TO THE CAUCHY PROBLEM FOR A WAVE EQUATION WITH A WEAKLY NONLINEAR DISSIPATION

TO THE CAUCHY PROBLEM FOR A WAVE EQUATION WITH A WEAKLY NONLINEAR DISSIPATION

ABB `ES BENAISSA AND SOUFIANE MOKEDDEM Received 21 November 2003

We prove the global existence and study decay properties of the solutions to the wave equation with a weak nonlinear dissipative term by constructing a stable set inH¹(Rⁿ).

1. Introduction

We consider the Cauchy problem for the nonlinear wave equation with a weak nonlinear dissipation and source terms of the type

u−∆xu+λ²(x)u+σ(t)g(u)= |u|^p−1u inRⁿ×[0, +∞[,

u(x, 0)=u0(x), u(x, 0)=u1(x) inRⁿ, (1.1) whereg:R→Ris a continuous nondecreasing function andλandσare positive func- tions.

When we have a bounded domain instead ofRⁿ, and for the caseg(x)₌δx(δ >0) (without the termλ²(x)u), Ikehata and Suzuki [8] investigated the dynamics, they have shown that for sufficiently small initial data (u0,u1), the trajectory (u(t),u(t)) tends to (0, 0) inH0¹(Ω)×L²(Ω) ast→+∞. Wheng(x)=δ|x|^m−1x(m≥1,λ≡0,σ≡1), Georgiev and Todorova [4] introduced a new method and determined suitable relations between mandp, for which there is global existence or alternatively finite-time blow up. Precisely they showed that the solutions continue to exist globally in time ifm≥p and blow up in finite time ifm < pand the initial energy is sufficiently negative. This result was later generalized to an abstract setting by Levine and Serrin [12] and Levine et al. [11]. In these papers, the authors showed that no solution with negative initial energy can be extended on [0,∞[, if the source term dominates over the damping term (p > m). This generaliza- tion allowed them also to apply their result to quasilinear wave equations (see [1,17]).

Quite recently, Ikehata [7] proved that a global solution exists with no relation between pandmby the use of a stable set method due to Sattinger [18].

For the Cauchy problem (1.1) withλ≡1 andσ≡1, wheng(x)=δ|x|^m−1x(m≥1) Todorova [21] (see [16]) proved that the energy decay rate isE(t)≤(1 +t)^{−(2−n(m−1))/(m−1)}

Copyright©2004 Hindawi Publishing Corporation Abstract and Applied Analysis 2004:11 (2004) 935–955 2000 Mathematics Subject Classification: 35B40, 35L70 URL:http://dx.doi.org/10.1155/S1085337504401031

fort≥0. She used a general method introduced by Nakao [14] on condition that the data have compact support. Unfortunately, this method does not seem to be applicable in the case of more general functionsλandσ.

Our purpose in this paper is to give a global solvability in the classH¹and energy decay estimates of the solutions to the Cauchy problem (1.1) for a weak linear perturbation and a weak nonlinear dissipation.

We use a new method recently introduced by Martinez [13] (see also [2]) to study the decay rate of solutions to the wave equationu−∆xu+g(u)=0 inΩ×R⁺, whereΩis a bounded domain ofRⁿ. This method is based on a new integral inequality that generalizes a result of Haraux [6]. So we proceed with the argument combining the method in [13]

with the concept of modified stable set onH¹(Rⁿ). Here the modified stable set is the extendedRⁿversion of Sattinger’s stable set.

2. Preliminaries and main results

λ(x),σ(t), andgsatisfy the following hypotheses.

(i)λ(x) is a locally bounded measurable function defined onRⁿand satisfies λ(x)≥d|x|

, (2.1)

wheredis a decreasing function such that limy→∞d(y)=0.

(ii)σ:R+→R+is a nonincreasing function of classC¹onR+.

Considerg:R→Ra nondecreasingC⁰ function and suppose that there existC_i>0, i=1, 2, 3, 4, such that

c₃|v|^m≤g(v)≤c₄|v|^1/m if|v| ≤1, (2.2) c1|v| ≤g(v)≤c2|v|^r ∀|v| ≥1, (2.3) wherem≥1 and 1≤r≤(n+ 2)/(n−2)⁺.

We first state two well-known lemmas, and then we state and prove two other lemmas that will be needed later.

Lemma2.1. Letqbe a number with2≤q <+∞(n=1, 2)or2≤q≤2n/(n−2)(n≥3).

Then there is a constantc_∗=c(q)such that

u q≤c_∗ u H¹(Rⁿ) foru∈H¹Rⁿ. (2.4) Lemma2.2 (Gagliardo-Nirenberg). Let1≤r < q≤+∞andp≥2. Then, the inequality

u p≤C∇^mxu^θ₂ u ¹_r^−θ foru∈Ᏸ(−∆)^m/2)L^r (2.5) holds with some constantC >0and

θ= 1

r⁻ 1 p

m n +1

r ⁻ 1 2

−1

(2.6) provided that0< θ≤1(assuming that0< θ <1ifm−n/2is a nonnegative integer).

Lemma2.3 [10]. LetE:R+→R+be a nonincreasing function and assume that there are two constantsp≥1andA >0such that

+∞

S E^(p+1)/2(t)dt≤AE(S), 0≤S <+∞, (2.7)

then

E(t)≤cE(0)(1 +t)^−2/(p−1) ∀t≥0,ifp >1,

E(t)≤cE(0)e^−ωt ∀t≥0,ifp=1, (2.8) wherecandωare positive constants independent of the initial energyE(0).

Lemma2.4 [13]. LetE:R+→R+be a nonincreasing function andφ:R+→R+an increas- ingC²function such that

φ(0)=0, φ(t)−→+∞ ast−→+∞. (2.9)

Assume that there existp≥1andA >0such that +_∞

S E(t)^(p+1)/2(t)φ(t)dt≤AE(S), 0≤S <+∞, (2.10) then

E(t)≤cE(0)1 +φ(t)^−2/(p−1) ∀t≥0,ifp >1,

E(t)≤cE(0)e^−ωφ(t) ∀t≥0, ifp=1, (2.11) wherecandωare positive constants independent of the initial energyE(0).

Proof ofLemma 2.4. Let f :R+→R+be defined by f(x) :=E(φ⁻¹(x)), (we remark that φ⁻¹has a sense by the hypotheses assumed onφ). f is nonincreasing,f(0)=E(0), and if we setx:=φ(t), we obtain

_φ(T)

φ(S) f(x)^(p+1)/2dx= _φ(T)

φ(S) Eφ⁻¹(x)^(p+1)/2dx= _T

S E(t)^(p+1)/2φ(t)dt

≤AE(S)=A fφ(S), 0≤S < T <+∞.

(2.12)

Settings:=φ(S) and lettingT→+∞, we deduce that +∞

s f(x)^(p+1)/2dx≤A f(s), 0≤s <+∞. (2.13)

Thanks toLemma 2.3, we deduce the desired results.

Before stating the global existence theorem and decay property of problem (1.1), we will introduce the notion of the modified stable set. Let

K(u)=∇xu²₂+ u ²₂− u ^p+1_p+1 ifλ≡1,

I(u)=∇xu²₂− u ^p+1_p+1 ifλ≡const, (2.14)

foru∈H¹(Rⁿ). Then we define the modified stable setᐃ^∗andᐃ^∗∗by ᐃ^∗≡ u∈H¹Rⁿ\K(u)>0∪ {0} ifλ≡1,

ᐃ^∗∗≡ u∈H¹Rⁿ\I(u)>0∪ {0} ifλ≡const. (2.15) Next, letJ(u) andE(t) be the potential and energy associated with problem (1.1), respec- tively:

J(u)=1

2^∇xu²₂+1

2λ(x)u²₂− 1

p+ 1 u ^p+1_p+1 foru∈H¹Rⁿ, E(t)=1

2 u ²₂+J(u).

(2.16)

We get the local existence solution.

Theorem2.5. Let1< p≤(n+ 2)/(n−2) (1< p <∞ifn=1, 2)and assume that(u0,u1)

∈H¹(Rⁿ)×L²(Rⁿ)andu0belong to the modified stable setᐃ^∗. Then there existsT >0 such that the Cauchy problem (1.1) has a unique solutionu(t)onRⁿ×[0,T)in the class

u(t,x)∈C[0,T);H¹Rⁿ∩C¹[0,T);L²Rⁿ, (2.17) satisfying

u(t)∈ᐃ^∗, (2.18)

and this solution can be continued in time as long asu(t)∈ᐃ^∗.

Whenλ≡const, we use the following theorem of local existence in the spaceH²×H¹, and the decay property of the energyE(t) is necessarily required for the local solution to remain inᐃ^∗∗ast→ ∞; this fact of course guarantees the global existence inH²×H¹ and by approximation, we obtain global existence inH¹×L¹.

Theorem2.6 [15]. Let(u0,u1)∈H²×H¹. Suppose that 1≤p≤ n

n−4 (1≤ ∞ifN≤4). (2.19)

Then under the hypotheses (2.1), (2.2), and (2.3), problem (1.1) admits a unique local solu- tionu(t)on some interval[0,T[,T≡T(u0,u1)>0, in the classW^2,∞([0,T[;L²)∩W^1,∞([0, T[;H¹)∩L^∞([0,T[;H²), satisfying the finite propagation property.

Proof ofTheorem 2.5(see[15,18]). Since the argument is standard, we only sketch the main idea of the proof. Let (u0,u1)∈H¹×L²andu0∈ᐃ^∗. Then we have a unique local solutionu(t) for someT >0. Indeed, taking suitable approximate functions fjsuch that (see [20])

fj(u)= f(u) if|u| ≤j, fj(u)≤f(u), fj(u)≤cj|u|, (2.20) problem (1.1) with f(u)≡ |u|^p−1ureplaced by f_j(u) admits a unique solutionu_j(t)∈ C([0,T);H¹(Rⁿ))∩C¹([0,T);L²(Rⁿ)). Further, we can prove thatu_j(t)∈ᐃ^∗, 0< t < T,

for sufficiently large j, and there exists a subsequence of{u_j(t)}which converges to a function ˜u(t) in certain senses. ˜u(t) is, in fact, a weak solution in C([0,T);H¹(Rⁿ))∩ C¹([0,T);L²(Rⁿ)) (see [19,20]) and such a solution is unique by Ginibre and Velo [5]

and Brenner [3]. We can also construct such a solution which meets moreover the finite propagation property, if we assume that the initial datau0(x) andu1(x) are of compact support:

suppu0∪suppu1⊂ x∈Rⁿ,|x|< L, for someL >0. (2.21) Applying [9, Appendix 1] of John, then the solution is also of compact support: suppuj(t)

⊂ {x∈Rⁿ,|x|< L+t}. So, we have supp ˜u(t)⊂ {x∈Rⁿ,|x|< L+t}. We denote the life span of the solutionu(t,x) of the Cauchy problem (1.1) byTmax. First we consider the caseλ(x)≡const (λ(x)≡1 without loss of generality). And con- struct a stable set inH¹(Rⁿ).

Setting

C0≡K

2(p+ 1) (p−1)

(p−1)/2

, (2.22)

_∞

0 σ(τ)dτ=+∞ ifm=1, (2.23)

_∞

0 (1 +τ)^{−n(m−1)/2}σ(τ)dτ=+∞ ifm >1. (2.24) Theorem2.7. Letu(t,x)be a local solution of problem (1.1) on[0,Tmax)with initial data u0∈ᐃ^∗,u1∈L²(Rⁿ)with sufficiently small initial energy E(0) so that

C0E(0)^(p−1)/2<1. (2.25)

ThenTmax= ∞. Furthermore, the global solution of the Cauchy problem (1.1) has the fol- lowing energy decay property. Under (2.22), (2.3), and (2.23),

E(t)≤E(0) exp

1−ω _t

0σ(τ)dτ

∀t >0. (2.26) Under (2.2), (2.3), and (2.24),

E(t)≤

CE(0) _t

0(1 +τ)^{−n(m−1)/2}σ(τ)dτ

2/(m−1)

∀t >0. (2.27) Secondly, we consider the caseλ(x)≡const and we assume that

n+ 4

n ^≤p≤ n

n−2. (2.28)

(1) Ifσ(t)=ᏻ( ˜d(t)), where ˜d(t)=d(L+t).

Ifm=1, we suppose that

_∞

0 σ(τ)dτ=+∞ (2.29)

with

d(t)˜ ^{−(4−(n−2)(p−1))/2}exp

1−ω _t

0σ(τ)dτ (p−1)/2

<∞, d(t)˜ ⁻¹exp

1 2⁻

ω 2

_t

0σ(τ)dτ

<∞.

(2.30)

Ifm >1, we suppose that _∞

0 (1 +τ)^{−n(m−1)/2}σ(τ)dτ= ∞ (2.31) with

d(t)˜ ^{−(4−(n−2)(p−1))/2} _t

0(1 +τ)^{−n(m−1)/2}σ(τ)dτ^{(p−1)/(m−1)}

<∞, d(t)˜ ⁻¹

_t

0(1 +τ)^{−n(m−1)/2}σ(τ)dτ^1/(m−1)

<∞.

(2.32)

(2) If ˜d(t)=ᏻ(σ(t)).

Ifm=1, we suppose that for some 0≤α <1, _∞

0

d˜²(τ)

σ^α(τ)dτ=+∞ (2.33)

with

d(t)˜ ^{−(4−(n−2)(p−1))/2}exp

1−ω _t

0

d˜²(τ) σ^α(τ)dτ

(p−1)/2

<∞, d(t)˜ ⁻¹exp

1 2⁻

ω 2

_t

0

d˜²(τ) σ^α(τ)dτ

<∞.

(2.34)

Ifm >1, we suppose that for some 0≤α <1, _∞

0 (1 +τ)^{−n(m−1)/2}σ⁻((1+α)(1+m)−2)/2(τ) ˜d^m+1(τ)dτ= ∞ (2.35)

with

d(t)˜ ^{−(4−(n−2)(p−1))/2} _t

0(1 +τ)^{−n(m−1)/2}σ⁻((1+α)(1+m)−2)/2(τ) ˜d^m+1(τ)dτ^{(p−1)/(m−1)}

<∞, d(t)˜ ⁻¹

_t

0(1 +τ)^{−n(m−1)/2}σ⁻((1+α)(1+m)−2)/2(τ) ˜d^m+1(τ)dτ^1/(m−1)

<_∞.

(2.36)

We have the following theorem.

Theorem2.8. Let(u0,u1)∈H¹×L²,u0∈ᐃ^∗∗, and let the initial energyE(0)be suffi- ciently small. The following cases are considered.

(i)σ(t)=ᏻ( ˜d(t)).

Suppose (2.2), (2.3), (2.29), and (2.30) or (2.2), (2.3), (2.31), and (2.32). Then problem (1.1) admits a unique solutionu(t)∈C([0,∞);H¹)∩C¹([0,∞);L²)and has the same decay property asTheorem 2.7.

(ii) ˜d(t)=ᏻ(σ(t)).

Suppose (2.2), (2.3), (2.33), and (2.34) or (2.2), (2.3), (2.35), and (2.36). Then prob- lem (1.1) admits a unique solutionu(t)∈C([0,∞);H¹)∩C¹([0,∞);L²). Furthermore, the global solution of the Cauchy problem (1.1) has the following energy decay property:

E(t)≤E(0) exp

1−ω _t

0

d˜²(τ) σ^α(τ)dτ

∀t >0ifm=1, (2.37)

E(t)≤

CE(0) _t

0(1 +τ)^{−n(m−1)/2}σ⁻((1+α)(1+m)−2)/2(τ) ˜d^m+1(τ)dτ

2/(m−1)

∀t >0ifm >1.

(2.38) Remark 2.9. InTheorem 2.7, the global existence and energy decay are independent, but inTheorem 2.8, we need the estimation of the energy decay for a local solution to prove global existence.

Examples 2.10. (1) Ifσ(t)=1/t^θ, by applyingTheorem 2.7we obtain E(t)≤E(0)e^{1−ωt1−θ} ifm=1,

E(t)≤CE(0)(1 +t)^{−(2−n(m−1)−2θ)/(m−1)} if 1< m <1 +2−2θ

n , 0< θ <1, E(t)≤CE(0)(lnt)^−2/(m−1) ifm=1 +2−2θ

n , 0< θ <1.

(2.39)

(2) Ifσ(t)=1/t^θlntln2t···lnpt, by applyingTheorem 2.7, we obtain

E(t)≤E(0)lnpt^−ω ifm=1,θ=1. (2.40) For example, ifn(m₋1)/2 +θ₌1, that is, 1< m <1 + 2/n,

E(t)≤CE(0)lnpt^−2/(m−1). (2.41)

(3) Ifσ(t)=1/t^θandd(r)=1/r^γwithθ≥γby applyingTheorem 2.8, we obtain E(t)≤CE(0)(1 +t)^{−(2−n(m−1)−2θ)/(m−1)} if 1< m <1 +2−2θ

2γ+n, 0< θ <1, E(t)≤CE(0)(lnt)^−2/(m−1) ifm=1 +2−2θ

2γ+n, 0< θ <1.

(2.42)

In order to show the global existence, it suffices to obtain the a priori estimates forE(t) and u(t) 2in the interval of existence.

To proveTheorem 2.7we first have the following energy identity to problem (1.1).

Lemma2.11 (energy identity). Letu(t,x)be a local solution to problem (1.1) on[0,Tmax) as inTheorem 2.5. Then

E(t) +

Rⁿ

_t

0σ(s)u(s)gu(s)ds dx=E(0) (2.43) for allt∈[0,Tmax).

Next we state several facts about the modified stable setᐃ^∗. Lemma2.12. Suppose that

1< p≤ n+ 2

n−2. (2.44)

Then

(i)ᐃ^∗is a neighborhood of0inH¹(Rⁿ), (ii)foru∈ᐃ^∗,

J(u)≥ p−1

2(p+ 1)^∇xu²₂+ u ²₂. (2.45) Proof ofLemma 2.12. (i) FromLemma 2.1we have

u ^p+1_p+1≤K u ^p+1_H1 ≤K u _H^p−1¹

u ²₂+∇xu²₂. (2.46) Let

U(0)≡

u∈H¹R^N u _H^p−1¹< 1 K

. (2.47)

Then, for anyu∈U(0)\{0}, we deduce from (2.46) that

u ^p+1_p+1< u ²2+∇xu²₂, (2.48) that is,K(u)>0. This impliesU(0)⊂ᐃ^∗.

(ii) By the definition ofK(u) andJ(u), we have the identity (p+ 1)J(u)=K(u) +(p−1)

2 ^∇xu²₂+ u ²₂. (2.49)

Sinceu∈ᐃ^∗, we haveK(u)≥0. Therefore from (2.44) we get the desired in-equality

(2.45).

Lemma2.13. Letu(t)be a solution to problem (1.1) on[0,Tmax). Suppose (2.44) holds. If u0∈ᐃ^∗andu1∈L²(Rⁿ)satisfy

C0E(0)^(p−1)/2<1, (2.50)

then

(i)u(t)∈ᐃ^∗on[0,Tmax), (ii) u(t) 2≤I0on[0,Tmax).

Proof ofLemma 2.13. Suppose that there exists a numbert^∗∈[0,Tmax[ such thatu(t)∈ ᐃ^∗on [0,t^∗[ andu(t^∗)∈ᐃ^∗. Then we have

Kut^∗=0, ut^∗=0. (2.51)

Sinceu(t)∈ᐃ^∗on [0,t^∗[, it holds that p−1

2(p+ 1)^∇xu²₂+ u ²₂≤J(u)≤E(t); (2.52) it follows from the nonincreasing of the energy that

∇xu²₂+ u ²₂≤2(p+ 1)

p−1 E(0)≡I₀². (2.53)

Hence, we obtain

u ²₂≤2(p+ 1)

p−1 E(0)≡I₀² on [0,t^∗]. (2.54) Next, fromLemma 2.1and (2.54) we have

u ^p+1_p+1≤Ku(t)^p+1_H1(Rⁿ)

≤Ku(t)^p_H⁻1¹(Rⁿ)∇xu²₂+ u ²₂

≤KI₀^p−1∇xu²₂+ u ²₂

≤C0E(0)^(p−1)/2u(t)²₂+∇xu(t)²₂

(2.55)

for allt∈[0,t^∗], whereC0is the constant defined by (2.22). Note that from (2.55) and our hypothesis

η0≡C0E(0)^(p−1)/2<1, (2.56) it follows that

u(t)^p+1_p+1≤

1−η0u(t)²₂+∇xu(t)²₂. (2.57)

Therefore, we obtain

Kut^∗≥η0ut^∗2₂+∇xut^∗2₂ (2.58) which contradicts (2.51). Thus, we conclude thatu(t)∈ᐃ^∗on [0,Tmax[. The assertion (ii) can be obtained by the same argument as for (2.54). This completes the proof of

Lemma 2.13.

Lemma2.14. Under the same assumptions as inLemma 2.13, there exists a constant M2

depending on u0 H¹and u1 2such that

u(t)²_H1+u(t)²₂≤M2² (2.59) for allt∈[0,Tmax[.

Proof ofLemma 2.14. It follows fromLemma 2.13thatu(t)∈ᐃ^∗on [0,Tmax[. SoLemma 2.12(ii) implies that

J(u)≥ p−1

2(p+ 1)u(t)²₂+∇xu(t)²₂ on [0,Tmax[. (2.60) Hence, fromLemma 2.11and (2.60) we get

1

2u(t)²₂+ p−1 2(p+ 1)

u ²₂+∇xu(t)²₂≤E(t)≤E(0). (2.61)

So we get

u(t)²_H1+u(t)²₂≤M₂², (2.62) for someM2>0.

The above inequality and the continuation principle lead to the global existence of the

solution, that is,Tmax= ∞.

Proof of the energy decay. From now on, we denote bycvarious positive constants which may be different at different occurrences. We multiply the first equation of (1.1) byE^qφu, whereφis a function satisfying all the hypotheses ofLemma 2.4. We obtain

0= _T

S E^qφ

Rⁿuu−∆u+u+σ(t)g(u)− |u|^p−1udx dt

=

E^qφ

Rⁿuudx T

S − _T

S

qEE^q−1φ+E^qφ

Rⁿuudx dt−2 _T

S E^qφ

Rⁿu²dx dt +

_T

S E^qφ

Rⁿ

u²+|u|²+|∇u|²− 2 p+ 1^|u|^p+1

dx dt+

_T

S E^qφ

Rⁿσ(t)ug(u)dx dt _T

S E^qφ

Rⁿ

2 p+ 1⁻1

|u|^p+1dx dt.

(2.63)

Since

1− 2

p+ 1

Rⁿ|u|^p+1dx≤ 1−η0

p−1

p+ 1u(t)²_H1(Rⁿ)dx

≤ 1−η0

p−1 p+ 1

2(p+ 1) p−1 E(t)

=21−η0

E(t),

(2.64)

we deduce that

2η0

_T

S E^q+1φdt≤ −

E^qφ

Rⁿuudx T

S

+ _T

S

qEE^q−1φ+E^qφ

Rⁿuudx dt + 2

_T

S E^qφ

Rⁿu²dx dt− _T

S E^qφ

Rⁿσ(t)ug(u)dx dt

≤ −

E^qφ

Rⁿuudx T

S+ _T

S

qEE^q−1φ+E^qφ

Rⁿuudx dt + 2

_T

S E^qφ

Rⁿu²dx dt+c(ε) _T

S E^qφ

|u|≤1g(u)²dx dt +ε

_T

S E^qφ

Rⁿu²dx dt+ _T

S E^qφ

|u|≥1σ(t)ug(u)dx dt

(2.65)

for everyε >0. Also, applying H¨older’s and Young’s inequalities, we have _T

S E^qφ

|u|>1σ(t)ug(u)dx dt

≤ _T

S E^qφσ(t)

Ω|u|^r+1dx

1/(r+1)

|u|>1

g(u)^(r+1)/rdx r/(r+1)

dt

≤c _T

S E^(2q+1)/2φσ^1/(r+1)(t)

|u|>1σ(t)ug(u)dx r/(r+1)

dt

≤ _T

S φσ^1/(r+1)(t)E^(2q+1)/2(−E)^r/(r+1)dt

≤c _T

S φσ^1/(r+1)(t)E^{(2q+1)/2−r/(r+1)}(−E)^r/(r+1)E^r/(r+1)dt

≤c(ε) _T

S φ(−EE)dt+ε _T

S φσ(t)E(r+1)((2q+1)/2−r/(r+1))dt

≤c(ε)E(S)²+εσ(0)E(0)^{(2rq−r−1)/2} _T

S φE^q+1dt

(2.66)

for everyε>0. Choosingεandεsmall enough, we obtain _T

S E^q+1φdt≤ −

E^qφ

Rⁿuudx T

S+ _T

S

qEE^q−1φ+E^qφ

Rⁿuudx dt +

|u|≥1σ(t)ug(u)dx dt+c _T

S Eφ

Rⁿu²dx dt

≤cE(S) +c _T

S Eφ

Rⁿu²dx dt.

(2.67)

Sincexg(x)≥0 for allx∈R, it follows that the energy is nonincreasing, locally absolutely continuous andE(t)= −

Rⁿσ(t)ug(u)dxa.c. inR+.

Proof of (2.26). We consider the casem=1, that is,

c3|v| ≤g(v)≤c4|v| for all|v| ≤1. (2.68) Then we have

u²≤ c13

σ(t)uρ(t,u) ∀t∈R,∀x∈Rⁿ, (2.69) whereρ(t,s)=σ(t)g(s) for alls∈R. Therefore we deduce from (2.67) (applied withq=0) that

_T

S E(t)φ(t)dt≤CE(S) + 2C _T

S φ(t)

Rⁿ

1

σ(t)uρ(t,u)dx dt. (2.70) Define

φ(t)= _t

0σ(τ)dτ. (2.71)

It is clear thatφis a nondecreasing function of classC² onR+. The hypothesis (2.23) ensures that

φ(t)−→+∞ ast−→+∞. (2.72)

Then we deduce from (2.70) that _T

S E(t)φ(t)dt≤CE(S) + 2C _T

S

Rⁿuρ(t,u)dx dt≤3CE(S), (2.73) and thanks toLemma 2.4we obtain

E(t)≤E(0)e^{(1−φ(t))/(3C)}. (2.74) Proof of (2.27). Now we assume thatm >1 in (2.2). Defineφby (2.71). We applyLemma 2.4withq=(m−1)/2.