Hoshiga, Akira

EPrint Series of Department of Mathematics, Hokkaido University

The existence of the global solutions to semilinear wave
equations with cubic nonlinearities in 2-dimensional
space
Akira HOSHIGA (Shizuoka University)
1 Introduction
　 Let us consider the following Cauchy problem:
2u = ∂2t u−4u = F (∂u) (x, t) ∈ R2 × (0,∞) (1.1)
u(x, 0) = εf(x), ∂tu(x, 0) = εg(x) x ∈ R2. (1.2)
Here, ∂ = (∂0, ∂1, ∂2), ∂0 = ∂t = ∂/∂t, ∂j = ∂/∂xj (j = 1, 2), ε is a positive small
parameter. We assume f, g ∈ C∞0 (R2), |f |+ |g| 6≡ 0 and
supp{f, g} ⊂ {x ∈ R2 ; |x| ≤M}. We also assume that
F ∈ C∞(R3)
F (∂u) = O(|∂u|3) near ∂u = 0.
More presicely, we assume
F (∂u) =
3∑
α,β,γ=0
Aαβγ∂αu∂βu∂γu+O(|∂u|4) near ∂u = 0,
where Aαβγ are constants.
　The aim of this talk is to estimate the lifespan Tε of the smooth solution to the Cauchy
problem (1.1), (1.2), which is defined as follows;
Tε = sup{ T > 0 ; there exists a smooth solution to (1.1), (1.2) in C∞(R2 × [0, T )) }
for each ε. In order to state the results which we have already known about the lifespan,
we introduce some notations.
　 For vectors X = (X0, X1, X2) ∈ R3, we define
C(X) =
2∑
α,β,γ=0
AαβγXαXβXγ,
1
which expresses the coefficient of the essential term of F (∂u).
　On the other hand, let u0 = u0(x, t) be the solution to the Cauchy problem;
2u0 = 0
u0(x, 0) = f(x), ∂tu
0(x, 0) = g(x)
and set r = |x| ≥ 0, ω = x
r
∈ S1 and ρ = r − t ∈ R．Then we define
F(ω, ρ) = lim
r→∞ r
1
2u0(rω, ρ+ r),
which is called the Friedlander radiation field. Ho¨rmander showed in [2] the following
properties of F．
|∂kρF(ω, ρ)| ≤ C(1 + |ρ|)−
1
2
−k ρ ∈ R (1.3)
F(ω, ρ) = 0 for ρ ≥M. (1.4)
By (1.3) and (1.4), we find that the constant
H = max
ρ∈R, ω∈S1
{
−1
2
C(−1, ω)(∂ρF(ω, ρ))2
}
(1.5)
is well-defined and nonnegative.
　Then Godin proved in [1] the following.
[Known results] (a) If H > 0, then
lim inf
ε→+0 ε
2 log Tε ≥ 1
H
holds.
　 (b) If C(−1, ω) ≡ 0 holds for ω ∈ S1, then Tε =∞ holds for sufficiently small ε > 0.
　The condition C(−1, ω) ≡ 0 is called ”null-condition.” The only function satisfying the
null-condition is
F (∂u) = C((∂0u)
2 − |∇u|2).
It follows from the definition of H that the null-condition is a sufficient condition of
H = 0 and not a necessary condition. In the case where |f |+ |g| 6≡ 0, we find that H = 0
is equivalent to the condition
C(−1, ω) ≥ 0 for any ω ∈ S1. (1.6)
This means that there is a nonlinearity F which does not satisfy the both assumption of
(a) and (b). Thus it is usefull to consider how long the solutions to (1.1) and (1.2) exists,
when the condition (1.6) holds.
2
2 Statement of the main theorem
We introduce generalaize differential operators;
Ω = x1∂2 − x2∂1, Li = t∂i + xi∂0, S = t∂0 + x1∂1 + x2∂2
and denote
Γ0 = ∂0, Γ1 = ∂1, Γ2 = ∂2, Γ3 = Ω, Γ4 = L1, Γ5 = L2, Γ6 = S.
We also write Γau = Γa00 Γ
a1
1 · · ·Γa66 u for a multi-index a = (a0, a1, . . . , a6).
Next we define some generalized Sobolev norms as
|v(t)|k =
∑
|a|≤k
||Γav(·, t)||L∞x (R2)
[v(t)]k =
∑
|a|≤k
||(1 + | · |+ t) 12 (1 + || · | − t|) 12Γav(·, t)||L∞x (R2)
||v(t)||k =
∑
|a|≤k
||Γav(·, t)||L2x(R2)
Now we state the main theorem.
Theorem 2.1 　Assume that (1.6) holds. Then there exists a constant ε0 > 0 such that
Tε =∞ holds for ε ∈ (0, ε0)．Moreover, the solution satisfies
[∂u(t)]k ≤ Ckε(1 + ε2 log t)Mk (2.1)
for some constants Mk > 0 which increases monotonely with respect to k.
(Remark 1) The decay estimate (2.1) means that the solution to (1.1) and (1.2) might
decay slower than solutions of linear wave eqations.
(Remark 2) Kubo proved the same result in [3] in which he also showed an asymptotic
behavior of solutions.
References
[1] P. Godin, Lifespan of solutions of semilinear wave equations in two space
dimensions, Comm. in P. D. E., 18 (5 and 6), pp. 895-916, (1993).
[2] L. Ho¨rmander, The lifespan of classical solutions of nonlinear hyperbolic equations,
Lecture Note in Math., 1256, pp. 214-280, (1987).
[3] H. Kubo, Large time behavior of solutions to semilinear wave equations with
dissipative structure, preprint
3


The existence of the global solutions to semilinear wave equations with cubic nonlinearities in 2-dimensional space

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EPrint Series of Department of Mathematics, Hokkaido University