Kubo, Hideo

EPrint Series of Department of Mathematics, Hokkaido University

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO
SEMILINEAR SYSTEMS OF WAVE EQUATIONS
HIDEO KUBO
DEPARTMENT OF MATHEMATICS, GRADUATE SCHOOL OF SCIENCE,
OSAKA UNIVERSITY, OSAKA TOYONAKA 560-0043, JAPAN
1. Introduction
In this note we consider asymptotic behavior of solutions to the Cauchy problem
for semilinear systems of wave equations :
∂2t ui − c2i∆ui = Fi(∂u) in R3 × (0,∞), (1.1)
where i = 1, · · · , N , ci > 0, ∆ =
∑3
j=1 ∂
2
j , ∂ = (∂0, ∂1, ∂2, ∂3), ∂j = ∂/∂xj ,
∂0 = ∂t = ∂/∂t and u(x, t) = (u1(x, t), · · · , uN(x, t)) is a real-valued unknown
function. Besides, Fi ∈ C1(R4N ) is a given function satisfying
Fi(0) = ∇Fi(0) = 0.
Our purpose here is to show that there are examples of nonlinearities F such that
the corresponding equation (1.1) cannot be regarded as a perturbation from the
system of homogeneous wave equations, even if we restrict our attention to small
amplitude solutions. The results presented in the section 2 was obtained by a joint
work with Professors Koˆji Kubota and Hideaki Sunagawa, and the results in the
section 3 was done by a joint work with Professor Soichiro Katayama.
We wish to explain the precise meaning of our purpose. Suppose that the Cauchy
problem for (1.1) admits a unique global solution u. We say the equation (1.1) can
be regarded as a perturbation from the system of homogeneous wave equations :
∂2t vi − c2i∆vi = 0 in R3 × (0,∞), (1.2)
if the global solution u tend to the solution v = (v1, · · · , vN ) of (1.2) as t → ∞.
This kind of asymptotic behavior is well studied in connection with the so-called
nonlinear scattering theory in the energy space. (see, e.g. [19, Chapter 6] and the
references cited therein). Nevertheless, there is another possibility that the eﬀect of
the nonlinearity remains so strong in suﬃciently large time that the global solution
u cannot approach to any free solutions. To our knowledge, there are only few
results which suggest that such a phenomenon occurs for nonlinear wave equations
(see e.g. Alinhac [3, 4], Lindblad–Rodnainski [16, 17]). Therefore our main goal
of this note is to show that there exist small amplitude solutions to the Cauchy
1
2 HIDEO KUBO
problem for (1.1) with a certain F whose large time behavior might be diﬀerent
from that of any free solutions.
We conclude this section by recalling a suﬃcient condition to ensure the small
data global existence for (1.1) when nonlinearity Fi is suﬃciently smooth. For the
case c1 = · · · = cN , such a condition was introduced by Klainerman [11]. We say
F (∂u) satisﬁes the null condition, if and only if the quadratic part of it can be
written as a linear combination of the following null forms
Q0(uj , uk; ci) =(∂tuj)(∂tuk)− c2i (∇uj) · (∇uk), (1.3)
Qab(uj , uk) =(∂auj)(∂buk)− (∂buj)(∂auk) (0 ≤ a < b ≤ 3). (1.4)
We remark that Christodoulou [5] also established the same result, independently.
Moreover, the global solution u to the Cauchy problem for (1.1) satisfying the null
condition approaches to some free solution (see Kubo–Ohta [14, Section 6]). On
the contrary, the null condition is necessary to ensure small data global existence if
we consider the scalar case, i.e., N = 1. In fact, the blow-up result was obtained by
Alinhac [2].
The null condition is extended to the multiple speeds case (i.e., the speeds
c1, . . . , cN do not necessarily coincide with each other) so that the small data global
existence for (1.1) holds (see Kovalyov [12], Agemi–Yokoyama [1], Yokoyama [21],
Sideris – Tu [18], Kubota – Yokoyama [15], Katayama [7], [8], [9], Katayama –
Yokoyama [10] and so on). For example, in addition to null forms, terms like
(∂auj)(∂buk) with cj = ck are allowed to be included for the multiple speeds case.
The precise conditions for the multiple speeds case are somewhat complicated, and
we do not go into details here. Instead of this, we shall discuss an extension of the
null condition for the case of the common propagation speeds with N ≥ 2.
2. Example, I
This section is concerned with the Cauchy problem for semilinear systems of
wave equations :{
∂2t u1 − c21∆u1 = |∂tu2|p in R3 × (0,∞),
∂2t u2 − c22∆u2 = |∂tu1|q in R3 × (0,∞),
(2.1)
where c1, c2 > 0, 1 < p ≤ q. First we recall known results concerning the small
data global existence and blowup for the Cauchy problem for (2.1). Yokoyama [21]
proved that when c1 = c2, the problem admits a unique global smooth solution
when p = q = 2 and the initial data are in C∞0 (R
3) and suﬃciently small. On the
other hand, Deng showed in Theorem 3.3 of [6] that if c1 = c2 and q(p−1) ≤ 2, then,
in general, a classical solution to the problem blows up in ﬁnite time however small
the initial data are. It is remarkable that the above condition is valid for p = q = 2.
Recently, Xu [20] proved the blowup result when c1 = c2 and 6(pq−1)/(p+q+2) ≤ 1.
ASYMPTOTIC BEHAVIOR FOR SEMILINEAR SYSTEMS OF WAVE EQUATIONS 3
Thus we see from these results that the feature of the problem (2.1) depend not
only on the exponents p, q but also on the propagation speeds c1, c2.
In order to extend the existence result due to [21] for general p, q > 1, we
consider only radially symmetric solution to the Cauchy problem for (2.1). To be
more speciﬁc, we seek solutions to the problem in X × X , where X is deﬁned by
X = {w(x, t) ∈ C(R3 × [0,∞)) ; there is u(r, t) ∈ X2 such that (2.2)
w(x, t) = u(|x|, t) for (x, t) ∈ R3 × [0,∞) and lim
|x|→∞
w(x,0) = 0}
with
X2 = {u(r, t) ∈ C1(R× [0,∞)) ; ru(r, t) ∈ C2(R× [0,∞)), (2.3)
u(−r, t) = u(r, t) for (r, t) ∈ R× [0,∞)}.
Note that X ⊂ C1(R3 × [0,∞)) ∩ C2((R3 \ {0}) × [0,∞)), because ∂ru(r, t) = 0
for r = 0 if u ∈ X2. Therefore the solution which we shall obtain is an “almost”
classical solution.
While, we consider the following type of initial condition :
uj(x,0) = fj(|x|), (∂tuj)(x,0) = gj(|x|) for x ∈ R3 (j = 1, 2), (2.4)
and introduce a class of the initial data Y as follows :
Y = {(f, g) ∈ C1(R)× C(R) ; rf(r) ∈ C2(R), rg(r) ∈ C1(R), (2.5)
f(−r) = f(r), g(−r) = g(r) for r ∈ R}.
This space is consistent with X2 in the sense that the solution v to the Cauchy
problem for the homogeneous wave equation
∂2t v − c2∆v = 0 in R3 × (0,∞) (2.6)
belongs to X2, if the initial data (f, g) ∈ Y satisfy such a decay condition as
Mκ(f, g) := sup
r>0
(1 + r)κ|||(f(r), g(r))||| <∞,
where κ > 0 and
|||(f(r), g(r))||| = |f(r)| + (1 + r)(|f ′(r)| + |g(r)|) + r(|f ′′(r)| + |g′(r)|).
Moreover we have the following estimate :
[v(r, t)](1 + |r − ct|)κ ≤ CMκ(f, g)
for (r, t) ∈ R× [0,∞), where we put
[v(r, t)] = |v(r, t)| + (1 + r)
∑
|α|=1
|∂αr,tv(r, t)|+ r
∑
|α|=2
|∂αr,tv(r, t)|.
4 HIDEO KUBO
In the application we choose κ as κ1 or κ2 which are deﬁned as follows:
κ1 = p− 1, (2.7)
κ2 = min(q − 1, q(p− 1)) if c1 = c2 or p > 2, (2.8)
κ2 = q(p− 1)− 1 if c1 = c2 and 1 < p < 2. (2.9)
In addition, if c1 = c2, p = 2 and q(p − 1) > 2 holds, then we take such κ2 as
1 < κ2 < q − 1. Then we have the following existence result.
Theorem 2.1. Let 1 < p ≤ q and suppose that
q(p− 1) > 2, (2.10)
if c1 = c2, and that
q(p− 1) > 1, (2.11)
if c1 = c2. Assume (fj , gj) ∈ Y and Mκj (fj , gj) ≤ ε for j = 1, 2 and ε > 0.
Then there are positive constants ε0 and C0 (depending only on c1, c2, p and q)
such that for any ε with 0 < ε ≤ ε0, there exists uniquely a solution (u1, u2) ∈ X×X
of the Cauchy problem (2.1) and (2.4) satisfying
[u˜1(r, t)](1 + |r − c1t|)κ1 + [u˜2(r, t)](1 + |r − c2t|)κ2 ≤ 2C0ε, (2.12)
if c1 = c2 or p > 2, and
[u˜1(r, t)]{(1 + r + t)[1−κ1]+}−1(1 + |r − c1t|) (2.13)
+[u˜2(r, t)](1 + |r − c2t|)κ2 ≤ 2C0ε,
if c1 = c2 and 1 < p ≤ 2. Here we denoted u1(x, t) = u˜1(|x|, t), u2(x, t) = u˜2(|x|, t),
and A[a]+ = Aa if a > 0 ; A[0]+ = 1 + logA for A ≥ 1.
This result shows that the condition given by [6] is sharp if c1 = c2 and that it
can be relaxed if c1 = c2. But it is still an open question what will happen when
c1 = c2, q(p− 1) ≤ 1 and the condition given by [20] does not fulﬁlled.
From now on we denote by (u1, u2) the global solution of the Cauchy problem
(2.1) and (2.4) obtained in Theorem 2.1 and assume that 0 < ε ≤ ε0. Our next
step is to examine the large time behavior of (u1, u2). We deﬁne θ1, θ2 by
θj = κj − 1 if c1 = c2 ; θj = κj − (1/2) if c1 = c2, (2.14)
where κ1 and κ2 are deﬁned by (2.7), (2.8) and (2.9). Since κ2 > 1/2 by the
deﬁnition and (2.11), we ﬁnd that there exists uniquely a solution v2 ∈ X of (2.6)
with c = c2 satisfying
‖u2(t)− v2(t)‖E(c2) ≤ Cεq(1 + t)−θ2 for t ≥ 0, (2.15)
and ‖v2(0)‖E(c2) < ∞, where C = C(c1, c2, p, q) is a positive constant and
‖u(t)‖2E(c) =
1
2
∫
R3
(|∂tu(x, t)|2 + c2|∇u(x, t)|2)dx.
ASYMPTOTIC BEHAVIOR FOR SEMILINEAR SYSTEMS OF WAVE EQUATIONS 5
Now we are in a position to state our main result. Suppose that θ1 ≥ 0, i.e., if
c1 = c2, then p ≥ 2 ; otherwise p ≥ 3/2 (for the remaining case, we refer to [13]).
As an unperturbed system, we choose{
∂2t v1 − c21∆v1 = |∂tv2|p in R3 × (0,∞),
∂2t v2 − c22∆v2 = 0 in R3 × (0,∞).
(2.16)
In other words, our proposal is to regard (2.1) as a perturbation from the “modified
free system ” (2.16), but in general not from the free system{
∂2t w1 − c21∆w1 = 0 in R3 × (0,∞),
∂2t w2 − c22∆w2 = 0 in R3 × (0,∞).
(2.17)
Theorem 2.2. Assume that p, q and (fj , gj) (j = 1, 2) fulfill the hypotheses of
Theorem 2.1. Suppose that θ1 ≥ 0. Then there exists uniquely a solution (v1, v2) ∈
X × X of (2.16) satisfying (2.15) and
‖u1(t)− v1(t)‖E(c1) ≤ Cεp+q−1(1 + t)−θ for t ≥ 0. (2.18)
Here θ is a positive number such that if c1 = c2, θ1 > 0 and q < 2, then θ =
θ1 +max{θ2 +(p− 1)(q− 2), 0} ; otherwise θ = θ1+ θ2, where θ2 (> 0) is defined by
(2.14) with j = 2. Besides, C is a constant depending only on c1, c2, p and q.
If we suppose in addition that θ1 > 0, then there exists uniquely a solution
(w1, w2) ∈ X × X of (2.17) satisfying
‖v1(t)−w1(t)‖E(c1) ≤ Cεp(1 + t)−θ1 for t ≥ 0. (2.19)
Therefore, combining (2.18) with (2.19), we see that u1 tends to w1 in the energy
norm as t →∞, hence (2.1) can be regarded simply as a perturbation from the free
system (2.17) in this case.
Therefore, the case θ1 = 0 is of our special interest. To simplify the situation,
we assume that the initial data are linear in ε. Namely,
fj(r) = εϕj(r), gj(r) = εψj(r) for r ∈ R (2.20)
with (ϕj , ψj) ∈ Y and Mκj (ϕj , ψj) ≤ 1. Then we have the following.
Theorem 2.3. Let c1 = c2, p = 2 and (2.10) hold. Suppose that (fj , gj) are as in
the above and that
rψ2(r) − (rϕ2(r))′ = 0 at r = r0 (2.21)
for a positive number r0. Then there are positive numbers C, ε1 and t0 such that
for 0 < ε ≤ ε1 and t ≥ t0 we have
C−1εp(log t) − ‖u1(0)‖E(c1) ≤ ‖u1(t)‖E(c1) ≤ ‖u1(0)‖E(c1) +Cεp(log t). (2.22)
6 HIDEO KUBO
Under the assumptions in Theorem 2.3, it is impossible that u1 has a free proﬁle
w1 with ‖w1(0)‖E(c1) < ∞. Indeed, if not, then limt→∞ ‖u1(t)‖E(c1) = ‖w1(0)‖E(c1).
Clearly, this contradicts (2.22).
Remark. We can extend the theorems presented in this section to the case where
the nonlinearity of the ﬁrst equation in (2.1) is replaced by |∂tu2|p−1∂tu2 or |∇u2|p.
In addition, we can admit the linear combination of these terms as the nonlinearity
in the theorems except Theorem 2.3, as well.
3. Example, II
The aim of this section is to show that the following semilinear system :{
∂2t u1 −∆u1 = (∂1u1)(∂1u2 − ∂2u1) in R3 × (0,∞),
∂2t u2 −∆u2 = (∂2u1)(∂1u2 − ∂2u1) in R3 × (0,∞),
(3.1)
cannot be regarded as a perturbation from the free system (2.17). Observe that the
quadratic nonlinearity is of critical order concerning the small data global existence
and blowup due to [6] and that the nonlinearities in (3.1) does not satisfy the null
condition. Therefore it seems hopeless to have a global solution for the problem.
Nevertheless, Alinhac [4] introduced some algebraic condition for (1.1) including
the null condition, and proved the global existence result for (1.1) satisfying his
condition with small initial data
uj(0, x) = εfj(x), (∂tuj)(0, x) = εgj(x) for x ∈ R3. (3.2)
The system (3.1) is nothing else an example satisfying the condition, hence the
Cauchy problem (3.1) and (3.2) admits a unique global smooth solution (u1, u2).
We underline that he suggests, without any rigorous proof, that his global solutions
does not tends to any solution of the free system in general.
The key of the proof given in [4] is to introduce an auxiliary function w =
∂1u2 − ∂2u1. Then we have
∂2t w −∆w = Q12(w, u1), (3.3)
where Q12(w, u1) = (∂1w)(∂2u1) − (∂2w)(∂1u1), which is one of the null forms.
Now, using (3.3), we can rewrite the system (3.1) as
∂2t u1 −∆u1 = w(∂1u1) in R3 × (0,∞),
∂2t u2 −∆u2 = w(∂2u1) in R3 × (0,∞),
∂2t w −∆w = Q12(w, u1) in R3 × (0,∞)
(3.4)
with initial data (3.2) for j = 1, 2 and
w(x,0) = εf3(x), (∂tw)(x,0) = εg3(x) for x ∈ R3. (3.5)
where
f3 = ∂1f2 − ∂2f1, g3 = ∂1g2 − ∂2g1. (3.6)
ASYMPTOTIC BEHAVIOR FOR SEMILINEAR SYSTEMS OF WAVE EQUATIONS 7
Note that the system (3.4) still does not satisfy the null condition, because the ﬁrst
and second equations in (3.4) are not written in terns of the null forms. While,
the third equation in (3.4) is written in terns of the null form, hence there exists
uniquely a solution v3 of (2.6) with c = 1 satisfying
lim
t→∞ ‖w(t)− v3(t)‖E(1) = 0 (3.7)
and ‖w2(0)‖E(1) < ∞. Having this in mind, we suppose that (3.4) can be regarded
as a perturbation from
∂2t v1 −∆v1 = v3 (∂1v1) in R3 × (0,∞),
∂2t v2 −∆v2 = v3 (∂2v1) in R3 × (0,∞),
∂2t v3 −∆v3 = 0 in R3 × (0,∞).
(3.8)
Actually we have the following result.
Theorem 3.1. For any initial data f1, f2, g1 and g2 ∈ C∞0 (R3), there exists
uniquely a solution (v1, v2, v3) of (3.8) satisfying (3.7) and
lim
t→∞ ‖uj(t)− vj(t)‖E(1) = 0 (j = 1, 2), (3.9)
where (u1, u2) is the solution to the Cauchy problem (3.1) and (3.2).
Finally we state a result which shows that the asymptotic proﬁle of (u1, u2) is
actually diﬀerent from any solutions of the free system.
Theorem 3.2. There exist initial data f1, f2, g1 and g2 ∈ C∞0 (R3) such that
lim
t→∞ ‖uj(t)‖E(1) =∞ (j = 1, 2) (3.10)
holds for the solution (u1, u2) to the Cauchy problem (3.1) and (3.2).
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Asymptotic behavior of solutions to semilinear systems of wave equations

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