Small-data scattering for nonlinear waves with potential and initial data of critical decay

We study the scattering problem for the nonlinear wave equation with potential. In the absence of the potential, one has sharp existence results for the Cauchy problem with small initial data; those require the data to decay at a rate greater than or equal to a critical decay rate which depends on the order of the nonlinearity. However, scattering results have appeared only for the supercritical case. In this paper, we extend the scattering results to the critical case and we also allow the presence of a short-range potential.


Introduction
We study the scattering problem for the nonlinear wave equation with potential where F (u) behaves like |u| p for some p > 1. When it comes to the special case V (x) ≡ 0, it is known that both the size of p and the decay rate k of the initial data play a crucial role in the existence theory of the associated Cauchy problem for small initial data. In fact, the condition k ≥ 2/(p − 1) is one of the sharp conditions needed to ensure the global existence of small-amplitude solutions. The scattering operator for (1.1), on the other hand, has been constructed only in the supercritical case k > 2/(p − 1), where a zero potential was assumed.
In this paper, we construct the scattering operator for the critical case as well, and we also allow the presence of a small, rapidly decaying potential. First, consider the Cauchy problem for (1.1) when V (x) ≡ 0 and the small initial data are prescribed at time t = 0. In what follows, we denote by p n the positive root of the equation (n − 1)p 2 = (n + 1)p + 2, (1.2) using the convention that p 1 = ∞. If 1 < p ≤ p n and the data are of compact support, then blow-up is known to generally occur. And even if p > p n , blow-up may still occur for slowly decaying data of noncompact support. Namely, under the assumption that blow-up does occur for any ε > 0, provided that 0 ≤ k < 2/(p − 1). As it turns out, however, the conditions p > p n and k ≥ 2/(p − 1) are not only necessary but also sufficient for the existence of global solutions in the following sense. Under the assumption that ∂ α x u(x, 0), ∂ β x ∂ t u(x, 0) = O(|x| −k−1 ) as |x| → ∞; |α| ≤ 3, |β| ≤ 2, small-amplitude solutions exist for all times when p > p n and k ≥ 2/(p − 1), provided that either n = 2, 3; or n ≥ 4 and the data are of compact support; or n ≥ 4 and the data are radially symmetric with noncompact support. For a precise list of references on the Cauchy problem, we refer the reader to [1].
Next, we focus on the scattering problem for (1.1). When it comes to the case V (x) ≡ 0, the existence of the scattering operator for small data was established by Pecher [7] in n = 3 space dimensions under the almost optimal assumptions p > p n and k > 2/(p − 1). Under the exact same assumptions, this result was extended to the case n = 2 by Tsutaya [10] and independently by Kubota and Mochizuki [6]. As for the higher-dimensional case n ≥ 4, the corresponding results were obtained by Kubo and Kubota [4,5] for radially symmetric data.
Before we state our main results, however, let us first introduce some hypotheses. When it comes to the nonlinear term F (u), we shall impose the conditions for some A > 0 and some p > 1. When it comes to the potential term V (r), we require that for some small V 0 > 0, where the bracket notation s = 1 + |s| is used for convenience. As for the initial data, we shall assume that for some small ε > 0 and some k > 0. Although our approach does not really depend on the parity of n, the decay estimates that one can obtain for solutions to (1.3) inevitably do. In particular, it will be convenient to introduce the parameters we shall frequently use in what follows. Let us remark that m ≥ 1 when n ≥ 4 and that the sum a + m = (n − 1)/2 is independent of the parity of n.
Our plan is to construct a solution of (1.3) that is continuously differentiable and belongs to the Banach space Here, the norm || · || is defined by where m is given by (1.7) and the weight function W k is of the form This weighted norm is partly dictated by the available estimates regarding the solutions to the homogeneous problem For a proof of the following lemma, we refer the reader to [2,4,5].
Regarding the solutions to the nonlinear equation, the existence result of this paper can now be stated as follows. Theorem 1.3. Let n ≥ 4 be an integer. Let ϕ ∈ C 2 (R + ) and ψ ∈ C 1 (R + ) satisfy (1.6) for some ε, k > 0. Suppose the nonlinear term F (u) satisfies (1.4) for some A > 0 and some where p n is the positive root of (1.2). Also, assume V (r) is subject to (1.5) for some V 0 > 0 and κ > 2. If k ≥ 2/(p − 1) and if V 0 , ε are sufficiently small, then the following conclusions hold with D = (∂ r , ∂ t ) and u − 0 as in Lemma 1.1. (a) There exists a unique solution u ∈ X to the nonlinear equation (1.3) such that for some constant C which is independent of r, t. Besides, one has ||u|| ≤ 1. (b) There exists a unique solution u + 0 ∈ X to the homogeneous equation for some constant C which is independent of r, t.
Remark 1.4. Using the norm dictated by (1.13) instead of our norm (1.9), one can extend Theorem 1.3 to the case n = 3 without imposing radial symmetry. When n = 1, 2, on the other hand, there exist arbitrarily small, rapidly decaying potentials for which small-data blow-up occurs; see section 6 in [1] for more details. Nevertheless, it is still possible to treat the case n = 2 when V (x) ≡ 0; for supercritical decay rates, we refer the reader to [10].
Remark 1.5. Under our assumption that k ≥ 2/(p − 1), the upper bound in (1.14) implies where a, m are defined by (1.7). It is possible to weaken the upper bound (1.14) on p in the spirit of [1] and thus obtain existence for decay rates k ≤ (n − 1)/2 as well. For such decay rates, however, we are unable to show that the solutions are asymptotically free, as we do in our next corollary.
Corollary 1.6. Let the assumptions of the previous theorem hold and define the energy norm If it happens that k > m + 1, then one has Here, the constant C is independent of t and we have also set for convenience.
The remaining of this paper is organized as follows. In section 2, we collect some facts about the inhomogeneous wave equation and reduce the proof of our existence result to the estimation of certain integrals involving our weight function (1.10). In section 3, we gather the necessary estimates to treat those integrals, while section 4 is devoted to the proofs of our main results, Theorem 1.3 and Corollary 1.6.

Preliminaries
In this section, we prepare a few basic lemmas that will be needed in the proof of our existence theorem regarding the nonlinear wave equation with potential Recall that we seek a solution to (2.1) for initial data of decay rate k ≥ 2/(p − 1). Since there is no loss of generality in decreasing this decay rate, we may take k to be smaller than any quantity that exceeds 2/(p − 1). Now, our assumption (1.14) that p is larger than the critical power p n of (1.2) can be written as with a, m as in (1.7); and it can also be written as This allows us to decrease the value of k and henceforth assume 2 without loss of generality. Since we also have p > 1, this automatically implies Finally, it is convenient to decrease the decay rate κ > 2 of the potential V (r) so that We can do this without loss of generality whenever m > 0, namely, whenever n ≥ 4.
Remark 2.1. The last simplification is not available when n = 2, 3, yet we shall only need it to control certain integrals that do not arise in odd dimensions due to the strong form of Huygens' principle; see Lemma 3.15. This is exactly the point where our proof would fail to work in two space dimensions, yet this is merely because of the potential term.
The following elementary fact is quite standard.
Lemma 2.2. Let L denote the Riemann operator for the wave equation in the radial case. Given a function G of two variables, we define the Duhamel operator L as When G ∈ C 1 (Ω), one then has L G ∈ C 1 (Ω) and this function provides a solution to Although we shall not need an explicit representation for the Riemann operator here, the following basic estimate for the Duhamel operator is important. Its proof essentially repeats that of Proposition 2.3 in [1], so we are going to omit it. Proposition 2.3. Let n ≥ 4 be an integer and define a, m by (1.7). Suppose G ∈ C 1 (Ω) satisfies the singularity condition for some fixed δ > 0. With D = (∂ r , ∂ t ) and λ ± = t − τ ± r, one then has and also j s=0 Proof. Because of our assumption (1.4), the fundamental theorem of calculus ensures that In particular, it ensures that Recalling the definition (1.9) of our norm, the last equation easily leads to In view of our assumptions that j ≤ 1 < p, this also implies Moreover, s − j ≤ j 0 − j within the last sum, so our first assertion (2.7) follows. Since our second assertion (2.8) is easier to establish, we shall omit the details.

A Priori Estimates
In this section, we prepare the main estimates needed in the proof of our existence result. Those are contained in the following Theorem 3.1. Let n ≥ 4 be an integer and a, m be as in (1.7). Suppose F (u) satisfies (1.4) for some p subject to (1.14) and that V (r) satisfies (1.5). Assume the decay rates k, κ are subject to (1.15), (2.2) and (2.4). Define the Duhamel operator L by (2.5) and let u be an element of the Banach space (1.8). With D = (∂ r , ∂ t ), one then has and also 2) as long as (r, t) ∈ Ω and |β| ≤ 1. Besides, the constant C 1 is independent of r, t. Remark 3.2. As an immediate consequence of (1.14), one finds that the conditions also hold under the assumptions of this theorem.
The proof of Theorem 3.1 is given in the next section. Here, we shall merely study certain integrals which arise in the course of the proof. Those involve our weight function (1.10) and some other parameters we have introduced (1.7). Throughout this section, in particular, we shall assume Let us remark that the two leftmost inequalities follow by (1.15) and (2.2), respectively. Lemma 3.3. Let r, t > 0 be arbitrary. Assuming that a, ν > 0, one has where the weight function W k is given by (1.10).
Proof. If either r ≤ 1 or t ≥ 2r, then t − r is equivalent to t + r , so we get because a > 0. This does imply the desired estimate under the present assumptions.
If r ≥ 1 and t ≤ 2r, on the other hand, an integration by parts gives Since a > 0 and since r − t + y ≤ 2y within the region of integration, we now get As r is equivalent to t + r when r ≥ max(t/2, 1), the result follows.
Lemma 3.4. Let z ≥ 0 be arbitrary. Given constants a > 0 and b < 1, one then has Proof. If it happens that 0 ≤ z ≤ 1, then we easily get because a > 0. Moreover, this does imply the desired (3.5) whenever z is bounded.
If it happens that z ≥ 1, on the other hand, we get In view of our assumptions that a > 0 and b < 1, the desired estimate (3.5) follows.
Proof. We only prove our second assertion, as our first assertion is trivial. Write When it comes to the first part, x is equivalent to z and we easily get by (3.4) and (2.2), we may combine the last two equations to deduce the desired (3.6).
When it comes to J 2 , on the other hand, we have 2x ≥ x + 2|y| ≥ x + y ≥ 1 within the region of integration, so . In view of (3.7), the desired (3.6) now follows.
where λ ± = t − τ ± r, W k is given by (1.10) and the constant C is independent of r, t.
Proof. According to Lemma 3.5 in [1], we do have the estimate t min(0,t) when t ≥ 0. Since this estimate holds trivially when t ≤ 0, it remains to show that Let us first recall our definition (1.10) of our weight function W k and write As τ ≤ t within the region of integration, we have λ ≥ |λ − | ≥ t − τ − r. As τ ≤ 0, we also have λ ≥ |λ − | ≥ |t − r| + τ . Changing variables by x = λ − τ and y = λ + τ , we then get by Lemma 3.5 with z = max(|y|, |t − r|).
Note that ν < νp < 1 by (3.4). To show that the last integral satisfies the desired (3.8), we shall now establish the more general estimate for any constants b 1 , b 2 ∈ R with b 1 < 1 and b 1 + b 2 = a + ν.
Case 1: When t ≥ 0, we have t − r ≤ |t − r| ≤ t + r, so our definition (3.9) reads To treat the latter integral, we need only invoke Lemma 3.3. To treat the former integral, we may assume that r ≥ t. In view of our assumption that b 1 < 1, we then get If r ≥ t and r ≤ 1, this is easy to see because r − t ≤ r and r − t is equivalent to r + t . If r ≥ t and r ≥ 1, on the other hand, r is equivalent to r + t and we similarly get Case 2: When t ≤ 0, we have |r + t| ≤ r + |t| = r − t, so our definition (3.9) reads (3.10) Subcase 2a: If it happens that |t| ≤ 3r, we proceed as in the previous case to obtain using Lemma 3.4. Since r − t = r + |t| ≤ 4r for this subcase, the result follows easily. Subcase 2b: If it happens that −t = |t| ≥ 3r, then r + t is equivalent to r − t because |r + t| ≤ r + |t| = r − t ≤ −2(r + t) for this subcase. In particular, equation (3.10) trivially leads to since a > 0. This does imply the desired (3.9) whenever r + t is equivalent to r − t .
where λ ± = t − τ ± r, W k is given by (1.10) and the constant C is independent of r, t.
Proof. According to Lemma 3.6 in [1], we do have the estimate t min(0,t) when t ≥ 0. Since this estimate holds trivially when t ≤ 0, it remains to show that Let us now proceed as in the proof of the previous lemma. Changing variables by x = λ − τ and y = λ + τ , we use Lemma 3.5 to arrive at Since νp < 1 by (3.4), we may then invoke our estimate (3.9) to complete the proof.
Lemma 3.8. Let y ∈ R. Assuming that κ > 2 and 0 ≤ b < 1, one has Assuming that 2 − mp + m > 0 and that b < 1 < m(p − 1) + b, one also has Proof. We shall only prove our second assertion, as the proof of our first assertion is quite similar. When x ≥ 2|y| + 1, one has 2 ≤ x + 1 ≤ 2(x + y) ≤ 3(x + 1) and this implies Using our assumption that m(p − 1) + b > 1, we now arrive at Since x is equivalent to y whenever |y| ≤ x ≤ 2|y| + 1, this easily leads us to

by assumption. To finish the proof, it thus remains to show that
whenever y ≥ 0. If it happens that 0 ≤ y ≤ 1, this estimate is easy to establish. Let us then assume that y ≥ 1. Since x is equivalent to y for each −y ≤ x ≤ −y/2 and since x + y is equivalent to y for each −y/2 ≤ x ≤ y, we find In view of our assumptions that 2 − mp + m > 0 and b < 1, the desired (3.11) follows.

Remark 3.9.
In what follows, we shall need to apply Lemma 3.8 when b = a(p − 1) and also when b = νp. To ensure the lemma is applicable, we shall thus need to know that a(p − 1) < 1 < (a + m)(p − 1), νp < 1 < m(p − 1) + νp. Since the remaining assertions of (3.12) are also easy to verify, we shall omit the details.

13)
where λ ± = t − τ ± r, W k is given by (1.10) and the constant C is independent of r, t.
Let us now turn to the proof of (3.14). Employing (3.15), we find that whenever t ≤ 2r. In order to establish (3.14), it thus suffices to show that Since λ + = t − τ + r is non-negative here, I ′′ + is rather easy to treat. To treat I ′′ − , on the other hand, it is convenient to write Using our assumption κ > 2 for the former integral and the substitution x = 2τ + r − t for the latter, we find Once we now recall that 0 < ν < νp < 1 by (3.4), an application of Lemma 3.8 gives This already establishes (3.16), so the proof is complete.
where λ ± = t − τ ± r, W k is given by (1.10) and the constant C is independent of r, t.
Proof. According to Lemma 3.13 in [1], we do have the estimate t max(t−2r,0) |λ ± | a+1−mp+m λ ± −1 · W k (|λ ± |, |τ |) −p dτ ≤ Cr a W k (r, |t|) −1 whenever t ≥ 0. In particular, it suffices to show that whenever t ≤ 2r. Let us now proceed as in the proof of the previous lemma. Using (3.15), one may deduce the last inequality as soon as we know that In what follows, we only concern ourselves with J ′ − , as J ′ + is easier to treat. Write Changing variables by x = t − r − 2τ for the former integral and by x = r − t + 2τ for the latter, we now arrive at In view of (3.12), we may then apply Lemma 3.8 with b = a(p − 1) and b = νp to get (3.7). Since this already implies the desired (3.17), the proof is complete.
Lemma 3.12. Let w ≤ 0 be arbitrary. Assuming that a, ν > 0, one has Proof. Let us change variables by z = −y and write the given integral as Since z is equivalent to w within the former integral and w + z is equivalent to z within the latter, we find that In view of our assumptions that a, ν > 0, the desired estimate is now easy to deduce.

18)
where λ − = t − τ − r, W k is given by (1.10) and the constant C is independent of r, t.
Proof. We shall only prove our second assertion, as the proof of our first assertion is quite similar. Since λ ≤ λ − within the region of integration, we trivially get and we may switch to characteristic coordinates x = λ − τ , y = λ + τ to arrive at Once we now invoke Lemma 3.5 to treat the inner integral, we find Using Lemma 3.12 for the former integral and Lemma 3.4 for the latter, we get (3.18).
Proof. We only concern ourselves with our second assertion, as our first assertion is easier to establish. Changing variables by y = λ − = t − r − τ , let us write When it comes to the former integral, λ is equivalent to y and we easily get because a > 0 by (3.4) and 2 − mp + m > 0 by (3.3). When it comes to the latter integral, on the other hand, y − λ is equivalent to y, so we find 4) and (3.3), the desired estimate follows.
and also where λ ± = t − τ ± r, W k is given by (1.10) and the constant C is independent of r, t.
Proof. We shall only prove our second assertion, as the proof of our first assertion is quite similar. According to Lemma 3.10 in [1], we do have the estimate t−r min(t−r,0) when t ≥ r. Since this estimate holds trivially when t ≤ r, it remains to show that Case 1: When t ≥ 0, it is convenient to express the last integral as the sum of and Let us treat J ′ 21 first. If it happens that t + r ≥ 1, then λ + = t + r − τ ≥ C t + r within the region of integration. Using this fact along with Lemma 3.13, we deduce the desired Assume now that t + r ≤ 1. Since we still have λ + = t + r − τ ≥ t + r + 1 for each τ ≤ −1, our argument above ensures that Besides, |t − r| ≤ t + r ≤ 1 here, so we need only show that the right hand side is bounded. Using a trivial estimate and then Lemma 3.14, we arrive at the desired Next, we treat J ′ 22 . In treating this integral (3.21), we may assume that t ≥ r and write Using a trivial estimate and then Lemma 3.14, one easily settles the case t + r ≤ 1 as above. Assume now that t + r ≥ 1. Since λ ± = t ± r − τ ≥ t ± r within the region of integration, we find Switching to characteristic coordinates x = λ − τ and y = λ + τ , we thus find Note that 2(x + 1) ≥ 2x ≥ x + y here, while (3.3) implies 2 − mp + 2m − ap > m − a ≥ 0 because we are assuming that m ≥ a. Combining these facts, we now get Besides, νp < 1 by (3.4), so we may apply Lemma 3.4 with z = t − r ≥ 0 to arrive at In view of our assumption (3.3) and our inequality (3.7), we also have so we may combine the last two equations to deduce the desired estimate (3.19). Case 2: When t ≤ 0, we have t < r and we shall express the integral (3.19) as the sum of To treat the first part (3.23), we note that λ + = t + r − τ ≥ 3r − t + 1 > r − t + 1 within the region of integration. In view of Lemma 3.13, this gives which implies the desired (3.19) because |r + t| ≤ r + |t| = r − t whenever t ≤ 0. To treat the second part (3.24), we use the fact that λ + ≥ λ − to trivially obtain Since λ − τ ≥ −τ ≥ r − t and λ + τ ≤ t − r < 0 within the region of integration, we get by Lemma 3.14. In view of our inequality (3.7), the desired estimate (3.19) follows.

Existence of solutions
In this section, we prove the two main results of this paper. Before we do that, however, let us first combine the estimates of the previous section and give the Proof of Theorem 3.1. There are two estimates we need to verify and we shall only verify the first, as the proof of the second is similar. To estimate the Duhamel integral (3.1), we apply Proposition 2.3. Given a function G ∈ C 1 (Ω) that satisfies the singularity condition as λ → 0 (4.1) for some fixed δ > 0, Proposition 2.3 provides the estimate whenever (r, t) ∈ Ω and |β| ≤ j ≤ 1.
We now use this fact with G(λ, τ ) = F (u(λ, τ )). By Lemma 2.4 with j = j 0 = 0, we have where δ 1 = 2 − mp + 2m is positive by (3.3). In particular, our estimate (4.2) does hold for the special case G = F (u). Besides, the sums that appear in the right hand side are those of Lemma 2.4, according to which Once we now insert this fact in (4.2), we obtain an estimate of the form where J 1 , J 2 and J ± are as in Lemmas 3.7, 3.15 and 3.11, respectively. The assumptions we imposed in those lemmas are not different from the ones imposed in this theorem, except for the inequality a ≤ 1 ≤ m that appears in Lemma 3.15. Since our definition (1.7) shows that a ≤ 1 ≤ m whenever n ≥ 4, however, we may employ Lemmas 3.7, 3.11 and 3.15 to get 3) Finally, we claim that this also implies our first assertion (3.1), namely that Indeed, if r ≤ 1, one may obtain the last inequality through the special case j = 1 of (4.3).
If r ≥ 1, on the other hand, one may obtain it through the special case j = |β| ≤ 1.
To prove the second part of the theorem, we define where L is the Riemann operator and we have set G(u) = F (u) − V (r) · u. Then u + 0 is clearly a solution to the homogeneous equation (1.11). Recalling the integral equation (4.5) and the definition (2.5) of the Duhamel operator, we can rewrite the last equation as u + 0 (r, t) = u(r, t) + ∞ t [LG(u(· , τ ))](r, t − τ ) dτ.
Besides, the right hand side bears a close resemblance to the Duhamel operator (2.5), so one can estimate this expression in exactly the same way that we used to estimate (4.5).