Kirchhoff systems with dynamic boundary conditions

Contents lists available atScienceDirect

Nonlinear Analysis

journal homepage:www.elsevier.com/locate/na

Kirchhoff systems with dynamic boundary conditions

Giuseppina Autuori, Patrizia Pucci

Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy

a r t i c l e i n f o

Article history:

Received 11 March 2010 Accepted 10 May 2010

MSC:

primary 35L70 35L20

secondary 35Q72 Keywords:

Kirchhoff systems

Nonlinear source and boundary damping terms

Non-continuation

a b s t r a c t

We are interested in the study of the global non-existence of solutions of hyperbolic nonlinear problems, governed by the p-Kirchhoff operator, under dynamic boundary conditions, when p>^pnwith p_n<2. The systems involve nonlinear external forces and may be affected by a perturbation of the type|u|^p−2u. Several models already treated in the literature are covered in special subcases, and concrete examples are provided for the source term f and the external nonlinear boundary damping Q .

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The aim of this paper is to present new results concerning the global non-existence of solutions of the p-Kirchhoff system







u_tt−M kDu(^t, ·)k^pp

∆pu+µ|^u|^p−2u=f(^t,^x,^u), in R⁺₀ ×_Ω,

u(^t,^x) =⁰, on R⁺₀ ×_Γ0,

u_tt = −

M(k^Du(^t, ·)k^pp)|^Du|^p−2∂νu+Q(^t,^x,^u,^ut) , on R⁺₀ ×_Γ1.

(1.1)

Here p>^pn, where p_nis a critical value smaller than 2. In particular, we extend the global non-existence theorems of Sections 3 and 4 of [1], as well the results of [2,3]. For general systems of type(1.1), as far as we know, this paper is the first tentative approach in which a global non-existence theorem is proved.

In(1.1)the function u=(^u1, . . . ,^uN) =^u(^t,^x)is the vectorial displacement, N≥_{1, R}⁺₀ = [0, ∞)^{, and}Ωis a regular and bounded domain of Rⁿ, with boundary∂Ω =_Γ0∪_Γ1,Γ0∩_Γ1= ∅,µn−1(Γ0) >^{0, where}µn−1denotes the(ⁿ−1)^- dimensional Lebesgue measure on∂Ω^{, while}µnis the n-dimensional Lebesgue measure onΩ^{. Finally}νis the outward normal vector field on∂Ω^.

The function f represents an internal nonlinear source force, and Q an external nonlinear boundary damping term. Suppose hereafter that

(^Q(^t,^x,^u, v), v) ≥^{0 for all} (^t,^x,^u, v) ∈R⁺₀ ×_Γ1×_R^N×_R^N, Q ∈C(R⁺0 ×_Γ1×_R^N×_R^N→_R^N), ^f ∈C(R⁺0 ×_Ω×_R^N→_R^N), f(^t,^x,^u) =^Fu(^t,^x,^u), ^F(^t,^x,⁰) =⁰,

so F(^t,^x,^u) = R₀¹(^f(^t,^x, τ^u),^u)^dτis a potential for f in u.

∗Corresponding author. Tel.: +39 075 585 5038; fax: +39 75 585 5024.

E-mail addresses:autuori@dmi.unipg.it(G. Autuori),pucci@dmi.unipg.it,pucci@dipmat.unipg.it(P. Pucci).

0362-546X/$ – see front matter©2010 Elsevier Ltd. All rights reserved.

doi:10.1016/j.na.2010.05.024

The Kirchhoff dissipative term M is assumed to be of the standard form

M(τ) =^a+bγ τ^{γ −1}, ^a,^b≥0,^a+b>⁰, γ >^{1 if b}>⁰. ^(1.2) System(1.1)is said to be non-degenerate when a >^{0 and b} ≥0, while it is called degenerate if a =0 and b > ^{0. When} a> ^{0, b} = 0 andµ = ^{0, problem(1.1)}reduces to the usual well known semilinear wave system. For simplicity we set M(τ) =^aτ +^bτ^{γ, so}γM(τ) ≥ τ^M(τ)^{, where}γ >^{1 if b}>0, and we takeγ =^{1 when b}=0. The caseγ ≥1 was first considered in [4] in very special cases of(1.1). For further comments on the asymptotic behavior of solutions, as well as for previous references, we refer the reader to [4].

The boundary conditions considered in(1.1)are usually called dynamic boundary conditions and they arise in several physical applications. In one dimension and in the scalar case, problem(1.1)can model the dynamic evolution of a viscoelastic route fixed at one end and with a tip mass attached to its free end. The dynamic boundary conditions represent Newton’s law for the attached mass; see [5–7] for more details. In two-dimensional space and for N =1, as shown in [8]

and references therein, these boundary conditions arise if we consider transverse motions of a flexible membraneΩ^whose boundary may be affected by vibrations only in a region. Also some dynamic boundary conditions as in problem(1.1), called acoustic boundary conditions, appear whenΩis an exterior domain of R³in which homogeneous fluids are at rest except for sound waves. Each point of the boundary is subjected to small normal displacements into the obstacle; cf. [9].

Damping terms Q , acting in the interior of the body and depending on(^t,^x,^u, v), are considered for related problems, under homogeneous boundary conditions, in [10–12].

Of course, when u_tt ≡_{0 on R}⁺₀ ×_Γ1, system(1.1)reduces to







u_tt−M kDu(^t, ·)k^pp

∆pu+µ|^u|^p−2u=f(^t,^x,^u), in R⁺₀ ×_Ω,

u(^t,^x) =⁰, on R⁺₀ ×_Γ0,

M kDu(^t, ·)k^pp
|_Du|^p−2∂νu= −Q(^t,^x,^u,^ut), on R⁺₀ ×_Γ1.

(1.3)

Problems of the above form (1.3) are mathematical models occurring in studies of p-Laplace systems, generalized reaction–diffusion theory, non-Newtonian fluid theory [13,14], non-Newtonian filtration [15] and turbulent flows of a gas in a porous medium [16]. In the non-Newtonian fluid theory, the quantity p is characteristic of the medium. Media with p>2 are called dilatant fluids and those with p<2 are called pseudoplastics. If p=2, they are Newtonian fluids.

The main result of this paper for problem(1.1)isTheorem 3.2, in which we show that any local solution u of(1.1)cannot be continued in R⁺₀×_Ω, whenever the initial energy Eu(⁰)is controlled above by a critical value. The proof uses embeddings given in [17] and is to some extent based on underlying ideas of [11,1,3]. When the dissipation term acts on the boundary of the domain, as in problem(1.1), blow up of the solutions could occur provided that the initial data belong to an appropriate region in the phase plane. In [3] similar problems were first treated for semilinear wave equations, included in(1.3)for p=2, using the potential well theory. The existence of infinitely many solutions for quasilinear elliptic p-Kirchhoff equations has been proved recently in [18].

Useful applications ofTheorem 3.2are given for special subcases of the external force f and the damping function Q . The prototype for f is the following:

f(^t,^x,^u) =^g(^t,^x)|^u|^σ−2u+c(^x)|^u|^q−2u,

where c ∈ L^∞(Ω)is a non-negative function, g ∈ C(R⁺0 ×_Ω)is non-positive, differentiable with respect to t and g_t ∈C(R⁺₀ ×_Ω); see conditions(4.1)–(4.4)for details. It is worth nothing that the negative term of (^f(^t,^x,^u),^u)^{, governed} by g, makes the analysis more delicate than in [19,20,3], since it works against the blow up. For dissipative wave systems this case was first considered in [21], but for the local stability problem. From here on(·, ·)denotes the usual scalar product in R^N.

On the other hand, a concrete example of damping given in the paper is a dissipative function dominated by the growth condition

|Q(^t,^x,^u, v)| ≤ (^d1(^t,^x)|^u|^κ)^1/m(^Q(^t,^x,^u, v), v)^1/m0+d₂(^t,^x,^u)^1/℘(^Q(^t,^x,^u, v), v)^1/℘0,

where d₁and d₂are non-negative continuous functions, satisfying integrability conditions with respect to the space variable, and m and℘are appropriate positive constants; see condition(^Q)^{ofLemma 4.2.}

Geometric features of the model(1.1)are presented inLemma 4.4by means of energy type estimates, and a related global non-existence region, which the initial data belong to, is considered inTheorem 4.5. In particular, inCorollary 4.7we prove the result, assuming that Eu(⁰) < ^E0, where E₀is an appropriate critical value determined by the geometry of the problem.Corollary 4.7extends and generalizes in several directions Theorem 7 of [3], in which the author investigates the global non-continuation of solutions in the scalar wave case of(1.3), when p=2. Moreover, in [3] only strong solutions are considered, while here we are able to establish refined results, even dealing with a wider class of solutions verifying a weak conservation law.

Finally, in this paper also the limit case Eu(⁰) =^E0is treated for(1.1)inTheorem 4.9. This new issue was not established before for systems with boundary dampings. Last but not least,Theorem 4.9is new even for problem(1.3), when p=2.

2. Preliminaries

For simplicity, we assume 1<^p<n and denote by L^p(Ω)^{the spaces}[L^p(Ω)]^Nand[L^p(Ω)]^nN, endowed with the usual normk · k_p. Indeed, we drop the exponents N and nN in all the functional spaces involved in the treatment, as was done before. Furthermore,

W_Γ¹₀^,p(Ω) = {^u∈W^1,p(Ω) = [^W1,p(Ω)]^N:u|_Γ0=0},

equipped with the normkuk = kDuk_p, where u|_Γ0 = 0 is understood in the trace sense. The normk · kis equivalent to k · k_[W1,^p(Ω)]^Nby the Poincaré inequality; see [22, Corollary 4.5.3 and Theorem 2.6.16]. In particular, inequality (4.5.2) of [22]

reduces simply to

kuk_p∗≤C_p∗kDuk_p for all u∈W_Γ^1,p

0(Ω), ^(2.1)

where p^∗ = np/(ⁿ−p)^{, C}p∗ = C(ⁿ,^N,^p,Ω) · [^B1,^p(Γ0)]^−1/p, and the Bessel capacity B_1,p(Γ0) >^{0 since}µn−1(Γ0) > ^0;

cf. [22, Theorem 2.6.16].

In L²(Ω)we consider the canonical normkϕk2 = R

Ω|ϕ(^x)|^2dx
1/2

, while the elementary bracket pairinghϕ, ψi = R

Ω(ϕ(^x), ψ(^x))dx is clearly well defined for all ϕ^, ψ ^{such that} (ϕ, ψ) ∈ ^L1(Ω). Analogously, also hω, φiΓ1 = R

Γ1(ω(^x), φ(^x))^dµn−1is well defined for allω^,φ^{such that}(ω, φ) ∈^L1(Γ1)^{. Let} K=C(R⁺0 →W_Γ^1,p

0 (Ω)) ∩^C1(R⁺0 →L²(Ω)) denote the main solution and test function space of the paper.

From here on assume that for allφ ∈^K

F(^t, ·, φ(^t, ·)), (^f(^t, ·, φ(^t, ·)), φ(^t, ·)) ∈^L1(Ω) ^{for all t}∈_R⁺₀; hf(^t, ·, φ(^t, ·)), φ(^t, ·)i ∈^L1loc(R⁺₀). ⁽F1) The potential energy of the fieldφ ∈K is given by

Fφ(^t) =F(^t, φ) =Z

Ω

F(^t,^x, φ(^t,^x))^dx,

and it is well defined by(F1), while the natural total energy of the fieldφ ∈K , associated with(1.1), is Eφ(^t) =¹

2(kφt(^t, ·)k²2+ kφt(^t, ·)k²2,Γ1) +Aφ(^t) −Fφ(^t), p_Aφ(^t) =M(k^Dφ(^t, ·)k^pp) + µkφ(^t, ·)k^pp≥0,

(2.2)

by(1.2), asµ ≥0. Of course Eφis well defined in K by(F1)^{. For all}φ ∈^{K and}(^t,^x) ∈R⁺0 ×_Ωput pointwise,

Aφ(^t,^x) = −^M(k^Dφ(^t, ·)k^pp)∆pφ(^t,^x) + µ|φ(^t,^x)|^p−2φ(^t,^x), ^(2.3) so A is the Fréchet derivative of_Awith respect toφ^{, and}

hhAφ(^t, ·), φ(^t, ·)ii : = h^Aφ(^t, ·), φ(^t, ·)i_(W^1,p

Γ0(Ω),[W¹,p Γ0(Ω)]⁰)

=M(k^Dφ(^t, ·)k^pp)k^Dφ(^t, ·)k^pp+µkφ(^t, ·)k^pp

≤γ^pAφ(^t), ^(2.4)

by(1.2),(2.2), asµ ≥ ^{0 and}γ ≥ 1. Before introducing the definition of solution of(1.1), we assume the following monotonicity condition:

Ft≥0 in R⁺₀ ×W_Γ^1,p

0 (Ω), ⁽F2)

where_Ftis the partial derivative with respect to t of_F=_F(^t, w)^{, with}(^t, w)in R⁺₀ ×W_Γ^1,p

0 (Ω)^. Following [23], we say that u is a (weak) solution of (1.1)if u∈K satisfies the following two properties:

(A) Distribution Identity

[hu_t, φi^]t0 = Z t

0

n

hu_t, φti −M(k^Du(τ, ·)k^pp) · h|^Du|^p−2Du,^Dφi − µh|^u|^p−2u, φi + hf(τ, ·,^u), φi − h^Q(τ, ·,^u,^ut) +^utt, φiΓ1

o dτ for all t∈_R⁺₀ andφ ∈^{K .}

(B) Energy Conservation

(ⁱ) Du(^t) = h^Q(^t, ·,^u(^t, ·),^ut(^t, ·)),^ut(^t, ·)iΓ1+_Ftu(^t) ∈^L1loc(R⁺₀), (ⁱⁱ) ^Eu(^t) ≤^Eu(⁰) −Z t

0

Du(τ)^dτ ^{for all t}∈_R⁺₀.

The Distribution Identity (A) is meaningful provided thathf(^t, ·,^u), φi ∈ ^L1loc(R⁺0)^andhQ(^t, ·,^u,^ut), φiΓ1 ∈ L¹_loc(R⁺0)^, along the fieldφ ∈K . The first condition is valid whenever(F1)is ‘in charge’. On the other hand, we assume that the terms hQ(τ, ·,^u,^ut), φiΓ1andhu_tt, φiΓ1are∈L¹_loc(R⁺0), along any fieldφ ∈K . The other terms in the Distribution Identity are well defined thanks to the choice of the space K . These assumptions, together with condition(F1), make the definition of the solution well posed.

In general it is important to consider (weak) solutions instead of strong solutions, namely functions u∈K satisfying (A), (B)-(i), with (B)-(ii) replaced by the Strong Energy Conservation (B)_s-(ii), that is Eu(^t) =^Eu(⁰) − R₀^tDu(τ)^dτ^{for all t} ∈_R⁺₀. The main reason for this was first given in [23, Remark 4 on page 199]; see also [21, Remark 2 on page 49] and the discussion in [19, page 345]. Of course, if u is a strong solution, then Eu is non-increasing in R⁺₀ and this makes the analysis much simpler.

Remark 2.1. If u∈K is a solution of(1.1)in R⁺₀ ×_Ω, then by(2.2)₂there always existsw1≥0 such that_Au(^t) ≥ w1for all t∈_R⁺₀. Hence by(2.2)₁, (B)-(ii) and(F2)^{we get}Fu(^t) ≥ w1−Eu(⁰) ≥ −^Eu(⁰)^{for all t} ∈_R⁺₀, in other words_Fu is bounded below in R⁺₀ along any solution u∈K .

In order to state the main result of Section3we consider the following condition:

(F3)There exists a positive number q, satisfying the restriction max{2, γ^p}<^q≤p^∗, ^p∗= ^np

n−p, ^(2.5)

with the property that for all F > ^{0 and}φ ∈ K for which inf_t∈

R⁺₀ Fφ(^t) ≥ F, there exist c₁ = c₁(^F, φ) > ^{0 and} ε0=ε0(^F, φ) >0 such that

(ⁱ) Fφ(^t) ≤^c1kφ(^t, ·)k^qq for all t∈_R⁺₀,

and for allε ∈ (⁰, ε0)there exists c₂=c₂(^F, φ, ε) >0 such that

(ⁱⁱ) h^f(^t, ·, φ(^t, ·)), φ(^t, ·)i − (^q−ε)Fφ(^t) ≥^c2kφ(^t, ·)k^qq for all t∈_R⁺₀.

By(2.5)the embedding W_Γ¹₀^,p(Ω) ,→^Lq(Ω)is continuous, that is there exists a constant C_qsuch that for all u∈K

ku(^t, ·)kq≤C_qkDu(^t, ·)kp. ^(2.6)

Furthermore the validity of(2.5)implies that 1≤γ <ⁿ/(ⁿ−p)^{and p}>²ⁿ/(ⁿ+2). In general the constant q introduced in(F3)verifies more restrictions than(2.5)in order to obtain the validity of(F1)^and(F2)^.

Proposition 2.2. Assume(F1)^and(F2)^{. If u}∈K is a solution of(1.1)in R⁺₀ ×_Ω, thenw2=inf_t∈

R⁺₀ Fu(^t) > −∞^{. If there} exists$ ≤ 1 such that Eu(⁰) < $ w1, wherew1 =inf_t∈R⁺

0 Au(^t) ≥^{0, then}w2 >0. Moreover, if also(F3)^-(ⁱ)holds, then w1>^0.

Proof. Let u∈K be a solution of(1.1)in R⁺0 ×_Ω. Clearly_Au and_Fu are bounded below in R⁺0 as shown inRemark 2.1. In particular,w2> −∞^andw1≥0.

If Eu(⁰) < $ w1, for some$ ≤^{1, then}Fu(^t) ≥ w1−Eu(⁰) > (¹−$ )w1, which givesw2> (¹−$ )w1≥0, and so w2>^0.

Suppose now that also(F3)-(i) holds. In correspondence with F=w2>^0,φ =^u∈K , there exist c₁=c₁(w2,^u) >⁰ andε0=ε0(w2,^u) >0 for which(F3)-(i) is valid along u, so for all t∈_R⁺₀,

ku(^t, ·)kq≥ ˜c₁>^{0 and} kDu(^t, ·)kp≥ ˜c₁/^Cq, ^(2.7) by(2.6), where_c˜₁=(w2/^c1)^1/q>0. Hence, by(1.2)and(2.2),

p_Au(^t) ≥ ^a+bkDu(^t, ·)k^pp^{(γ −1)}
k_Du(^t, ·)k^pp≥a₁kDu(^t, ·)k^pp, ^(2.8) where a₁=a+b(˜^c1/^Cq)^{p(γ −1)}>^{0, as a}+b>0. In particular,

w1≥^a1 p inf

t∈_R⁺₀

kDu(^t, ·)k^pp>⁰, and the proposition is proved. 

3. The main theorem

In this section we show the main abstract result for problem(1.1). In what follows p∗=p(ⁿ−1)/(ⁿ−p)^and(^pn)^∞n=1, with

2n

n+2 <^pn= ¹

2[p(ⁿ+1)²+4n+1−n]<².

Clearly,(^pn)^∞n=1is a strictly increasing sequence, with p₃=. 1,^{65 and lim}n→∞p_n=2. We extend the global non-existence results of Section3of [1] to the system(1.1), under the main structural assumptions given in the Introduction and assuming p>^pn. We start with a preliminary result.

Proposition 3.1. If p>^pn, then

℘0= ^pq(ⁿ−1+p) −^p2(ⁿ−1)

n(^q−p) +^p2 ∈(^max{2,^p},^min{p∗,^q}). ^(3.1)

Proof. We start with proving that℘0>2. This is equivalent to(^q−p)[ⁿ(^p−2) +^p(^p−1)] >^p2(²−p), which is clearly satisfied whenever p≥2, since q>^{p by(2.5).}

It remains to consider the case p_n<^p<2. The relation℘0>2 can be written in the equivalent form:[_p²+(ⁿ−₁)^p− 2n]q>^p[p(ⁿ+1) −²ⁿ], where the coefficient of q is strictly positive as p>^pn. An easy calculation shows that the function

p7→ψ(^p) = ^p[p(ⁿ+1) −²ⁿ] [n(^p−2) +^p(^p−1)]

is strictly decreasing in the interval(℘1, ℘2)^{, where}

℘1< ²ⁿ

n+2 <^pn<²< ℘2. In particular, for any p∈(^pn,²)^{we obtain}

ψ(^p) < ψ(²ⁿ/(ⁿ+2)) =²ⁿ/(ⁿ+6) <²<^q, by(2.5). Hence℘0>2 whenever p>^pn.

Now℘0>^{p since q}(^p−1) +ⁿ(^q−q−p+p) −^p(^p−1) >^{0, as}(^p−1)(^q−p) >^{0 by(2.5).}

On the other hand,℘0<q if and only if pq(ⁿ−1) −^p2(ⁿ−1) <^nq(^q−p), that is p(ⁿ−1) < nq, which holds once more by(2.5).

It remains to prove that℘0 < ^p∗. Indeed this is equivalent to−p(ⁿ−1)(^q−p) +^pq(ⁿ−p) < ^p2(ⁿ−1), that is to q(ⁿ−n+p−1) >0, which trivially holds, as p>^1. 

Observe that inProposition 3.1the assumption p>^pnis needed only to show that℘0 >2. Of course, since 1<^p<^n, in order to have p_n<n it is enough to have n>³/2, that is n≥2.

Theorem 3.2. Assume(F1)^–(F3)^{and take p}∈(^pn,ⁿ). Then there are no solutions u∈K of(1.1)in R⁺₀ ×_Ω, for which Eu(⁰) <

 1−γ^p

q



w1=E₁, ^(3.2)

wherew1=inf_t∈R⁺

0 Au(^t), and for which there exist T≥0, q₁>^{0, m,}℘^{, with}

1<^m≤℘ − κ, ⁰≤κ ≤^p(¹−m/℘) ^{and 2}≤℘ < ℘0, ^(3.3)

with℘0defined in(3.1), and non-negative functionsδ1, δ2∈L^∞_loc(^J)^,ψ,^k∈W_loc^1,1(^J)^{, J}= [T, ∞)^{, with k0}≥0,ψ >^{0 in J and} ψ⁰(^t) =^o(ψ(^t))^{as t} → ∞, such that for all t∈_J,

hQ(^t, ·,^u,^ut),^ui_Γ1 ≤q₁{δ1(^t)^1/mku(^t, ·)k^κ/_℘,^m_Γ1Du(^t)^1/m0+δ2(^t)^1/℘Du(^t)^1/℘0}ku(^t, ·)k℘,Γ1, ^(3.4) and

δ1^1/(m−1)+δ2^1/(℘−1)≤k/ψ ^{in J}, Z ∞

ψ(^t)^[max{k(^t), ψ(^t)}^]−(1+θ)dt= ∞, ^(3.5) for some appropriate constantθ ∈ (⁰, θ0)^{, where}

θ0=min

q−2 q+2, ^r

1−r

 ,

r= ¹

℘ −

1−s

q + ^s

p



∈(⁰,¹) ^{and s}= ⁿ

p−ⁿ−1

℘0

∈(⁰,¹).

(3.6)

Proof. Suppose as a contradiction that there exists a global solution u∈K of(1.1)as in the statement. We follow the proof of Theorem 3.1 of [1] to some extent, reporting here all the necessary steps for greater clarity.

ByProposition 2.2and(2.5)we have thatw1>0 and consequently E₁>^{0. Fix E}2such that max{0,^Eu(⁰)} <^E2 <^E1

and takeε ∈ (⁰, ε0)so small that

εw1≤(^q−γ^p)w1−qE₂. ^(3.7)

This choice is possible sincew1>^{0 and E}2<^E1. For each t∈_R⁺₀ put H(^t) =^E2−_Eu(⁰) +Z t

0

Du(τ)^dτ.

Of course_H is well defined and non-decreasing by (B)-(i) and(F2)^asD≥0 and finite along u. Hence, by (B)-(ii),

E₂−Eu(^t) ≥H(^t) ≥H0=E₂−Eu(⁰) >^{0 for t}∈_R⁺₀, ^(3.8) where_H0=_H(⁰). Moreover, by(2.2),(3.8), the choice of E₂, the definition of E₁and the inequalityw2> γ^pw1/q, it follows that for all t∈_R⁺₀,

H(^t) ≤ ^E2−Eu(^t) <^E1+_Fu(^t) <

 q γ^p−1



Fu(^t) +Fu(^t)

= ^q

γ^pFu(^t). ^(3.9)

On the other hand, if we putφ =u in the Distribution Identity, we obtain d

dt n

hu_t(^t, ·),^u(^t, ·)i + h^ut(^t, ·),^u(^t, ·)iΓ1

o

= ku_t(^t, ·)k²2− hhAu(^t, ·),^u(^t, ·)ii + h^f(^t, ·,^u),^u(^t, ·)i

− hQ(^t, ·,^u(^t, ·),^ut(^t, ·)),^u(^t, ·)iΓ1+ ku_t(^t, ·)k²2,Γ1

=c₃ ku_t(^t, ·)k²2+ ku_t(^t, ·)k²2,Γ1
+(^q−ε)Au(^t) − hh^Au(^t, ·),^u(^t, ·)ii

+ hf(^t, ·,^u(^t, ·)),^u(^t, ·)i − (^q−ε)Fu(^t) − (^q−ε)^Eu(^t) − h^Q(^t, ·,^u(^t, ·),^ut(^t, ·)),^u(^t, ·)iΓ1,

where c₃ = 1+(^q−ε)/²> 0 by the choice ofε^{. Using(2.4)and}(F3)-(ii) with c₂ =c₂(w2,^u, ε) >0, we obtain for all t∈_R⁺₀,

d dt

n

hu_t(^t, ·),^u(^t, ·)i + h^ut(^t, ·),^u(^t, ·)iΓ1

o

≥c₃ ku_t(^t, ·)k²2+ ku_t(^t, ·)k²2,Γ1
+_c2ku(^t, ·)k^qq−(^q−ε)^Eu(^t)

− hQ(^t, ·,^u(^t, ·),^ut(^t, ·)),^u(^t, ·)iΓ1+(^q−ε − γ^p)Au(^t).

Sinceε <^q−γ^{p by(3.7)and Eu}≤E₂−_H by(3.8), we have d

dt n

hu_t(^t, ·),^u(^t, ·)i + h^ut(^t, ·),^u(^t, ·)iΓ1

o

≥c₃ ku_t(^t, ·)k²2+ ku_t(^t, ·)k²2,Γ1
+_c2ku(^t, ·)k^qq

+(^q−ε − γ^p)Au(^t) − h^Q(^t, ·,^u(^t, ·),^ut(^t, ·)),^u(^t, ·)iΓ1+γ^pH(^t) − (^q−ε)^E2, by(2.2). Now set C₂=ε^a1(^q−ε − γ^p)/^pq>^{0, so}

(^q−ε − γ^p)Au(^t) − (^q−ε)^E2 ≥(^q−ε − γ^p)



1−^q−ε q



Au(^t) + (^q−ε − γ^p)^q−ε

q w1−(^q−ε)^E2

≥C₂kDu(^t, ·)k^pp,

by(2.8)and the fact thatε ∈ (⁰, ε0)^implies(^q−ε)[(^q−ε − γ^p)w1−qE₂]/^q≥0 thanks to(3.7). Consequently, putting c˜₂=min{c₂,^C2}>^{0, we get}

d dt

n

hu_t(^t, ·),^u(^t, ·)i + h^ut(^t, ·),^u(^t, ·)iΓ1

o

≥c₃ ku_t(^t, ·)k²2+ ku_t(^t, ·)k²2,Γ1

+ ˜c₂ ku(^t, ·)k^qq+ kDu(^t, ·
k^pp) + γ^pH(^t) − h^Q(^t, ·,^u(^t, ·),^ut(^t, ·)),^u(^t, ·)iΓ1. ^(3.10) Since℘ < ℘0there exists S₀>0 such that

ku(^t, ·)k℘,Γ1 ≤S₀ku(^t, ·)k℘0,Γ1.

On the other hand, by the choice of s in(3.6), as℘0 > ^{p by}Proposition 3.1, by virtue of (Theorem 7.58, withχ = ^0, k=n−1 [24]) the embedding W_Γ^s,p

0(Ω) ,→^L℘0(∂Ω)is continuous. Hence, ku(^t, ·)k℘0,Γ1≤S₁ku(^t, ·)kW^s,p

Γ0(Ω), ^(3.11)

where S₁is an appropriate positive constant. Moreover, by (Corollary 3.2-(a), with s₁ = 0, s₂ = 1, p₁ = p₂ = p and θ =¹−s [17]), there exists S₂>0 such that

ku(^t, ·)kW^s,p

Γ0(Ω)≤S₂ku(^t, ·)k¹p^−skDu(^t, ·)k^sp≤S₃ku(^t, ·)k¹q^−skDu(^t, ·)k^sp, ^(3.12) where S₃=S₂µn(Ω)^{(1−s)(q−p)/pq}and in the last step we have used the fact that p<q. Combining the last three estimates, we get

ku(^t, ·)k℘,Γ1 ≤Sku(^t, ·)k¹q^−skDu(^t, ·)k^sp, ^(3.13) with S=S₀S₁S₃. Furthermore,

n

p−ⁿ−1

℘ <^s<

q

℘ −1

 q p−1



, ^(3.14)

as℘ < ℘0by(3.3). For brevity, letα1,α2andβ1,β2denote the numbers 1

α1

= ¹ m− ^s

p

 1+ κ

m , β1=(¹−s) 1+ κ

m



−q

1 m− ^s

p

 1+ κ

m

 , 1

α2

= ¹

℘− ^s

p, β2=1−s−q

1

℘ − ^s p

 .

We claim that 1< α1≤α2. Indeed, s<^p/℘^{whenever 1}< ℘ ≤^{p as s}∈(⁰,¹); otherwise it is a direct consequence of (3.14), and so s<^p/℘ ≤^p/(^m+κ)^by(3.3). This gives the claim by direct calculation, whileβ1 ≤β2 <^{0 again by(3.3)} and(3.14). Hence, using(3.4)and(3.13), we get for all t ∈J,

hQ(^t, ·,^u(^t, ·),^ut(^t, ·)),^u(^t, ·)iΓ1 ≤q₂ (

δ1(^t)^1/(m−1)Du(^t)
^1/m0· ku(^t, ·)k⁽q^{1−s)(1+κ/m)}kDu(^t, ·)k^sp^(1+κ/m)

+ δ2(^t)^1/(℘−1)Du(^t)
^1/℘0ku(^t, ·)k¹q^−skDu(^t, ·)k^sp

)

=q₂ (

h δ1(^t)^1/(m−1)Du(^t)
^1/m0ku(^t, ·)k^qq^/α1kDu(^t, ·)k^sp^(1+κ/m)

i

ku(^t, ·)k^βq¹

+h δ2(^t)^1/(℘−1)Du(^t)
^1/℘0ku(^t, ·)k^qq^/α2kDu(^t, ·)k^sp

i

ku(^t, ·)k^βq²

)

≤q₂



[(²δ1(^t)/`)^1/(m−1)Du(^t) +¹

2`k^u(^t, ·)k^qq+¹

2`k^Du(^t, ·)k^pp] · ku(^t, ·)k^βq¹

+[(²δ2(^t)/`)^1/(℘−1)Du(^t) +¹

2`k^u(^t, ·)k^qq+¹

2`k^Du(^t, ·)k^pp] · ku(^t, ·)k^βq²

 , where q₂=q₁max{S,^S1+κ/m}and in the last step we have applied Young’s inequality, with` ∈ (⁰,¹)to be fixed later. In conclusion,

hQ(^t, ·,^u(^t, ·),^ut(^t, ·)),^u(^t, ·)iΓ1 ≤ ˜q₂n`<sup>−m0/m</sup>[δ1(<sup>t</sup>)<sup>1/(m−1)</sup>+δ2(<sup>t</sup>)<sup>1/(℘−1)</sup>]<sub>D</sub>u(<sup>t</sup>) +` k^u(^t, ·)k^qq+ kDu(^t, ·)k^pp

o

· ku(^t, ·)k^βq², ^(3.15) where_q˜₂=2^1/(m−1)q₂max{1, (˜^c1)^β1−β2}>^{0 by}(2.7). Clearly

r= −β2/^q∈(⁰,¹) ^(3.16)

by(3.14). But by(F3)^{and(3.9)we have}

ku(^t, ·)k^βq²= ku(^t, ·)k⁻q^qr ≤c₁^r[_Fu(^t)]^−r ≤ [c₁q/γ^p]^r[_Hu(^t)]^−r. Therefore

hQ(^t, ·,^u(^t, ·),^ut(^t, ·)),^u(^t, ·)iΓ1 ≤c₄n`<sup>−m0/m</sup>[δ1(<sup>t</sup>)<sup>1/(m−1)</sup>+δ2(<sup>t</sup>)<sup>1/(℘−1)</sup>]<sub>D</sub>u(<sup>t</sup>) +` k^u(^t, ·)k^qq+ kDu(^t, ·)k^pp

o

[_H(^t)]^−r