1. Introduction. In this paper we first investigate the non–existence of global solutions of the p–Kirchhoff system

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Volume 9, Number 5, September 2010 pp. 1161–1188

KIRCHHOFF SYSTEMS WITH NONLINEAR SOURCE AND BOUNDARY DAMPING TERMS

Giuseppina Autuori and Patrizia Pucci

Dipartimento di Matematica e Informatica Universit`a degli Studi di Perugia Via Vanvitelli 1, I–06123 Perugia, Italy

Abstract. In this paper we treat the question of the non–existence of global solutions, or their long time behavior, of nonlinear hyperbolic Kirchhoff systems. The main p–Kirchhoff operator may be affected by a perturbation which behaves like |u|^p−2uand the systems also involve an external force f and a nonlinear boundary damping Q. When p = 2, we consider some problems involving a higher order dissipation term, under dynamic boundary conditions.

For them we give criteria in order that ku(t, ·)kq → ∞ as t → ∞ along any global solution u = u(t, x), where q is a parameter related to the growth of f in u. Special subcases of f and Q, interesting in applications, are presented in Sections4,5and6.

1. Introduction. In this paper we first investigate the non–existence of global solutions of the p–Kirchhoff system



 

 

u

tt

− M kDu(t, ·)k

^p_p

∆

p

u + µ|u|

^p−2

u = f (t, x, u), in R

⁺0

× Ω,

u(t, x) = 0, on R

⁺0

× Γ

0

,

M kDu(t, ·)k

^p_p

|Du|

^p−2

∂

ν

u = −Q(t, x, u, u

t

), on R

⁺0

× Γ

1

,

(1.1)

where u = (u

1

, . . . , u

N

) = u(t, x) is the vectorial displacement, N ≥ 1, R

⁺0

= [0, ∞), Ω is a regular and bounded domain of R

ⁿ

, with boundary ∂Ω = Γ

0

∪Γ

1

, Γ

0

∩Γ

1

= ∅, µ

n−1

(Γ

0

) > 0, where µ

n−1

denotes the (n − 1)–dimensional Lebesgue measure on

∂Ω, while µ

n

is the n–dimensional Lebesgue measure on Ω. Finally ν is the outward normal vector field on ∂Ω.

The Kirchhoff dissipative term M is assumed to be of the standard form M (τ ) = a + bγτ

^γ−1

, a, b ≥ 0, a + b > 0, γ > 1 if b > 0. (1.2) System (1.1) is said non–degenerate when a > 0 and b ≥ 0, while it is called degenerate if a = 0 and b > 0. When a > 0 and b = 0, problem (1.1) reduces to the usual well known semi–linear wave system. For simplicity we set M (τ ) = aτ + bτ

^γ

, so that γM (τ ) ≥ τ M (τ ), being γ > 1 if b > 0, and taking γ = 1 when b = 0.

In (1.1) the parameter µ ≥ 0, 1 < p < n and ∆

p

denotes the vectorial p–Laplacian operator defined as div(|Du|

^p−2

Du), where div is the vectorial divergence and Du the Jacobian matrix of u.

2000 Mathematics Subject Classification. Primary: 35L70, 35L20; Secondary: 35Q70.

Key words and phrases. Kirchhoff systems, nonlinear source and boundary damping terms, non continuation, blow up.

1161

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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

× Γ

1

× R

^N

× R

^N

, Q ∈ C(R

⁺0

× Γ

1

× R

^N

× R

^N

→ R

^N

) and f ∈ C(R

⁺0

× Ω × R

^N

→ R

^N

),

f (t, x, u) = F

u

(t, x, u), F (t, x, 0) = 0, so that F (t, x, u) = R

1

0

(f (t, x, τ u), u)dτ is a potential for f in u.

Problems of the above form (1.1) are mathematical models occuring in studies of p–Laplace systems, generalized reaction–diffusion theory, non–Newtonian fluid theory [3, 17], non–Newtonian filtration [15] and turbulent flows of a gas in a porous medium [11]. 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.

Our main result concerning problem (1.1) is Theorem 3.1, 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 [7] and is somehow based on underlying ideas of [5, 23]. When the dissipation term acts on the boundary of the domain, as in problem (1.1), the blow up of the solutions could occur provided that the initial data belong to an appropriate region in the phase plane. In [23] similar problems were first treated for semilinear wave equations when p = 2 by the potential well theory.

Useful applications of Theorem 3.1 are 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−2

u + c(x)|u|

^q−2

u,

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, since it works against the blow up.

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)| ≤ (d

1

(t, x)|u|

^κ

)

^1/m

(Q(t, x, u, v), v)

^1/m^′

+ d

2

(t, x, u)

^1/℘

(Q(t, x, u, v), v)

^1/℘^′

,

where d

1

and d

2

are non–negative continuous functions, satisfying integrability conditions with respect to the space variable, and m and ℘ are appropriate positive constants, see condition (Q

1

) of Lemma 4.2.

Geometric features of the model (1.1) are presented in Lemma 4.4 by means of energy type estimates, and a related non–existence region, which the initial data belong to, is considered in Theorem 4.5 and in the in the subsequent Corollary 4.6.

With Corollary 4.8 we extend and generalize in several directions Theorem 7 of [23], in which the author investigates the blow up of solutions of wave equations involving boundary damping term in the case p = 2. Moreover, in [23] 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, and assuming Eu(0) smaller than a critical value E

0

.

Also the limit case Eu(0) = E

0

is treated in Theorem 4.10, under a further

assumption on f and Q, thanks to a deep qualitative analysis concerning the main

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operators involved in the problem and the norm of the solutions. This new issue was not established before for systems with boundary dampings, while we refer to [5] for similar results, obtained under homogeneous Dirichlet boundary conditions, when the function Q acts in the interior of the domain.

In the second part of the paper we treat the problem



 

 

u

tt

− M kDu(t, ·)k

²₂

∆u − ̺(t)∆u

t

+ µu = f (t, x, u), in R

⁺0

× Ω,

u(t, x) = 0, on R

⁺0

× Γ

0

,

u

tt

= − M (kDu(t, ·)k

²₂

)∂

ν

u+̺(t)∂

ν

u

t

+Q(t, x, u, u

t

), on R

⁺₀

× Γ

1

,

(1.3)

where ̺ ∈ C(R

⁺0

) is a nonnegative function, differentiable in R

⁺

, and M , f and Q are as before. Problem (1.3) involves higher dissipation terms very interesting from an applicative point of view and of course includes the model (1.1) when p = 2,

̺ ≡ 0, that is when no higher dissipation terms are involved, and u

tt

≡ 0 on Γ

1

. The term −̺(t)∆u

t

represents the internal material damping of Kelvin–Voight type of the body structure, which is always present, even if small, in real material as long as the system vibrates. We mention [10] for a model describing nonlinear viscoelastic materials with short memory in the special scalar case of (1.3), when

̺ ≡ 1, f ≡ 0 and Q ≡ 0. For a further detailed physical discussion the reader is referred to [4, 14, 18], as well as the references therein.

The boundary conditions considered in (1.3) are usually called dynamic boundary conditions and they arise in several physical applications. In one dimension and in the scalar case, problem (1.3) can model the dynamic evolution of a viscoelastic road 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 [2, 8, 9]

for more details). In two dimensional space and for N = 1, as showed in [13]

and references therein, these boundary conditions arise if we consider transverse motions of a flexible membrane Ω whose boundary may be affected by vibrations only in a region. Also some dynamic boundary conditions as in problem (1.3), called acoustic boundary conditions , appear when Ω is an exterior domain of R

³

in 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. [6].

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 [4, 5, 19].

In Theorem 5.1 we show that ku(t, ·)k

q

→ ∞ as t → ∞ along global solutions u = u(t, x) of (1.3), where q is a parameter related to the growth of f in u, see condition (F

3

), provided that the initial energy is bounded above by a critical value.

Corollary 5.4 extends and generalizes to the vectorial case Theorem 3.1 of [12], in which the authors only consider a standard wave equation with N = 1, a = 1 and b = 0. Moreover, in [12] the dissipation function ̺ is assumed constant with respect to t and both f and Q are of very special form; see the remarks at the end of Section 5. In particular, in Corollary 5.4 we show that the norm ku(t, ·)k

q

approaches infinity with a polynomial or exponential rate, depending on the degree

of the growth assumed on the damping terms. Finally, thanks to Proposition 5.5

and Theorem 5.6, we cover also the case Eu(0) = E

0

for (1.3), not before established

in the literature.

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A further model is considered in Section 6, involving the Kirchhoff function M in front of the internal dissipation ̺. More precisely, we treat the system



 

 

u

tt

− M kDu(t, ·)k

²₂

∆(u + ̺(t)u

t

) + µu = f (t, x, u), in R

⁺0

× Ω,

u(t, x) = 0, on R

⁺₀

× Γ

0

,

u

tt

= − M (kDu(t, ·)k

²₂

)(∂

ν

u+̺(t)∂

ν

u

t

)+Q(t, x, u, u

t

), on R

⁺0

× Γ

1

,

(1.4)

under the same structural assumptions on g taken for (1.3). Of course when a = 1 and b = 0 system (1.4) reduces to (1.3). Otherwise, the two models constitute differ- ent problems of independent interest. As for (1.3), we show that the ku(t, ·)k

q

→ ∞ when times goes to infinity, if Eu(0) is bounded above. Clearly, we here must take into account the more delicate setting given by the presence of M in front of ̺.

In particular, in Section 6 we outline the main results concerning (1.4) and the principal differences with respect to (1.3).

2. Preliminaries. For simplicity we denote by L

^p

(Ω) the space [L

^p

(Ω)]

^N

, endowed with the usual norm k · k

p

. Furthermore

W

_Γ^1,p₀

(Ω) = {u ∈ [W

^1,p

(Ω)]

^N

: u|

Γ0

= 0},

equipped with the norm kuk = kDuk

p

, where u|

Γ0

= 0 is understood in the trace sense. The norm k · k is equivalent to k · k

[W^1,p(Ω)]^N

by the Poincar´e inequality, see [24, Corollary 4.5.3 and Theorem 2.6.16]. In particular inequality (4.5.2) of [24]

reduces simply to

kuk

p^∗

≤ C

p^∗

kDuk

p

for all u ∈ W

_Γ^1,p₀

(Ω), (2.1) where p

^∗

= np/(n − p), C

p^∗

= C(n, N, p, Ω) · [B

1,p

(Γ

0

)]

^−1/p

, and the Bessel capacity B

1,p

(Γ

0

) > 0 since µ

n−1

(Γ

0

) > 0, cf. [24, Theorem 2.6.16].

The usual Lebesgue space L

²

(Ω) = [L

²

(Ω)]

^N

is equipped with the canonical norm kϕk

2

= R

Ω

|ϕ(x)|

²

dx

1/2

, while the elementary bracket pairing hϕ, ψi = R

Ω

(ϕ(x), ψ(x))dx is clearly well defined for all ϕ, ψ such that (ϕ, ψ) ∈ L

¹

(Ω), where (·, ·) denotes the usual scalar product in R

^N

. Analogously, also hω, φi

Γ1

= R

Γ1

(ω(x), φ(x))dµ

n−1

is well defined for all ω, φ such that (ω, φ) ∈ L

¹

(Γ

1

). Let K = C(R

⁺₀

→ W

_Γ^1,p₀

(Ω)) ∩ C

¹

(R

⁺₀

→ L

²

(Ω))

denote the main solution and test function space of the paper.

From here on assume that for all φ ∈ K

(F

1

) F (t, ·, φ(t, ·)), (f (t, ·, φ(t, ·)), φ(t, ·)) ∈ L

¹

(Ω) for all t ∈ R

⁺0

; t 7→ hf (t, ·, φ(t, ·)), φ(t, ·)i ∈ L

¹_loc

(R

⁺0

).

The potential energy of the field φ ∈ K is given by F φ(t) = F (t, φ) =

Z

Ω

F (t, x, φ(t, x)) dx,

and it is well defined by (F

1

), while the natural total energy of the field φ ∈ K, associated with (1.1), is

Eφ(t) =

¹₂

kφ

t

(t, ·)k

²₂

+ A φ(t) − F φ(t),

pA φ(t) = M (kDφ(t, ·)k

^p_p

) + µkφ(t, ·)k

^p_p

≥ 0, (2.2)

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by (1.2), being µ ≥ 0. Of course Eφ is well defined in K by (F

1

). For all φ ∈ K and (t, x) ∈ R

⁺0

× Ω put pointwise

Aφ(t, x) = −M (kDφ(t, ·)k

^p_p

)∆

p

φ(t, x) + µ|φ(t, x)|

^p−2

φ(t, x), (2.3) so that A is the Fr´echet derivative of A with respect to φ, and

hhAφ(t, ·), φ(t, ·)ii : = hAφ(t, ·), φ(t, ·)i

_(W^1,p

Γ0 (Ω),[W^1,p

Γ0(Ω)]^′)

= M (kDφ(t, ·)k

^p_p

)kDφ(t, ·)k

^p_p

+ µkφ(t, ·)k

^p_p

≤ γpA φ(t),

(2.4)

by (1.2), (2.2), being µ ≥ 0 and γ ≥ 1. Before introducing the definition of solution of (1.1), we assume the following monotonicity condition

(F

2

) F

_t

≥ 0 in R

⁺0

× W

_Γ^1,p

0

(Ω),

where F

t

is the partial derivative with respect to t of F = F (t, w), with (t, w) ∈ R

⁺₀

× W

_Γ^1,p₀

(Ω).

Following [20], we say that u is a (weak) solution of (1.1) if u ∈ K satisfies the two properties:

(A) Distribution Identity hu

t

, φi

t

0

= Z

t

0

n hu

t

, φ

t

i − M (kDu(τ, ·)k

^p_p

) · h|Du|

^p−2

Du, Dφi − µh|u|

^p−2

u, φi

+ hf (τ, ·, u), φi − hQ(τ, ·, u, u

t

), φi

Γ1

o dτ for all t ∈ R

⁺0

and φ ∈ K;

(B) Energy Conservation

(i) D u(t) = hQ(t, ·, u(t, ·), u

t

(t, ·)), u

t

(t, ·)i

Γ1

+ F

t

u(t) ∈ L

¹_loc

(R

⁺0

),

(ii) Eu(t) ≤ Eu(0) −

Z

t 0

D u(τ )dτ for all t ∈ R

⁺0

.

The Distribution Identity (A) is meaningful provided that hf (t, ·, u), φi ∈ L

¹_loc

(R

⁺₀

) and hQ(t, ·, u, u

t

), φi

Γ1

∈ L

¹_loc

(R

⁺0

), along the field φ ∈ K. The first condition is valid whenever (F

1

) is in charge. On the other hand, throughout this section and the following one, we assume that hQ(τ, ·, u, u

t

), φi

Γ1

∈ L

¹_loc

(R

⁺0

), along any field φ ∈ K. The other terms in the Distribution Identity are well defined thanks to the choice of the space K.

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) − R

t

0

D u(τ )dτ for all t ∈ R

⁺0

. The main reason was first given in [20, Remark 4 at page 199]; see also [21, Remark 2 at page 49] and the discussion in [16, 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)

2

there exists

always w

1

≥ 0 such that A u(t) ≥ w

1

for all t ∈ R

⁺0

. Hence by (2.2)

1

, (B)–(ii) and

(F

2

) we get F u(t) ≥ w

1

− Eu(0) ≥ −Eu(0) for all t ∈ R

⁺0

, in other words F u is

bounded below in R

⁺0

along any solution u ∈ K.

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In order to state the main result of Section 3 we consider the following condition (F

3

) There exists a positive number q, satisfying the restriction

max{2, γp} < 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

F φ(t) ≥ F, there exist c

1

= c

1

(F, φ) > 0 and ε

0

= ε

0

(F, φ) > 0 such that

(i) F φ(t) ≤ c

1

kφ(t, ·)k

^q_q

for all t ∈ R

⁺₀

, and for all ε ∈ (0, ε

0

) there exists c

2

= c

2

(F, φ, ε) > 0 such that

(ii) hf (t, ·, φ(t, ·)), φ(t, ·)i − (q − ε)F φ(t) ≥ c

2

kφ(t, ·)k

^q_q

for all t ∈ R

⁺0

.

By (2.5) the embedding W

_Γ^1,p₀

(Ω) ֒→ L

^q

(Ω) is continuous, that is there exists a constant C

q

such that for all u ∈ K

ku(t, ·)k

q

≤ C

q

kDu(t, ·)k

p

. (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 (F

3

) verifies further restrictions than (2.5) in order to get the validity of (F

1

) and (F

2

), see Section 4 for concrete examples, that is (4.1) and (4.2).

Proposition 2.2. Assume (F

1

) and (F

2

). If u ∈ K is a solution of (1.1) in R

⁺₀

× Ω, then w

2

= inf

_t∈R⁺

0

F u(t) > −∞. If there exists ̟ ≤ 1 such that Eu(0) <

̟w

1

, where w

1

= inf

_t∈R⁺

0

A u(t) ≥ 0, then w

2

> 0. Moreover, if also (F

3

) holds, then w

1

> 0.

Proof. Let u ∈ K be a solution of (1.1) in R

⁺0

×Ω. Clearly A u and F u are bounded below in R

⁺₀

as shown in Remark 2.1. In particular w

2

> −∞ and w

1

≥ 0. Suppose that Eu(0) < ̟w

1

, with ̟ ≤ 1. Then F u(t) ≥ w

1

− Eu(0) > (1 − ̟)w

1

, which gives w

2

> (1 − ̟)w

1

≥ 0, and so w

2

> 0.

Suppose now that also (F

3

) holds. In correspondence to F = w

2

> 0, φ = u ∈ K, there exists ε

0

= ε

0

(w

2

, u) > 0 such that (F

3

) holds true and in particular there exists c

1

= c

1

(w

2

, u) > 0 for which (F

3

)–(i) is valid along u, so that for all t ∈ R

⁺0

ku(t, ·)k

q

≥ ˜ c

1

> 0 and kDu(t, ·)k

p

≥ ˜ c

1

/C

q

, (2.7) by (2.6), where ˜ c

1

= (w

2

/c

1

)

^1/q

> 0. Hence by (1.2) and (2.2),

pA u(t) ≥

a + bkDu(t, ·)k

^p(γ−1)_p

kDu(t, ·)k

^p_p

≥ a

1

kDu(t, ·)k

^p_p

, (2.8) where a

1

= a + b(˜ c

1

/C

q

)

^p(γ−1)

> 0. In particular,

w

1

≥ a

1

p inf

t∈R⁺0

kDu(t, ·)k

^p_p

> 0, and the proposition is proved.

3. The main Theorem. In this section we show the main abstract global non–

existence result for solutions of problem (1.1), assuming condition (1.2) and the

general structure assumptions given in the Introduction. In what follows p

∗

=

p(n − 1)/(n − p).

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Theorem 3.1. Assume (F

1

)–(F

3

). Then there are no solutions u ∈ K of (1.1) in R

⁺0

× Ω, for which

Eu(0) <

1 − γp

q

w

1

= E

1

, (3.1)

where w

1

= inf

_t∈R⁺

0

A u(t), and for which there exist T ≥ 0, q

1

> 0, m, ℘, with 1 < m ≤ ℘ − κ, 0 ≤ κ ≤ p(1 − m/℘) and ℘ < ℘

0

= ℘

0

(n, p, q), where

℘

0

= pq(n − 1 + p) − p

²

(n − 1)

n(q − p) + p

²

∈ (p, p

∗,q

), p

∗,q

= min{p

∗

, q}, (3.2) and non–negative functions δ

1

, δ

2

∈ L

^∞_loc

(J), ψ, k ∈ W

_loc^1,1

(J), J = [T, ∞), with k

^′

≥ 0, ψ > 0 in J and ψ

^′

(t) = o(ψ(t)) as t → ∞, such that for all t ∈ J

(Q) hQ(t, ·, u, u

t

), ui

Γ1

≤ q

1

{δ

1

(t)

^1/m

ku(t, ·)k

^κ/m_℘,Γ₁

D u(t)

^1/m^′

+ δ

2

(t)

^1/℘

D u(t)

^1/℘^′

}ku(t, ·)k

℘,Γ1

, and

δ

^1/(m−1)₁

+ δ

^1/(℘−1)₂

≤ k/ψ in J, Z

∞

ψ(t) [max{k(t), ψ(t)}]

^−(1+θ)

dt = ∞, (3.3) for some appropriate constant θ ∈ (0, θ

0

), where

θ

0

= 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).

(3.4)

Proof. Proceed by contradiction and assume that there exists a solution u ∈ K of (1.1) in R

⁺0

× Ω, satisfying (3.1)–(3.4) as in the statement. Clearly w

1

> 0 by Proposition 2.2, so that also E

1

> 0, since q satisfies (2.5). Fix E

2

, with max{0, Eu(0)} < E

2

< E

1

and take ε ∈ (0, ε

0

) so small that

εw

1

≤ (q − γp)w

1

− qE

2

, (3.5)

which is possible since w

1

> 0 and E

2

< E

1

. For each t ∈ R

⁺₀

put H (t) = E

2

− Eu(0) +

Z

t 0

D u(τ )dτ.

Of course H is well defined and non–decreasing by (B)–(i) and (F

2

), being D ≥ 0 and finite along u. Hence, by (B)–(ii),

E

2

− Eu(t) ≥ H (t) ≥ H

0

= E

2

− Eu(0) > 0 for t ∈ R

⁺0

, (3.6) where H

0

= H (0). Moreover, by (3.6), (2.2), the choice of E

2

, the definition of E

1

and the inequality w

2

> γpw

1

/q, it follows that for all t ∈ R

⁺₀

H (t) ≤ E

2

− Eu(t) < E

1

+ F u(t) < q

γp − 1

F u(t) + F u(t)

= q

γp F u(t).

(3.7)

(8)

Take φ = u in the Distribution Identity (A). By (2.2) d

dt hu

t

(t, ·), u(t, ·)i = c

3

ku

t

(t, ·)k

²₂

+ (q − ε)A u(t) − hhAu(t, ·), u(t, ·)ii

+ hf (t, ·, u(t, ·)), u(t, ·)i − (q − ε)F u(t) − (q − ε)Eu(t)

− hQ(t, ·, u(t, ·), u

t

(t, ·)), u(t, ·)i

Γ1

,

where c

3

= 1 + (q − ε)/2 > 0 by the choice of ε. By (2.4), applying (F

3

)–(ii) with c

2

= c

2

(w

2

, u, ε) > 0, we obtain for all t ∈ R

⁺0

d

dt hu

t

(t, ·), u(t, ·)i ≥c

3

ku

t

(t, ·)k

²₂

+ c

2

ku(t, ·)k

^q_q

− (q − ε)Eu(t)

− hQ(t, ·, u(t, ·), u

t

(t, ·)), u(t, ·)i

Γ1

+ (q − ε − γp)A u(t).

Since ε < q − γp by (3.5) and Eu ≤ E

2

− H by (3.6), by (2.2) we have d

dt hu

t

(t, ·), u(t, ·)i ≥ c

3

ku

t

(t, ·)k

²₂

+ c

2

ku(t, ·)k

^q_q

+ (q − ε − γp)A u(t)

− hQ(t, ·, u(t, ·), u

t

(t, ·)), u(t, ·)i

Γ1

+ γpH (t) − (q − ε)E

2

. Now set C

2

= εa

1

(q − ε − γp)/pq > 0, so that

(q − ε − γp)A u(t) − (q − ε)E

2

≥ (q − ε − γp)

1 − q − ε q

A u(t) + (q − ε)

(q − ε − γp) w

1

q − E

2

≥ C

2

kDu(t, ·)k

^p_p

, by (2.8) and the fact that ε ∈ (0, ε

0

) implies (q − ε)[(q − ε − γp)w

1

− qE

2

]/q ≥ 0 thanks to (3.5). Consequently, putting ˜ c

2

= min{c

2

, C

2

} > 0, we get

d

dt hu

t

(t, ·), u(t, ·)i ≥ c

3

ku

t

(t, ·)k

²₂

+ ˜ c

2

ku(t, ·)k

^q_q

+ kDu(t, ·k

^p_p

) + γpH (t)

− hQ(t, ·, u(t, ·), u

t

(t, ·)), u(t, ·)i

Γ1

.

(3.8)

By [1, Theorem 7.58, with χ = 0, k = n − 1] the embedding W

^s,p

(Ω) ֒→ L

^℘⁰

(∂Ω) is continuous, by the choice of s in (3.4), being ℘

0

> p by (3.2). Furthermore,

n

p − n − 1

℘ < s < q

℘ − 1 q p − 1

, (3.9)

being ℘ < ℘

0

by (3.2). Clearly also the embedding W

^s,p

(Ω) ֒→ L

^℘

(∂Ω) is continuous. Moreover, by [7, Corollary 3.2–(a), with s

1

= 0, s

2

= 1, p

1

= p

2

= p and θ = 1 − s] there exists a constant S = S(s, p, ℘) > 0 such that

ku(t, ·)k

℘,Γ1

≤ Sku(t, ·)k

^1−s_q

kDu(t, ·)k

^s_p

, (3.10) since q > p. For brevity, let α

1

, α

2

and β

1

, β

2

denote the numbers

1 α

1

= 1 m − s

p

1 + κ m

, β

1

= (1 − s) 1 + κ

m

− q 1 m − s

p

1 + κ m

, 1

α

2

= 1

℘ − s

p , β

2

= 1 − s − q 1

℘ − s p

.

Hence 1 < α

1

≤ α

2

. Indeed, s < p/℘ whenever 1 < ℘ ≤ p being s ∈ (0, 1), otherwise

it is a direct consequence of (3.9), and so s < p/℘ ≤ p/(m + κ) by (3.2). This gives

(9)

the claim by direct calculation. While β

1

≤ β

2

< 0 again by (3.2) and (3.9). Hence, using (Q) and (3.10), we get for all t ∈ J

hQ(t, ·, u(t, ·), u

t

(t, ·)), u(t, ·)i

Γ1

≤ q

2

(

δ

1

(t)

^1/(m−1)

D u(t)

1/m^′

· ku(t, ·)k

(1−s)(1+κ/m)

q

kDu(t, ·)k

^s(1+κ/m)_p

+

δ

2

(t)

^1/(℘−1)

D u(t)

^1/℘^′

ku(t, ·)k

^1−s_q

kDu(t, ·)k

^s_p

)

= q

2

(

δ

1

(t)

^1/(m−1)

D u(t)

^1/m^′

ku(t, ·)k

^q/α_q ¹

kDu(t, ·)k

^s(1+κ/m)_p

ku(t, ·)k

^β_q¹

+

δ

2

(t)

^1/(℘−1)

D u(t)

1/℘^′

ku(t, ·)k

^q/α_q ²

kDu(t, ·)k

^s_p

ku(t, ·)k

^β_q²

)

≤ q

2

n [(2δ

1

(t)/ℓ)

^1/(m−1)

D u(t) +

¹₂

ℓku(t, ·)k

^q_q

+

¹₂

ℓkDu(t, ·)k

^p_p

] · ku(t, ·)k

^β_q¹

+[(2δ

2

(t)/ℓ)

^1/(℘−1)

D u(t) +

¹₂

ℓku(t, ·)k

^q_q

+

¹₂

ℓkDu(t, ·)k

^p_p

] · ku(t, ·)k

^β_q²

o

, where q

2

= q

1

max{S, S

^1+κ/m

} and in the last step we have applied Young’s inequality, with ℓ ∈ (0, 1) to be fixed later. In conclusion,

hQ(t, ·, u(t, ·), u

t

(t, ·)), u(t, ·)i

Γ1

≤ ˜ q

2

n ℓ

^−m^′^/m

[δ

1

(t)

^1/(m−1)

+ δ

2

(t)

^1/(℘−1)

]Du(t) + ℓ ku(t, ·)k

^q_q

+ kDu(t, ·)k

^p_p

o

· ku(t, ·)k

^β_q²

,

(3.11)

where ˜ q

2

= 2

^1/(m−1)

q

2

max{1, (˜ c

1

)

^β¹^−β²

} > 0 by (2.7). Clearly

r = −β

2

/q ∈ (0, 1) (3.12)

by (3.9). But by (F

3

) and (3.7) we have

ku(t, ·)k

^β_q²

= ku(t, ·)k

^−qr_q

≤ c

^r₁

[F u(t)]

^−r

≤ [c

1

q/γp]

^r

[H u(t)]

^−r

. Therefore

hQ(t, ·, u(t, ·), u

t

(t, ·)), u(t, ·)i

Γ1

≤ c

4

n ℓ

^−m^′^/m

[δ

1

(t)

^1/(m−1)

+ δ

2

(t)

^1/(℘−1)

]Du(t) + ℓ ku(t, ·)k

^q_q

+ kDu(t, ·)k

^p_p

o

[H (t)]

^−r

for all t ∈ J, where c

4

= ˜ q

2

(c

1

q/γp)

^r

. Put

r

0

= min 1 2 − 1

q , r

. (3.13)

Note that θ

0

in (3.4) can be expressed as θ

0

= r

0

/(1 − r

0

), and take from now on r = θ/(1 + θ), so that r ∈ (0, r

0

). Consequently, we get

hQ(t,·, u(t, ·), u

t

(t, ·)), u(t, ·)i

Γ1

≤ c

4

n ℓH

₀^−r

ku(t, ·)k

^q_q

+ kDu(t, ·)k

^p_p

+ ℓ

^−m^′^/m

H

^r−r

0

[δ

1

(t)

^1/(m−1)

+ δ

2

(t)

^1/(℘−1)

] · [H (t)]

^−r

D u(t) o ,

(3.14)

(10)

where in the last step we have used the facts that H ≥ H

0

by (3.6) and that 0 < r < r

0

≤ r by (3.13). Since Du = H

^′

, we see that (1 − r)H

^−r

H

^′

= [H

^1−r

]

^′

. Hence it is convenient to introduce the function

Z = Z (t) = λk(t) [H (t)]

^1−r

+ ψ(t)hu

t

, ui,

where λ > 0 is a constant to be fixed later. Clearly Z ∈ W

_loc^1,1

(J) and so, a.e. in J, Z

^′

= λk(1 − r)H

^−r

H

^′

+ λk

^′

H

^1−r

+ ψ d

dt hu

t

, ui + ψ

^′

hu

t

, ui.

By (3.8) and (3.14), a.e. in J

Z

^′

≥ λk(1 − r)H

^−r

H

^′

+ λk

^′

H

^1−r

+ ψ

^′

hu

t

, ui

+ ψ c

3

ku

t

k

²₂

+ ˜ c

2

ku(t, ·)k

^q_q

+ kDu(t, ·)k

^p_p

+ γpH − hQ(t, ·, u, u

t

), ui

Γ1

≥

λk(1 − r) − c

4

ℓ

^−m^′^/m

H

^r−r

0

[δ

1

(t)

^1/(m−1)

+δ

2

(t)

^1/(℘−1)

]ψ

H

^−r

H

^′

+ γpψH + λk

^′

H

^1−r

+ ψ

^′

hu

t

, ui + ψ c

3

ku

t

k

²₂

+ ˜ c

2

− c

4

ℓH

₀^−r

(kuk

^q_q

+ kDu(t, ·)k

^p_p

) . Thus, a.e. in J, by (3.3)

1

and the fact that λk

^′

H

^1−r

≥ 0, we find

Z

^′

≥ k n

λ(1 − r) − c

4

ℓ

^−m^′^/m

H

^r−r

0

o

H

^−r

H

^′

+ γpψH

+ ψ

^′

hu

t

, ui + ψ c

3

ku

t

k

²₂

+ ˜ c

2

− c

4

ℓH

₀^−r

(kuk

^qq

+ kDu(t, ·)k

^p_p

) . Next, from Cauchy’s and Young’s inequalities, and the definition of K, we have

|hu

t

(t, ·), u(t, ·)i| ≤ ku

t

(t, ·)k

2

ku(t, ·)k

2

≤ ku

t

(t, ·)k

²₂

+ ku(t, ·)k

²₂

. (3.15) Consider now the relation z

^ξ

≤ z + 1 ≤ (1 + 1/η)(z + η), which holds for all z ≥ 0, ξ ∈ [0, 1], η > 0, and take z = ku(t, ·)k

^q₂

, ξ = 2/q < 1, since q > 2 by (2.5), and η = H

0

, we obtain

ku(t, ·)k

²₂

≤ (1 + 1/H

0

)(ku(t, ·)k

^q₂

+ H

0

).

Since the embedding L

^q

(Ω) ֒→ L

²

(Ω) is continuous by (2.5), we get

ku(t, ·)k

²₂

≤ c

5

{ku(t, ·)k

^q_q

+ H (t)}, (3.16) where c

5

= (1 + 1/H

0

) max{1, µ

n

(Ω)

^(q−2)/2

} > 0, being H ≥ H

0

in J by (3.6).

Then, using (3.15) and (3.16) in the preceding estimate of Z

^′

, we find that Z

^′

≥ k n

λ(1 − r) − c

4

ℓ

^−m^′^/m

H

^r−r

0

o H

^−r

H

^′

+ ψ(c

3

− |ψ

^′

|/ψ)ku

t

k

²₂

+ ψ {γp − c

5

|ψ

^′

|/ψ} H + ψ ˜c

2

− c

5

|ψ

^′

|/ψ − c

4

ℓH

₀^−r

ku(t, ·)k

^q_q

+ kDu(t, ·)k

^p_p

.

(3.17)

There is T

1

∈ J such that 2|ψ

^′

|/ψ ≤ min{c

3

, γp/c

5

, ˜ c

2

/c

5

} for all t ∈ J

1

= [T

1

, ∞), since ψ

^′

= o(ψ) as t → ∞. Then we take ℓ > 0 so small that 4c

4

ℓ ≤ ˜ c

2

H

₀^r

and λ > 0 so large that λ ≥ max{c

4

H

^r−r

0

/ℓ

^m^′^/m

(1 − r), 1} and Z (T

1

) > 0. In conclusion, we have shown that for a.a. t ∈ J

1

Z

^′

(t) ≥ Cψ(t) H (t) + ku

t

(t, ·)k

²₂

+ ku(t, ·)k

^q_q

+ kDu(t, ·)k

^p_p

, (3.18) where 2C = min{˜ c

2

/2, c

3

, γp}. In particular Z (t) ≥ Z (T

1

) > 0 for all t ∈ J

1

, since k(T

1

), H (T

1

) > 0.

On the other hand, from the definition of Z , we obtain Z

^α

≤

λkH

^1/α

+ ψ|hu

t

, ui|

α

≤ 2

^α−1

{(λk)

^α

H + ψ

^α

ku

t

k

^α₂

kuk

^α₂

} , (3.19)

(11)

where α = 1/(1 − r). Of course, α ∈ (1, 2) by (3.13) and the choice of r. Put ν = 2/α, so that ν > 1. Furthermore,

1 αν

^′

= ν − 1 αν = 1

α − 1 2 = 1

2 − r > 1 q

by (3.13), and so αν

^′

< q. Thus, using the relation z

^ξ

≤ z + 1 ≤ (1 +1/η)(η + z) once more, with z = ku(t, ·)k

^q₂

, ξ = αν

^′

/q < 1 and η = H

0

, it follows that

ku(t, ·)k

^αν₂ ^′

≤ (1 + 1/H

0

)(H

0

+ ku(t, ·)k

^q₂

) ≤ c

5

(H (t) + ku(t, ·)k

^q_q

), (3.20) by (3.6), where c

5

is the same constant as in (3.16). Hence, from (3.19), Young’s inequality and (3.20), for all t ∈ J

1

Z (t)

^α

≤ 2

^α−1

[max{λk(t), ψ(t)}]

^α

n

H (t) + ku

t

(t, ·)k

^αν₂

+ ku(t, ·)k

^αν₂ ^′

o

≤ B [max{λk(t), ψ(t)}]

^α

H (t) + ku

t

(t, ·)k

²₂

+ ku(t, ·)k

^q_q

+ kDu(t, ·)k

^p_p

, where B = 2

^α−1

(c

5

+ 1). Combining this with (3.18) and λ ≥ 1, we have a.e. in J

1

Z

^−α

Z

^′

≥ C

B ψ [max{λk, ψ}]

^−α

≥ c

0

ψ [max{k, ψ}]

^−α

,

where c

0

= C/Bλ

^α

. Finally, since α = 1 + θ, being r = θ/(1 + θ), by (3.3)

2

we see that Z cannot be global. This completes the proof.

4. Applications. In this section we provide some concrete examples of functions f and Q, and give useful applications to the main Theorem 3.1. Take f of the form f (t, x, u) = g(t, x)|u|

^σ−2

u + c(x)|u|

^q−2

u, (4.1) where c ∈ L

^∞

(Ω) is a non–negative function, g ∈ C(R

⁺0

× Ω) is differentiable with respect to t and g

t

∈ C(R

⁺0

× Ω). Moreover, assume

σ ≤ q, max{2, γp} < q ≤ p

^∗

, kck

∞

> 0;

0 ≤ −g(t, x), g

t

(t, x) ≤ h(x) in R

⁺0

× Ω, for some h ∈ L

¹

(Ω), g(t, ·) ∈ L

^η

(Ω) in R

⁺₀

, where η =

( q/[q − σ], if σ < q,

∞, if σ = q.

(4.2)

As observed in the introduction, the negative term of (f (t, x, u), u), governed by g, plays the role of a nonlinear perturbation acting against blow up. Functions of type (4.1) were first considered in similar contexts in [22], and then extended in [5] to the case of variable exponents. Here

F φ(t) = F (t, φ) = Z

Ω

g(t, x) |φ(t, x)|

^σ

σ + c(x) |φ(t, x)|

^q

q

dx (4.3)

for any φ ∈ K.

Lemma 4.1 (Lemma 4.1 of [5]). Assume that the external force f is of the type given in (4.1) and (4.2). Then (F

1

), (F

2

) and (F

3

)–(i) hold. Furthermore, if in addition

σ < q and c = ess inf

_Ω

c(x) > 0, (4.4) then (F

3

)–(ii) is verified, and in particular

hf (t, ·, φ(t, ·)), φ(t, ·)i ≥ qF φ(t), F φ(t) ≤ c

1

ku(t, ·)k

^q_q

(4.5)

for all φ ∈ K and t ∈ R

⁺0

, where c

1

= kck

∞

/q.

(12)

Lemma 4.2. Assume that the continuous damping function Q given in the Intro- duction verifies also the pointwise condition

(Q

1

) There exist constants t

Q

≥ 0, m, ℘ and κ satisfying (3.2), and non–negative functions d

1

∈ C(R

⁺0

→ L

^℘¹

(Γ

1

)) and d

2

∈ C(R

⁺0

→ L

^∞

(Γ

1

), where ℘

1

= ∞ if

℘ = m + κ and ℘

1

= ℘/(℘ − κ − m) if ℘ > m + κ, such that

|Q(t, x, u, v)| ≤ (d

1

(t, x)|u|

^κ

)

^1/m

(Q(t, x, u, v), v)

^1/m^′

+ d

2

(t, x, u)

^1/℘

(Q(t, x, u, v), v)

^1/℘^′

(4.6) whenever (t, x, u, v) ∈ [t

Q

, ∞) × Γ

1

× R

^N

× R

^N

.

Then (Q) is satisfied along any solution u of the problem (1.1), with T ≥ t

Q

, q

1

= 1, δ

1

(t) = kd

1

(t, ·)k

℘1,Γ1

and δ

2

(t) = sup

_(x,ξ)∈Γ₁_×RN

d

2

(t, x, ξ), provided that (F

2

) holds.

Proof. Clearly hQ(t, ·, u(t, ·), u

t

(t, ·)), u

t

(t, ·)i

Γ1

≥ 0 for each t ≥ 0 along any solution u ∈ K of the problem (1.1), and so Du ≥ 0, since also F

t

u ≥ 0 in R

⁺₀

by (F

2

), and D u is finite in R

⁺0

along u by (B)–(i).

Let first assume ℘ > m + κ. By (4.6) and H¨ older’s inequality, for each t ≥ t

Q

, along any solution u of (1.1),

kQ(t, ·, u(t, ·),u

t

(t, ·))k

℘^′,Γ1

≤ k(d

1

(t, x)|u|

^κ

)

^1/m

(Q(t, x, u, u

t

), u

t

)

^1/m^′

k

℘^′,Γ1

+ kd

2

(t, x, u)

^1/℘

(Q(t, x, u, u

t

), u

t

)

^1/℘^′

k

℘^′,Γ1

≤

Z

Γ1

d

1

(t, x)|u(t, x)|

^κ

_℘−m^℘

dµ

n−1

^℘−m_m℘

hQ(t, ·, u, u

t

), u

t

(t, ·)i

^1/m_Γ₁ ^′

+ sup

(x,ξ)∈Γ1×R^N

d

2

(t, x, ξ)

!

1/℘

· hQ(t, ·, u, u

t

), u

t

(t, ·)i

^1/℘_Γ₁ ^′

. On the other hand, applying once again H¨ older’s inequality, we find that

Z

Γ1

d

1

(t, x)|u(t, x)|

^κ

_℘−m^℘

dµ

n−1

≤

Z

Γ1

d

℘

℘−m−κ

1

dµ

n−1

^℘−m−κ_℘−m

Z

Γ1

|u|

^℘

dµ

n−1

_℘−m^κ

. Hence, combining the last two inequalities, we get by (F

2

), also in the simpler case

℘ = m + κ, whatsoever κ ≥ 0 is,

kQ(t, ·, u(t, ·), u

t

(t, ·))k

℘^′,Γ1

≤ kd

1

(t, ·)k

^1/m_℘₁_,Γ₁

ku(t, ·)k

^κ/m_℘,Γ₁

hQ(t, ·, u, u

t

), u

t

(t, ·)i

^1/m_Γ₁ ^′

+ δ

2

(t)

^1/℘

· hQ(t, ·, u, u

t

), u

t

(t, ·)i

^1/℘_Γ₁ ^′

≤ δ

1

(t)

^1/m

ku(t, ·)k

^κ/m_℘,Γ₁

D u(t)

^1/m^′

+ δ

2

(t)

^1/℘

D u(t)

^1/℘^′

, where δ

1

(t) = kd

1

(t, ·)k

℘1,Γ1

and δ

2

(t) = sup

_(x,ξ)∈Γ₁_×RN

d

2

(t, x, ξ) are of class C(R

⁺0

), so that δ

1

, δ

2

∈ L

^∞_loc

(R

⁺0

). Finally, we obtain

|hQ(t,·, u, u

t

), u(t, ·)i

Γ1

| ≤ kQ(t, ·, u, u

t

)k

℘^′,Γ1

ku(t, ·)k

℘,Γ1

≤ n

δ

1

(t)

^1/m

ku(t, ·)k

^κ/m_℘,Γ

1

D u(t)

^1/m^′

+ δ

2

(t)

^1/℘

D u(t)

^1/℘^′

o

ku(t, ·)k

℘,Γ1

(4.7) for all t ≥ T = t

Q

.

Remark 4.3. Thanks to (4.7), we immediately get that hQ(t, ·, u, u

t

), u(t, ·)i

Γ1

is in L

¹_loc

(R

⁺0

), and this makes the definition of solution well posed; cf. also the

discussion made in Section 2 and the calculation provided in [21, page 35].

(13)

Let us now distinguish two cases, depending on the fact that b could be zero or not. Of course, when b > 0 and a = 0 we are in the so called degenerate case, which is in our context more interesting. Remind that we have assumed γ > 1 when b > 0, while γ = 1 if b = 0, and put s = b if b > 0, while s = a if b = 0.

Lemma 4.4. Assume (4.1) and (4.2). If u ∈ K is a solution of (1.1) in R

⁺0

× Ω, then for all t ∈ R

⁺0

Eu(t) ≥ ϕ(υ(t)) = s

p υ(t)

^γp

− c

q υ(t)

^q

, (4.8)

where υ(t) = kDu(t, ·)k

p

, c = C

^q_q

kck

∞

and C

q

is the embedding constant introduced in (2.6).

Proof. Let u ∈ K be a solution of (1.1) in R

⁺₀

× Ω. By (2.8) we have A u(t) ≥ 1

p akDu(t, ·)k

^p_p

+ bkDu(t, ·)k

^γp_p

= s

p kDu(t, ·)k

^γp_p

. (4.9) Therefore, since Eu(t) ≥ A u(t) − F u(t) for each t ∈ R

⁺0

by (2.2), the assertion follows at once by (4.9), (4.5)

2

and (2.6).

It is easy to see that the function ϕ : R

⁺₀

→ R introduced in Lemma 4.4 attains its maximum at

υ

0

= sγ c

1/(q−γp)

. (4.10)

Moreover ϕ is strictly decreasing for υ ≥ υ

0

, with ϕ(υ) → −∞ as υ → ∞. Finally, ϕ(υ

0

) =

1 − γp

q

w

0

= E

0

> 0, where w

0

= sυ

₀^γp

p > 0. (4.11)

Theorem 4.5. Assume (4.1), (4.2), (4.4) and (Q

1

). If u ∈ K is a solution of (1.1) in R

⁺0

× Ω, then w

2

= inf

_t∈R⁺

0

F u(t) > −∞. If, furthermore, Eu(0) < E

1

, with E

1

given in (3.1), then w

2

> 0, (υ(t), Eu(t)) ∈ ˜ Σ for all t ∈ R

⁺0

, where

Σ = {(υ, E) ∈ R ˜

²

: υ > υ

0

, E < E

1

}, (4.12) and w

1

= inf

_t∈R⁺

0

A u(t) ≥ w

0

, with υ

0

and w

0

defined in (4.10) and (4.11). Con- sequently, there are no solutions u ∈ K of the problem (1.1) in R

⁺0

× Ω, with Eu(0) < E

1

, for which there exist positive functions ψ, k verifying (3.3)–(3.4) as in Theorem 3.1.

Proof. The fact that w

2

is finite and positive is an immediate consequence of Propo- sition 2.2. By (F

2

), (Q

1

) and (B)–(ii) clearly Eu(t) ≤ Eu(0) < E

1

for all t ∈ R

⁺0

. Moreover, by (4.9), (2.2), (4.5)

2

and (2.6), for all t ∈ R

⁺₀

we get

A u(t) − γs

q υ(t)

^γp

≥

1 − γp

q

A u(t) ≥ E

1

> Eu(0) ≥ A u(t) − c

q υ(t)

^q

,

so that υ(t) > υ

0

for all t ∈ R

⁺₀

, by (4.10). Hence (υ(t), Eu(t)) ∈ ˜ Σ for all t ∈ R

⁺₀

,

as required. In particular, A u(t) ≥ sυ(t)

^γp

/p ≥ sυ

^γp₀

/p = w

0

for all t ∈ R

⁺0

, that is

w

1

≥ w

0

. The last part of the theorem is a direct consequence of Theorem 3.1.

(14)

In the next corollary we provide sufficient conditions under which assumptions (3.3)–(3.4) of Theorem 3.1 are satisfied. Let Q = Q(t, x, u, v) be a continuous damping function as in the Introduction and assume also that there exists t

^∗

>> 1 such that

Q(t, x, u, v) = d

1

(t, x)|u|

^κ

|v|

^m−2

v + d

2

(t, x, u)|v|

^℘−2

v (4.13) for all (t, x, u, v) ∈ [t

^∗

, ∞) × Γ

1

× R

^N

× R

^N

, where κ, m, ℘, d

1

and d

2

are as stated in (Q

1

).

Hence, for all (t, x, u, v) ∈ [t

^∗

, ∞) × Γ

1

× R

^N

× R

^N

,

|Q(t,x, u, v)|

≤ [d

1

(t, x)|u|

^κ

]

^1/m

[d

1

(t, x)|u|

^κ

|v|

^m

]

^1/m^′

+ d

2

(t, x, u)

^1/℘

[d

2

(t, x, u)|v|

^℘

]

^1/℘^′

≤ [d

1

(t, x)|u|

^κ

]

^1/m

(Q(t, x, u, v), v)

^1/m^′

+ d

2

(t, x, u)

^1/℘

(Q(t, x, u, v), v)

^1/℘^′

, so that (Q

1

) holds with t

Q

= t

^∗

. Now for all t ≥ t

^∗

define δ

1

(t) = kd

1

(t, ·)k

℘1,Γ1

, δ

2

(t) = sup

_(x,u)∈Γ₁_×RN

d

2

(t, x, u) and ℘

1

is defined in (Q

1

).

Corollary 4.6. Assume (4.1), (4.2), (4.4), (4.13) and that for each t ≥ t

^∗

δ

1

(t)

^1/(m−1)

+ δ

2

(t)

^1/(℘−1)

≤ K ·

( (1 + t)

^{ℓ/(℘−1)}

, if ℓ < 0,

(1 + t)

^ℓ/(m−1)

, if 0 ≤ ℓ ≤ m − 1, (4.14) for some appropriate number K ≥ 1. Then there are no solutions u ∈ K of (1.1) in R

⁺0

× Ω, with Eu(0) < E

1

.

Proof. Let u ∈ K be a solution of (1.1) in R

⁺₀

× Ω, with Eu(0) < E

1

. All the structural assumptions of Theorem 4.5 are available, and it remains to provide the auxiliary functions k and ψ verifying (3.3), with θ

0

as in (3.4) to reach the desired contradiction.

Case ℓ < 0. Take T ≥ t

^∗

and put k(t) = K

^1/℘

and ψ(t) = K

^−1/℘^′

(1 + t)

^{−ℓ/(℘−1)}

for each t ∈ J = [T, ∞), so that (3.3)

1

is verified in J. Now, taking T enough large to have T ≥ max{t

^∗

, K

^{(℘−1)/|ℓ|}

− 1}, then for all t ∈ J

ψ(t) ≥ ψ(T ) = K

^−1/℘^′

(1 + T )

^{|ℓ|/(℘−1)}

≥ K

^1/℘

= k(t).

Case 0 ≤ ℓ ≤ m − 1. Take k(t) = K

^1/m

and ψ(t) = K

^−1/m^′

(1 + t)

^{−ℓ/(m−1)}

for each t ∈ J, so that (3.3)

1

is verified in J and k(t) ≥ ψ(t) for each t ∈ J, being K ≥ 1.

Hence, in both cases, for each t ≥ T we have ψ(t)[max{k(t),ψ(t)}]

^−(1+θ)

=

( K

^θ/℘^′

(1 + t)

^{ℓθ/(℘−1)}

, if ℓ < 0, K

^−1−θ/m

(1 + t)

^{−ℓ/(m−1)}

, if 0 ≤ ℓ ≤ m − 1.

(4.15)

If ℓ < 0, then we choose θ ∈ (0, θ

0

), so small that θ ≤ (℘ − 1)/|ℓ|, that is so small that (3.3)

₂

holds. While (3.3)

₂

is trivially verified for all θ ∈ (0, θ

0

), with θ

0

as in (3.4), whenever 0 ≤ ℓ ≤ m − 1.

Theorem 4.7. Assume (4.1), (4.2) and (Q

1

). Let u ∈ K be a solution of (1.1) in R

⁺0

× Ω, such that Eu(0) < E

0

, with E

0

given in (4.11). Then υ

0

∈ υ(R /

⁺0

) and w

1

= inf

_t∈R⁺

0

A u(t) 6= w

0

, where υ

0

and w

0

are defined in (4.10) and (4.11),

respectively. Moreover, w

1

> w

0

if and only if υ(R

⁺0

) ⊂ (υ

0

, ∞).

(15)

Proof. Let u ∈ K be a solution of (1.1) in R

⁺0

× Ω with Eu(0) < E

0

. We first claim that υ

0

∈ υ(R /

⁺0

). Proceed by contradiction and suppose that υ

0

∈ υ(R

⁺0

). It follows that there exists a sequence (t

j

)

j

in R

⁺0

such that υ(t

j

) → υ

0

as j → ∞.

By (4.8) we have E

0

> Eu(0) ≥ Eu(t

j

) ≥ ϕ(υ(t

j

)), which provides E

0

> E

0

by the continuity of ϕ ◦ υ, and the claim is proved.

We show that w

1

6= w

0

. Otherwise, A u(t) ≥ w

0

for all t ∈ R

⁺₀

. Therefore, by (2.2), (4.9) and (4.11), we have

A u(t) − γs

q υ(t)

^γp

≥

1 − γp

q

A u(t) ≥ E

1

= E

0

> Eu(0) ≥ A u(t) − c q υ(t)

^q

, so that υ(t) > υ

0

for each t ∈ R

⁺0

. Consequently, the first part of the theorem yields υ(R

⁺0

) ⊂ (υ

0

, ∞). On the other hand, there exists a sequence (t

j

)

j

such that A u(t

j

) → w

1

= w

0

as j → ∞, so that lim sup

_j→∞

υ(t

j

) ≤ lim

j→∞

[pA u(t

j

)/s]

^1/γp

= υ

0

, which contradicts the fact that υ(R

⁺0

) ⊂ (υ

0

, ∞). Hence w

1

6= w

0

.

If w

1

> w

0

, then Eu(0) < E

1

and υ(t) > υ

0

for all t ∈ R

⁺₀

by Theorem 4.5, so that υ(R

⁺0

) ⊂ (υ

0

, ∞), since υ

0

∈ υ(R /

⁺0

).

On the other hand, if υ(R

⁺0

) ⊂ (υ

0

, ∞), then υ(t) > υ

0

and A u(t) > sυ

₀^γp

/p = w

0

for all t ∈ R

⁺0

by (4.9). Hence w

1

> w

0

, since the case w

1

= w

0

cannot occur by the argument above.

From now on in the section we assume for simplicity the structure assumptions (4.1), (4.2), (4.4), (4.13) and (4.14) for each t ∈ J, without further mentioning.

Corollary 4.8. Problem (1.1) does not possess solutions u ∈ K in R

⁺0

× Ω, with kDu(0, ·)k

p

> υ

0

, Eu(0) < E

0

, (4.16) where E

0

is defined in (4.11).

Proof. Assume by contradiction that u ∈ K is a solution of (1.1) in R

⁺0

× Ω, verifying (4.16). By Theorem 4.7 then w

1

> w

0

. Hence Eu(0) < E

0

< E

1

and the contradiction follows at once by an application of Corollary 4.6.

Corollary 4.8 extends and generalizes Theorem 7 of [23], in which a = 1, b = 0, p = 2, µ = 0, f = f (x, u), Q = Q(x, v). Moreover, in [23], the Conservation Law (B)–(ii) is assumed in the stronger form (B)

s

–(ii); cf. the remarks of Section 2 on strong solutions.

Proposition 4.9. If u ∈ K is a solution of (1.1) in R

⁺0

× Ω, with Eu(0) ≤ E

0

, where E

0

is defined in (4.11), then

w

1

≤ w

0

. (4.17)

Proof. Otherwise w

1

> w

0

, so that Eu(0) < E

1

, and u could not be global by Corollary 4.6.

In the sequel of the section we assume also (D) There exists t

∗

> 0 such that either

(i) g

t

(t, x) ≥ g

0

(t) > 0 for each (t, x) ∈ [0, t

∗

) × Ω, or

(ii) φ ∈ K and hQ(t, ·, φ, φ

t

), φ

t

i

Γ1

= 0 in [0, t

∗

] implies either φ(t, ·) ≡ 0 or

φ

t

(t, ·) ≡ 0 for all t ∈ [0, t

∗

].

(16)

In the next result we present a non–continuation theorem concerning the limit case Eu(0) = E

0

, not yet established for (1.1). For a nonlinear damped Kirchhoff system, with damping effective in Ω and homogeneous Dirichlet boundary conditions, a similar theorem was established in [5, Theorem 4.3]. This theorem already generalizes and extends previous results for damped wave equations, see [5, Re- mark 4.1] and the references therein.

Theorem 4.10. Problem (1.1) does not possess solutions u ∈ K in R

⁺0

× Ω, with kDu(0, ·)k

p

> υ

0

, Eu(0) = E

0

. (4.18) Proof. Assume by contradiction that u ∈ K is a global solution of (1.1) in R

⁺0

× Ω, verifying (4.18). By Proposition 4.9 we have w

1

≤ w

0

. We first claim that w

1

< w

0

cannot occur. As a matter of fact, if w

1

< w

0

there would exists t

0

such that A u(t

0

) < w

0

, and this is possible only if υ(t

0

) < υ

0

; indeed if υ(t

0

) ≥ υ

0

we immediately would have A u(t

0

) ≥ w

0

. Hence t

0

> 0 by (4.18) and by the continuity of υ there exists t

1

∈ (0, t

0

) such that υ(t

1

) = υ

0

. Thus E

0

= Eu(0) ≥ Eu(t

1

) ≥ w

0

− cυ

₀^q

/q = E

0

by (4.8). In other words, Eu(t

1

) = E

0

and R

t1

0

D u(τ )dτ = 0 by (B)–(ii). Consequently Du ≡ 0 in [0, t

1

] and so, by (F

2

) and (4.13), we obtain hQ(t, ·, u(t, ·), u

t

(t, ·)), u

t

(t, ·)i

Γ1

= 0 and F

t

u(t) = 0 for all t ∈ [0, t

1

].

Now, if (D)–(i) holds, then 0 = F

t

u(t) =

Z

Ω

g

t

(t, x) |u(t, x)|

^σ

σ dx ≥ g

0

(t)

σ ku(t, ·)k

^σ_σ

≥ 0

for each t ∈ [0, s

0

], where s

0

= min{t

∗

, t

1

}. Therefore ku(t, ·)k

σ

≡ 0 and in turn u ≡ 0 in [0, s

0

] × Ω. But this occurrence is impossible, since kDu(0, ·)k

p

= υ(0) >

υ

0

> 0 by (4.18)

1

, so that we reach a contradiction.

While, if (D)–(ii) holds, since hQ(t, ·, u(t, ·), u

t

(t, ·)), u

t

(t, ·)i

Γ1

= 0 for all t ∈ [0, s

0

], we get that either u(t, ·) = 0 or u

t

(t, ·) = 0 for all t ∈ [0, s

0

], where as above s

0

= min{t

∗

, t

1

}. Again, as already shown, the first case cannot occur since υ(0) > υ

0

. In the latter, u is clearly constant with respect to t in [0, s

0

], and so u(t, x) = u(0, x) for each t ∈ [0, s

0

]. Taking φ(t, x) = u(0, x) in the Distribution Identity (A), then for each t ∈ [0, s

0

] we have thAu(0, ·), u(0, ·)i = R

t

0

hf (τ, ·, u(0, ·)), u(0, ·)idτ , since hQ(t, ·, u(0, ·), 0), u(0, ·)i

Γ1

= 0 by (4.7), being D u = 0 in [0, s

0

]. Therefore hAu(0, ·), u(0, ·)i = hf (t, ·, u(0, ·)), u(0, ·)i for each t ∈ [0, s

0

], and so hA(u(0, ·)), u(0, ·)i = hf (0, ·, u(0, ·)), u(0, ·)i. Now γpA u(0) ≥ qF u(0) by (2.4) and (F

3

). On the other hand, E

0

= Eu(0) = A u(0) − F u(0) by (2.2), since u

t

(0, ·) = 0. By (4.9) and (4.11) we have A u(0) > w

0

> 0, and so

E

0

≥

1 − γp

q

A u(0) >

1 − γp

q

w

0

= E

0

by (4.11). This contradiction shows the claim.

Hence w

1

= w

0

. In particular A u(t) ≥ w

0

for all t ∈ R

⁺0

and we assert that equality cannot occur at a finite time. Indeed, if there is τ such that A u(τ ) = w

0

, then υ(τ ) ≤ υ

0

by (4.9). On the other hand, as shown in the proof of Theorem 4.7, we get υ(τ ) > υ

0

. This contradiction shows that it remains to consider only the case w

1

= w

0

, A u(t) > w

0

and υ(t) > υ

0

for all t ∈ R

⁺0

. A continuity argument shows at once that

lim inf

t→∞

A u(t) = w

0

and lim inf

t→∞

υ(t) = υ

0

. (4.19)

(17)

Indeed, inf

_t∈R⁺

0

A u(t) = w

1

= w

0

, so that any minimizing sequence (t

k

)

k

is un- bounded since A u cannot reach w

0

at a finite time. Hence, there exists a sequence t

k

→ ∞ such that lim

k

A u(t

k

) = w

0

. In particular, lim inf

t→∞

A u(t) ≤ w

0

. Of course, lim inf

t→∞

A u(t) = w

0

, being w

0

= inf

_t∈R⁺

0

A u(t). Clearly, υ

∗

= inf

_t∈R⁺

0

υ(t) ≥ υ

0

, since υ(t) > υ

0

for all t ∈ R

⁺0

, as shown above. On the other hand, A u(t) ≥ sυ(t)

^γp

/p ≥ sυ

^γp∗

/p for all t ∈ R

⁺₀

by (4.9), so that w

0

≥ sυ

^γp∗

/p, that is υ

∗

≤ υ

0

by (4.10). In conclusion, υ

∗

= υ

0

. Again, since υ(t) > υ

0

for all t ∈ R

⁺₀

, this shows that lim inf

t→∞

υ(t) = υ

0

. Hence (4.19) holds.

By (2.2) and (B)–(ii) we have w

0

− cυ(t)

^q

/q ≤ w

0

− F u(t) < Eu(t) ≤ E

0

, so that lim sup

_t→∞

Eu(t) = E

0

by (4.11). Hence R

∞

0

D u(τ )dτ = 0 by monotonicity.

In particular Du ≡ 0 in R

⁺0

, which is again impossible by (D) using the argument already produced. This completes the proof.

5. A model with higher dissipation terms. In this section we study some qualitative properties of global solutions of the system (1.3), under the same structural hypotheses on the boundary damping Q as in Section 3, together with (F

1

)–(F

3

), when p = 2. Moreover assume

̺ ∈ C(R

⁺0

) ∩ C

¹

(R

⁺

), with ̺, ̺

^′

≥ 0. (5.1) The natural energy associated to the field φ ∈ K of the problem (1.3) is the following Eφ(t) =

¹₂

kφ

t

(t, ·)k

²₂

+ kφ

t

(t, ·)k

²_2,Γ₁

+ A φ(t) − F φ(t) (5.2) where A u is the operator defined in (2.2) in the subcase p = 2 and F φ is the usual potential energy introduced before. In this context, u ∈ K is said to be a (weak) solution of (1.3) if it satisfies the two properties:

(A) Distribution Identity hu

t

, φi

t

0

= Z

t

0

n hu

t

, φ

t

i − M (kDu(τ, ·)k

²₂

) · hDu, Dφi − ̺(τ )hDu

t

, Dφi − µhu, φi

+ hf (τ, ·, u), φi − hQ(τ, ·, u, u

t

) + u

tt

, φi

Γ1

o dτ for all t ∈ R

⁺₀

and φ ∈ K;

(B) Energy Conservation

D u(t) = hQ(t, ·, u(t, ·), u

t

(t, ·)), u

t

(t, ·)i

Γ1

+ ̺(τ )kDu

t

(t, ·)k

²₂

+ F

t

u(t) (i) D u ∈ L

¹_loc

(R

⁺₀

),

(ii) Eu(t) ≤ Eu(0) − Z

t

0

D u(τ )dτ for all t ∈ R

⁺0

.

To make the definition of solution meaningful, we require that hQ(τ, ·, u, u

t

), φi

Γ1

and hu

tt

, φi

Γ1

are in L

¹_loc

(R

⁺₀

) along any field φ ∈ K. This makes the Distribution Identity well posed, since all the other terms in (A) are well defined thanks to the main assumption (F

1

) and the choice of the function space K.

Of course, the presence of the internal damping ̺ and of the dynamical boundary

conditions in the system (1.3) make the analysis more delicate with respect to

the previous case (1.1). Consequently, also the energy function, as well as the

Distribution Identity and the Energy Conservation in the definition of the solution,

change significantly to take into account the new dissipative term ̺ acting on Γ

1

.

Technical difficulties arise if p 6= 2 and this justifies the restriction p = 2.