The Pointwise Estimates of Solutions for Semilinear Dissipative Wave Equation

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 estimates of the solution. Comment: 20 pages

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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.
1 Introduction
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 ICKNO
[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
1aHO, 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 (lr1 lr1lr lr
5.1.
aeae
2.3 and then obtain the global-
pe pe
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)(ICIC
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) (IC IC
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.
ge ge
2.1 is proved by combining the local existence of solutions stated
le le
2.2 with a priori estimate in the following Propo-
aeae
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,
ge ge
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-
orem2.1, 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,
gege
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
4

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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
pe pe
2.4, we see that the order
gege
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.
3 The 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