ON THE POINTWISE DECAY ESTIMATE FOR THE WAVE EQUATION WITH COMPACTLY SUPPORTED FORCING TERM

. In this paper we derive a new type of pointwise decay estimates for solutions to the Cauchy problem for the wave equation in 2D, in the sense that one can diminish the weight in the time variable for the forcing term if it is compactly supported in the spatial variables. As an application of the estimate, we also establish an improved decay estimate for the solution to the exterior problem in 2D.


Introduction
In this paper we derive a new type of pointwise decay estimates for solutions to the following Cauchy problem for the wave equation: where u = u(t, x) is an unknown function, ∆ = ∂ 2 1 + ∂ 2 2 , ∂ t = ∂ 0 = ∂/∂t, and ∂ j = ∂/∂x j (j = 1, 2). In addition, we assume that v 0 and v 1 are smooth functions on R 2 , and that g is a smooth function on [0, T ) × R 2 . As is well known, different kind if pointwise estimates has been used for treating nonlinear wave equations. This approach goes back to the seminal work of John [5].
On the other hand, the point of the present study is to understand how we can feel influence coming from such an assumption on the forcing term g(t) that for each fixed t ∈ [0, T ), its support is contained in a ball with a fixed radius centered at the origin. This question is closely related to the exterior problem (see the section 4 below for details). But our pointwise estimate (1.3) itself is of interest, because it tells us that if the forcing term is compactly supported in the spatial variables, then one can diminish the weight in the time variable for the forcing term, compared with the standard pointwise decay estimate (2.4) below (see also Remark 1 in the below of Proposition 3.2).
This paper is organized as follows. In the next section we collect basic notations and recall known pointwise estimates for the problem (1.1)-(1.2). In the section 3 we give a proof of Theorem 1.1 in a slightly generalized form. The section 4 is devoted to establish an improved decay estimate for the exterior problem as an application of Theorem 1.1.

Preliminaries
2.1. Notation. Let us start with some standard notation.
• Let A = A(y) and B = B(y) be two positive functions of some variable y, such as y = (t, x) or y = x, on suitable domains. We write A B if there exists a positive constant C such that A(y) ≤ CB(y) for all y in the intersection of the domains of A and B. • The norm · L ∞ without any other index stands for · L ∞ (R 2 ) . • For a time-space depending function u satisfying u(t, ·) ∈ X for 0 ≤ t < T with a Banach space X, we put u L ∞ T X := sup 0≤t<T u(t, ·) X . For the brevity of the description, we sometimes use the expression h(s, y) L ∞ t L ∞ with dummy variables (s, y) for a function h on [0, t) × R 2 , which means sup 0≤s<t h(s, ·) L ∞ . • B r stands for an open ball with radius r centered at the origin of R 2 .
The following lemma is concerned with the inhomogeneous wave equation. For the proof, see [13,Lemma 3.4], also Di Flaviano [1].
3. Proof of Theorem 1.1 In this section we shall prove the following pointwise decay estimate which is involved with the generalized derivatives Γ's. This kind of generalization of Theorem 1.1 would be useful, when we wish to study the nonlinear wave equations. Theorem 3.1. Let a > 0, 1 < ν < 3/2 and m be a non-negative integer. If supp g(t, ·) ⊂ B a for any t ∈ [0, T ), then we have In order to prove (3.1), it suffices to consider the solution of the inhomogeneous wave equation, in view of (2.3).

Proposition 3.2.
Under the same assumptions of Theorem 3.1, we have In particular, if supp g(s, ·) ⊂ B a , then the right hand side is evaluated by Therefore, (3.2) is actually different from (2.4) in the sense that one can diminish the weight on the forcing term with respect to the time variable as long as the forcing term is compactly supported in the spatial variable.

Proof of Proposition 3.2. It follows from (2.2) and the uniqueness for the classical solution that
with suitable constants C β , C β . Therefore, by (2.3) with 1 < ν < 3/2, we get because of the assumption on g. Hence, from (3.4) we see that it is enough to show Let χ A be the characteristic function for a set A. Then, following the opening of the section 4 in [11], we get with r = |x|. Here we denoted , and they are estimated as follows: Indeed, these estimates follow from the proofs of (4.18) and (4.22) in [11], because √ λ is simply evaluated by √ α when α ≥ β, which is equivalent to s ≥ 0, in their proofs (notice that K 1 (λ, s, r, t) and K 2 (λ, s, r, t) in [11] are actually equal to K 1 (λ, r, t − s) and K 2 (λ, r, t − s)). Thus we have Therefore, in order to prove (3.5), we have only to show First we estimate I 1 (r, t). Since I 1 ≡ 0 if r − t ≥ a, we may assume t − r ≥ −a. Notice that if λ ≥ |(t − r) − s| and λ ≤ a, then s is equivalent to t − r and λ − s + r + t ≥ 2r. Therefore from (3.7) we have To proceed further, we divide the argument into four cases.
1. |t − r| ≤ a and 0 < r ≤ a: It follows that Since t ≤ |t − r| + r ≤ 2a in this case, the above estimate yields (3.9). 2. |t − r| ≤ a and r ≥ a: We change the order of the integration to obtain Since 4r ≥ 2r + a + (t − r) = r + t + a in this case, this estimate implies (3.9). 3. t − r ≥ a and 0 < r ≤ a: It follows from (3.11) that Since t ≥ r + a ≥ 2r in this case, t − r is equivalent to t + r , so that the above estimate yields (3.9). 4. t − r ≥ a and r ≥ a: From (3.11) we have which yields (3.9). Indeed, if t ≥ 2r, then t − r is equivalent to t + r . On the other hand, if t ≤ 2r, then r is equivalent to t + r . Therefore, the above estimate is enough to conclude that the desired one holds. Next we estimate I 2 (r, t). We may assume t − r > 0, and we divide the argument into three cases.

On the exterior problem for the wave equation
As an application of Theorem 3.1, we examine the pointwise decay estimate for the exterior problem in this section.
Let Ω be an unbounded domain in R 2 with compact and smooth boundary ∂Ω. We put O := R 2 \ Ω, which is called an obstacle and is assumed to be non-empty. Throughout this section we shall assume 0 ∈ O so that we have |x| ≥ c 0 for x ∈ Ω with some positive constant c 0 . We shall also assume that O ⊂ B 1 . Thus a function v = v(x) on Ω vanishing for |x| ≤ 1 can be naturally regarded as a function on R 2 .
Given T > 0, we consider the mixed problem for the wave equation : We assume u 0 , u 1 ∈ C ∞ 0 (Ω), namely, they are smooth functions on Ω vanishing outside some ball. On the other hand, f ∈ C ∞ ([0, T ) × Ω) is assumed to satisfy f (t, ·) ∈ C ∞ 0 (Ω) for any fixed t ∈ [0, T ). In order to obtain a smooth solution to (4.1)-(4.3), we need the compatibility condition to infinite order, i.e., u j (x) vanishes on ∂Ω for any non-negative integer j, where u j (x) is determined by Then we have the following decay estimates.
Theorem 4.1. Let 1 < ν < 3/2, κ > 1, and k be a non-negative integer. Assume that O is star-shaped, and Ξ = ( u 0 , f) ∈ X(T ). Then we have Here we put Our proof of the theorem is based on the cut-off method used in Shibata and Tsutsumi [19], where the nonlinear problem was handled when the space dimension is greater than 3 (see also [2,3], [4], [6], [8,9], [12], [16], [17], [18], and the references cited therein). Because the decaying rate of the solution to the wave equation is weaker and weaker as the space dimension is lower and lower, in our previous works [7,13,14] which treat the two-dimensional case, we could not obtain such a decay estimate of the solution itself as above. But, thanks to (3.2), we succeed to derive the same decay estimate as in the boundaryless case. The estimate (4.5) will be used in the forthcoming papers concerning the nonlinear wave equation in the exterior domain.

Decomposition of solutions.
For a > 0, we denote by ψ a a smooth radially symmetric function on R 2 satisfying ψ a (x) = 0, |x| ≤ a, ψ a (x) = 1, |x| ≥ a + 1. Besides, for a ≥ 1 we set Since O ⊂ B 1 , we see that Ω a = ∅ for any a ≥ 1. Then we have the following decomposition (for the proof, we refer to [19], [12, Lemma 3.1]).
Assume that supp f (t, ·) ⊂ Ω t+a and supp u 0 ⊂ Ω a , supp u 1 ⊂ Ω a holds for any t ∈ (0, T ). Then we have It is useful to prepare the following lemma for the proof of Theorem 4.1.
Finally we prove (4.15) by using (2.3) and (2.4) with (4.6). It follows that Observe that t − |x| is equivalent to t when x ∈ Ω b . Therefore we get (4.15). This completes the proof. Now we are in a position to prove Theorem 4.1.