡≥0.(3.21)
Taking into account Lemma 3.1 and the relations (2.6), (3.20)-(3.21), we see that 𝐿^(𝑡)≤12−4𝐶𝛿^𝑝₃𝜆₂𝐵𝑉[ℎ]_Ωℎ^□∇𝑢𝑑𝑥+𝜆₁+𝛿₃−ℎ_∗𝜆₂‖‖𝑢_𝑡‖‖²²+𝜆₂1−ℎ_∗𝛿₁+32_𝒬𝑡ℎ(𝑡−𝑠)𝑑𝑠+𝜆₃𝐻_𝛾(0)+𝜆₄Ψ_𝛾(0)−𝜆₁1−𝜅×‖∇𝑢‖₂₂2+𝜆¹2+𝜆²1−ℎ_∗2−𝜆³^𝑡0+𝑎ℎ(𝑡−𝑠)‖∇𝑢(𝑠)‖²²𝑑𝑠_(𝑡)2(𝑞+1)+𝛿⁵𝜆₂𝑎(𝑡)−𝜆₁𝑎(𝑡)‖∇𝑢‖^22(𝑞+1)+1+𝛿₂𝜆₂_𝒬𝑡ℎ(𝑡−𝑠)𝑑𝑠−𝜆¹×_Ω2(ℎ□∇𝑢)𝑑𝑥+𝜆₂𝜅1+1−ℎ^∗4𝛿₁+1𝛿₂_Ω_𝒜𝑡−𝜆₃ℎ(𝑡−𝑠)||∇𝑢(𝑡)−∇𝑢(𝑠)||²𝑑𝑠𝑑𝑥𝜂(𝑡)Φ₃(𝑡)−𝜆₄𝜂(𝑡)Φ₄(𝑡)+𝑏^(𝑡)𝑝+2−𝜆¹𝑏(𝑡)‖𝑢‖^𝑝+2𝑝+2+𝜆²𝐶_𝑝𝜅4𝛿₄_Ω+2(ℎ□∇𝑢)𝑑𝑥_𝑞−1𝜅𝑎(𝑡)𝛿₅(1−𝜅)^𝑞𝜆₂ℰ^𝑞+1(𝑡)+2^𝑝+1𝛿₄𝐶_𝑒𝑏²(𝑡)(1−𝜅)^𝑝+1𝜆₂ℰ^𝑝+1(𝑡)−𝜆₄^𝑡0+𝜆𝜉(𝑡−𝑠)‖∇𝑢(𝑠)‖²²𝑑𝑠₂𝐶_𝑝4𝛿₃_𝒬𝑡𝜉(𝑡−𝑠)𝑑𝑠_𝒬𝑡𝜉(𝑡−𝑠)‖∇𝑢(𝑡)−∇𝑢(𝑠)‖²²𝑑𝑠,𝑡≥𝑡_∗.(3.22) Next, as in [17], we introduce the sets 𝒜_𝑛∶=𝑠∈𝐑⁺∶𝑛ℎ^(𝑠)+ℎ(𝑠)≤0,𝑛∈𝐍,(3.23) and observe that _𝑛𝒜_𝑛=𝐑⁺⧵𝒬_ℎ∪𝒩_ℎ,(3.24) where 𝒩_ℎ is the null set where ℎ^ is not defined and 𝒬_ℎ is as in (3.2). Furthermore, if we denote 𝒬_𝑛∶=𝐑⁺⧵𝒜_𝑛, then lim_𝑛→∞ℎ(𝒬_𝑛)=ℎ(𝒬_ℎ) because 𝒬_𝑛+1⊂𝒬_𝑛 for all 𝑛 and ⋂_𝑛𝒬_𝑛=𝒬_ℎ∪𝒩_ℎ. Moreover, we designate by 𝐴_𝑛𝑡 the sets 𝐴_𝑛𝑡∶=𝑠∈𝐑⁺∶0≤𝑠≤𝑡,𝑛ℎ^(𝑡−𝑠)+ℎ(𝑡−𝑠)≤0,𝑛∈𝐍.(3.25) In (3.22), we take 𝒜_𝑡∶=𝐴_𝑛𝑡 and 𝒬_𝑡∶=𝒬_𝑛𝑡. Choosing 𝜆₁=(ℎ_∗−𝜀)𝜆₂, it is clear that 1+𝛿₂𝜆₂𝜅ℎ𝒬_𝑛−𝜆2¹≤0(3.26) for small 𝜀 and 𝛿₂, large 𝑛 and 𝑡_∗, if ℎ(𝒬)<1/4. We deduce that 1+𝛿₂𝜆₂𝒬_𝑛𝑡ℎ(𝑡−𝑠)𝑑𝑠−𝜆2¹<0.(3.27) Furthermore, if ℎ(𝒬)<1/4, then 31−ℎ_∗2𝒬_𝑛𝑡ℎ(𝑡−𝑠)𝑑𝑠<𝛿ℎ_∗1−𝜅2(3.28) with 𝛿=31−ℎ^∗𝜅4(2−𝜅)ℎ_∗+𝛽(3.29) and a small 𝛽>0. Pick 𝜆₃=12𝜆¹+𝜆₂1−ℎ_∗(3.30) and 𝐻_𝛾(0) such that 𝜆₃𝐻_𝛾(0)<𝜆₂(1−𝛿)ℎ_∗(2−𝜅)2.(3.31) Note that this is possible if 𝑡_∗ is so large that ℎ_∗>7𝜅/(8−𝜅) even though 𝐻_𝛾(0)=𝛾(0)⁻¹^∞0ℎ(𝑠)𝛾(𝑠)𝑑𝑠≥^∞0ℎ(𝑠)𝑑𝑠=𝜅.(3.32)
Taking the relations (3.22)–(3.30) into account and selecting 𝜆₂<𝛿₃/𝐶_𝑝𝐵𝑉[ℎ] so that 12−𝐶^𝑝𝜆₂4𝛿₃𝐵𝑉[ℎ]≥14,(3.33) and small enough so that 𝜆₂𝜅1+1−ℎ^∗4𝛿₁+1𝛿₂+𝐶^𝑝4𝛿₄<41𝑛,(3.34) we find for 𝛿₃=𝜀/2, large 𝛿₄, small Ψ_𝛾(0), and 𝑡≥𝑡_∗𝐿^(𝑡)≤−𝐶₁‖‖𝑢^𝑡‖‖²²+‖∇𝑢‖²²+_Ω(ℎ□∇𝑢)𝑑𝑥+2^𝑝+1𝛿₄𝐶_𝑒𝑏²(𝑡)(1−𝜅)^𝑝+1+𝑎𝜆₂ℰ^𝑝+1(𝑡)_(𝑡)2(𝑞+1)+𝛿⁵𝜆₂𝑎(𝑡)−𝜆₁𝑎(𝑡)‖∇𝑢‖^22(𝑞+1)+2^𝑞−1𝜅𝑎(𝑡)𝛿₅(1−𝜅)^𝑞−𝜆₄𝜆₂ℰ^𝑞+1(𝑡)−𝜆₃𝜂(𝑡)Φ₃(𝑡)𝜂(𝑡)Φ₄(𝑡)+121+𝜆²𝐶_𝑝2𝛿₃_𝒬𝑡𝜉(𝑡−𝑠)𝑑𝑠_Ω_𝒬𝑛𝑡+𝑏ℎ^(𝑡−𝑠)||∇𝑢(𝑡)−∇𝑢(𝑠)||²𝑑𝑠𝑑𝑥_(𝑡)𝑝+2−𝜆¹𝑏(𝑡)‖𝑢‖^𝑝+2𝑝+2−𝜆₄^𝑡0𝜉(𝑡−𝑠)‖∇𝑢(𝑠)‖²²𝑑𝑠(3.35) for some positive constant 𝐶₁. Take 𝜆₄>1+(𝜆₂𝐶_𝑝/2𝛿₃)∫_𝒬𝜉(𝑠)𝑑𝑠,𝛿₅,∫_𝒬𝜉(𝑠)𝑑𝑠 small, and 𝑎^(𝑡)2(𝑞+1)<𝜆¹−𝛼𝑎(𝑡)(3.36) (i.e., 𝐴=2(𝑞+1)(𝜆₁−𝛼) for some 0<𝛼<𝜆₁) and 𝑏^(𝑡)𝑝+2<𝜆¹−𝛽𝑏(𝑡)(3.37) (i.e., 𝐵=(𝑝+2)(𝜆₁−𝛽) for some 0<𝛽<𝜆₁) to derive that 𝐿^(𝑡)≤−𝐶₂ℰ(𝑡)+2^𝑝+1𝛿₄𝐶_𝑒𝑏²(𝑡)(1−𝜅)^𝑝+1𝜆₂ℰ^𝑝+1(𝑡)+2^𝑞−1𝜅𝑎(𝑡)𝛿₅(1−𝜅)^𝑞−𝜆₃𝜆₂ℰ^𝑞+1(𝑡)𝜂(𝑡)Φ₃(𝑡)−𝜆₄𝜂(𝑡)Φ₄(𝑡)(3.38) for some positive constant 𝐶₂.
If lim_𝑡→∞𝜂(𝑡)≠0, then there exist a 𝑡̂≥𝑡_∗ and 𝐶₃>0 such that 𝜂(𝑡)≥𝐶₃ for 𝑡≥𝑡̂. Thus, in virtue of Proposition 2.1, for 𝐶₃>0, we have 𝐿^(𝑡)≤−𝐶₃𝐿(𝑡)+𝐵(𝑡)𝐿^𝑝+1(𝑡)+𝐴(𝑡)𝐿^𝑞+1(𝑡),(3.39) where 𝐵(𝑡)∶=2^𝑝+1𝛿₄𝐶_𝑒𝜆₂(1−𝜅)^𝑝+1𝜌^1𝑝+1𝐴(𝑡)∶=2_𝑞−1𝑏²(𝑡),𝜅𝜆₂𝛿₅(1−𝜅)^𝑞𝜌^1𝑞+1𝑎(𝑡).(3.40) If there exists a 𝑇≥𝑡̂ such that 𝐿(𝑇)<[𝑝∫^∞0𝐵(𝑠)𝑑𝑠]^−1/𝑝, where 𝐵(𝑡)∶=𝐵(𝑡)𝑒^{−𝑝𝐶3(𝑡−𝑇)}, then from (3.39) 𝑒^{𝐶3(𝑡−𝑇)}𝐿(𝑡)^≤𝑒^{𝐶3(𝑡−𝑇)}𝐵(𝑡)𝐿^𝑝+1(𝑡)+𝑒^{𝐶3(𝑡−𝑇)}𝐴(𝑡)𝐿^𝑞+1(𝑡),(3.41) and it follows that for 𝑡≥𝑇𝐿(𝑡)≤𝐿(𝑇)+^𝑡𝑇𝑒^{−𝑝𝐶3(𝑠−𝑇)}𝐵(𝑠)𝐿^𝑝+1(𝑠)𝑑𝑠+^𝑡𝑇𝑒^{−𝑞𝐶3(𝑠−𝑇)}𝐴(𝑠)𝐿^𝑞+1(𝑠)𝑑𝑠(3.42) with 𝐿(𝑡)∶=𝑒^{𝐶3(𝑡−𝑇)}𝐿(𝑡). Now we apply Lemma 2.3 to get 𝐿(𝑡)≤𝑁^−𝑞−𝑞^𝑡0𝐴(𝑠)𝑑𝑠^−1/𝑞,𝑡≥𝑇,(3.43) where 𝐴(𝑡)∶=𝑒^{−𝑞𝐶3(𝑡−𝑇)}𝐴(𝑡) and 𝑁∶=[𝐿(𝑇)^−𝑝−𝑝∫^∞0𝐵(𝑠)𝑑𝑠]^−1/𝑝. If, in addition, 𝑞∫^∞0𝐴(𝑠)𝑑𝑠<𝐿(𝑇)^−𝑞, then 𝐿(𝑡) is uniformly bounded by a positive constant 𝐶₄. Thus 𝐿(𝑡)≤𝐶₄𝑒^−𝐶3𝑡,𝑡≥𝑇,(3.44) and by continuity (3.44) holds for all 𝑡≥0.
If lim_𝑡→∞𝜂(𝑡)=0, then for any 𝐶>0 there exists a 𝑡(𝐶)≥𝑡_∗ such that 𝜂(𝑡)≤𝐶 for 𝑡≥𝑡(𝐶). Therefore, 𝐿^(𝑡)≤−𝐶₅𝜂(𝑡)𝐿(𝑡)+𝐵(𝑡)𝐿^𝑝+1(𝑡)+𝐴(𝑡)𝐿^𝑞+1(𝑡),𝑡≥𝑡=𝑡𝐶₂(3.45) for some 𝐶₅>0. The previous argument carries out with 𝑒^{𝐶3(𝑡−𝑇)} replaced by 𝑒^{𝐶5∫𝑡𝑇𝜂(𝑑)𝑑𝑠}.
In case that 𝑞<𝑝, we reverse the roles of 𝑝 and 𝑞 in the argument above. The case 𝑝=𝑞 is clear.

Remark 3.3. The case where the derivative of the kernel does not approach zero on 𝒜 (as is the case, for instance, when ℎ^≤−𝐶ℎ on 𝒜) is interesting. Indeed, the right-hand side in condition (3.34) will be replaced by 𝐶/4 with a possibly large constant 𝐶.

Remark 3.4. The argument clearly works for all kinds of kernels previously treated where derivatives cannot be positive or even take the value zero. In these cases there will be no need for the smallness conditions on the kernels. This work shows that derivatives may be positive (i.e., kernels may be increasing) on some “small” subintervals and open the door for (optimal) estimations and improvements of these sets.

Remark 3.5. The assumptions 𝑎^(𝑡)<2(𝑞+1)(𝜆₁−𝛼)𝑎(𝑡) and 𝑏^(𝑡)<(𝑝+2)(𝜆₁−𝛽)𝑏(𝑡) may be relaxed to 𝑎^(𝑡)<2(𝑞+1)𝜆₁𝑎(𝑡) and 𝑏^(𝑡)<(𝑝+2)𝜆₁𝑏(𝑡), respectively. In this case 𝛼=𝛼(𝑡) and 𝛽=𝛽(𝑡) would depend on 𝑡.

Remark 3.6. The assertion in Theorem 3.2 is an “alternative” statement. As a next step it would be nice to discuss the (sufficient conditions of) occurrence of each case in addition to the global existence.

Acknowledgment

The author is grateful for the financial support and the facilities provided by the King Fahd University of Petroleum and Minerals.