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arXiv:0804.0298v3 [math.AP] 6 Jan 2010

The Pointwise Estimates of Solutions for

Semilinear Dissipative Wave Equation

Yongqin Liu∗

DepartmentofMathematics,FudanUniversity,Shanghai,China

DepartmentofMathematics,KyushuUniversity,Fukuoka,Japan

Abstract

In this paper we focus on the global-in-time existence and the

pointwise estimates of solutions to the initial value problem for the

semilinear dissipative wave equation in multi-dimensions. By using

the method of Green function combined with the energy estimates,

we obtain the pointwise decay estimates of solutions to the problem.

keywords: semilinear dissipative wave equation, pointwise esti-

mates, Green function.

MSC(2000): 35E15; 35L15.

1Introduction

In this paper we consider the initial value problem for the semilinear dissi-

pative wave equation in n(n ≥ 1) dimensions,

(? + ∂t)u(x,t) = f(u), x ∈ Rn, t > 0,

with initial condition

(1.1)

1a

(u,∂tu)(x,0) = (u0,u1)(x), x ∈ Rn,

t− △x+ ∂tis the dissipative wave operator with Lapla-

xj, f(u) = −|u|θu, θ > 0 is an integer. Equation (1a

often called the semilinear dissipative wave equation or semilinear telegraph

equation.

There have been many results on the equation (1a

responding to the different forms of f(u). By employing the weighted L2

(1.2)

IC

where ? + ∂t= ∂2

cian △x =

n ?

j=1∂2

1a

1.1) is

1a

1.1) and its variants cor-

∗email: yqliu2@yahoo.com.cn

1

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energy method and the explicit formula of solutions, Ikehata, Nishihara and

Zhao

pected to be same as that for the corresponding heat equation, Nishihara

Ni0

[15] studied the global asymptotic behaviors in three and four dimensions,

and Nishihara and Zhao[19] obtained the decay properties of solutions to

the problem (1a

1.2). Kawashima, Nakao and Ono

property of solutions to (1a

1.1) by using the energy method combined with

Lp− Lqestimates, and Ono

case of solutions to (1a

1.1) in unbounded domains in RNwithout the smallness

condition on initial data. Also, recently Nishihara, etc. in

ied the following semilinear damped wave equations with time or space-time

dependent damping term,

INZ

[7] obtained that the behavior of solutions to (1a1a

1.1) as t → ∞ is ex-

NZ

1a

1.1)(IC IC KNO

[9] studied the decay

1a

Ono1

[20] derived sharp decay rates in the subcritical

1a

Ni1, Ni2

[16, 17] stud-

utt− ∆u + b(t)ut+ |u|ρ−1u = 0,(1.3)

n1

and

utt− ∆u + b(t,x)ut+ |u|ρ−1u = 0,(1.4)

n2

where ρ > 1, b(t) = b0(1 + t)−βwith b0 > 0,−1 < β < 1, and b(t,x) =

b0(1 + |x|2)−α

obtained the global existence and the L2decay rate of the solution by using

the weighted energy method. (n1

yield (1a

1.1). For studies on the case f(u) = |u|θu, see

studies on the case f(u) = |u|θ+1, see

on the global attractors, see

[1, 10] and the references cited there.

The main purpose of this paper is to study the pointwise estimates of so-

lutions for (1a

1.2). In [11], Liu and Wang studied the corresponding linear

problem, i.e. (1a

mates of solutions. In this paper, we first obtain the global-in-time solutions

by energy method combined with the fixed point theorem of Banach, and

then obtain the optimal pointwise decay estimates of the solutions by using

the properties of the Green function proved in

analysis. One point worthy to be mentioned is that, different from that for

solutions to the corresponding linear problem, the order of derivatives with

respect to time variable t of solutions does not contribute to the decay rate

of solutions due to the presence of the semilinear term, which could be seen

from (pe1

2.4) in Theorem5.1) in Theorem

The rest of the paper is arranged as follows. In section 2, the main results

are stated. We give the proof of Proposition

in-time existence of solutions in section 3. In section 4 we give estimates on

the Green function by Fourier analysis which will be used in the last section

where the proof of Theorem

2.4 is given.

2(1 + t)−βwith b0 > 0,α ≥ 0,β ≥ 0,α + β ∈ [0,1), and

n1

1.3) and (n2

1.4) with the exponents α = β = 0

n2

1a HO, IMN, IO, Na, Ni, NO

[3, 6, 8, 13, 14, 18], for

Ik, LZ, Ono2, Ono3, TY, Z

[5, 12, 21, 22, 23, 25], and for studies

BP, KS

1a

1.1)(IC

1a

1.1) with f(u) = 0 and (IC

IC LW

IC

1.2), and obtained the pointwise esti-

LW

[11] combined with Fourier

pe1pepe

2.4 and (lr1lr1lrlr

5.1.

aeae

2.3 and then obtain the global-

pepe

2

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Before closing this section, we give some notations to be used below. Let

F[f] denote the Fourier transform of f defined by

F[f](ξ) =ˆf(ξ) :=

?

Rne−ix·ξf(x)dx,

and we denote its inverse transform by F−1.

For 1 ≤ p ≤ ∞, Lp= Lp(Rn) is the usual Lebesgue space with the norm

? · ?Lp. Let s be a nonnegative integer. Then Hs= Hs(Rn) denotes the

Sobolev space of L2functions, equipped with the norm

?f?Hs :=

?

s

?

k=0

?∂k

xf?2

L2

?1

2.

In particular, we use ? · ? = ? · ?L2, ? · ?s= ? · ?Hs. Here, for a multi-index

α, Dα

to x ∈ Rn. Also, Ck(I;Hs(Rn)) denotes the space of k-times continuously

differentiable functions on the interval I with values in the Sobolev space

Hs= Hs(Rn).

Finally, in this paper, we denote every positive constant by the same

symbol C or c without confusion. [·] is Gauss’ symbol.

xdenotes the totality of all the |α|-th order derivatives with respect

2Main theorems

The first main result is about the global existence of solutions to the initial

value problem (1a

1.2).

1a

1.1)(IC IC

ge

Theorem 2.1 (Global existence). Let θ > 0 be an integer. Assume that

(u0,u1) ∈ Hs+1(Rn) × Hs(Rn), s ≥ [n

E0:= ?u0?Hs+1 + ?u1?Hs.

Then if E0is suitably small, (1a

1.2) admits a unique solution

2] , put

1a

1.1) (ICIC

u ∈

s+1

?

i=0

Ci([0,∞);Hs+1−i(Rn)),

which satisfies

s+1

?

i=0

?∂i

tu(t)?2

s+1−i+

?t

0

(?∇u(τ)?2

s+

s+1

?

i=1

?∂i

τu(τ)?2

s+1−i)dτ ≤ CE2

0.(2.1)

ge1

3

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Theorem

in the following Theorem

sition

2.3.

gege

2.1 is proved by combining the local existence of solutions stated

le le

2.2 with a priori estimate in the following Propo-

ae ae

le

Theorem 2.2 (Local existence).

(u0,u1) ∈ Hs+1(Rn)×Hs(Rn), s ≥ [n

solution to (1a

1.2) satisfying

Let θ > 0 be an integer. Assume that

2], then there exists T > 0 and a unique

1a

1.1) (IC IC

u ∈

s+1

?

i=0

Ci([0,T);Hs+1−i(Rn)).

The proof of the local existence result is based on the fixed point theorem

of Banach and standard argument, so the detail is omitted.

Based on the a priori assumption

sup

0<t<T?u(t)?L∞ ≤¯δ,(2.2)

aa

where s > n is an integer and¯δ < 1 is a small constant, the following a priori

estimate is obtained.

ae

Proposition 2.3 (A priori estimate). Under the same assumptions as in The-

orem

and verifies (aa

2.2), then the following estimate holds,

gege

2.1, let u(x,t) be the solution to (1a

aa

1a

1.1)(ICIC

1.2) which is defined on [0,T]

sup

0<t<T{

s+1

?

i=0

?∂i

tu(t)?2

s+1−i} +

?T

0

(?∇u(τ)?2

s+

s+1

?

i=1

?∂i

τu(τ)?2

s+1−i)dτ ≤ CE2

0.

(2.3)

ae1

Remark 1. In (1a

makes it possible to close energy estimates. Otherwise, if f(u) = |u|θu, then

Theorem

2.1 does not hold, since the lower-order term present in the energy

estimates could not be controlled.

The second main result is about the pointwise estimate to the solution

obtained in Theorem

2.1.

1a

1.1), f(u) = −|u|θu is called absorption term which

gege

gege

pe

Theorem 2.4 (Pointwise estimate). Under the same assumptions as in The-

orem 2.1, if s > n, θ ≥ 2 + [1

there exists some constant r > max{n

|Dα

then for h ≥ 0 satisfying |α| + h < s − n, the solution to (1a

in Theorem

2.1 satisfies the following pointwise estimate,

ge ge

n], and for any multi-indexes α, |α| < s −n

2,1} such that

xu0(x)| + |Dα

2,

xu1(x)| ≤ C(1 + |x|2)−r,

1a

1.1)(ICIC

1.2) obtained

gege

|∂h

tDα

xu(x,t)| ≤ CE0(1 + t)−n+|α|

2 (1 +|x|2

1 + t)−r.(2.4)

pe1

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Remark 2.

of derivatives with respect to t of the solution obtained in Theorem

no effect on the decay rate of the solution, as is different from that for the

solution to the corresponding linear problem studied in

As a direct corollary of Theorem 2.4 we have

From the estimate in Theorem

pepe

2.4, we see that the order

ge ge

2.1 has

LW

[11].

pepe

44

Corollary 2.5. Assume that the same assumptions as in Theorem

then for p ∈ [1,∞], |α| + h < s − n, the solution to (1a

?∂h

pe pe

2.4 hold,

1a

1.1)(ICIC

1.2) satisfies,

tDα

xu(·,t)?Lp ≤ CE0(1 + t)−n

2(1−1

p)−|α|

2.

3The global existence of solutions

First we give a lemma which will be used in our next energy estimates.

L

Lemma 3.1. Let n ≥ 1, 1 ≤ p,q,r ≤ ∞ and1

estimate holds:

p=1

q+1

r. Then the following

?∂k

x(uv)?Lp ≤ C(?u?Lq?∂k

xv?Lr + ?v?Lq?∂k

xu?Lr)(3.1)

d1

for k ≥ 0.

Proof. The estimate (d1

proof. To prove (d1

k1+ k2= k, the following estimate holds:

d1

3.1) can be found in a literature but we give here a

3.1), it is enough to show that, for k1 ≥ 1, k2 ≥ 1 and

d1

?∂k1

xu∂k2

xv?Lp ≤ C(?u?Lq?∂k

xv?Lr + ?v?Lq?∂k

xu?Lr).

Let θj=

kj

k, j = 1,2, and define pj, j = 1,2, by

1

pj

−kj

n= (1 − θj)1

q+ θj(1

r−k

n).

Since θ1+ θ2= 1, we have

the Gagliardo-Nirenberg inequality, we have

1

p=

1

p1+

1

p2. By using the H¨ older inequality and

?∂k1

xu∂k2

xv?Lp ≤ ?∂k1

xu?Lp1?∂k2

≤ C(?u?1−θ1

≤ C(?u?Lq?∂k

≤ C(?u?Lq?∂k

xv?Lp2

xu?θ1

xv?Lr)θ2(?v?Lq?∂k

xv?Lr + ?v?Lq?∂k

Lq ?∂k

Lr)(?v?1−θ2

Lq ?∂k

xv?θ2

xu?Lr)θ1

xu?Lr).

Lr)

In the last inequality, we have used the Young inequality. Thus (d1

proved.

d1

3.1) is

5