Almost global existence for exterior Neumann problems of semilinear wave equations in 2D

The aim of this article is to prove an"almost"global existence result for some semilinear wave equations in the plane outside a bounded convex obstacle with the Neumann boundary condition.


Introduction
Let O be an open bounded convex domain with smooth boundary in R 2 and put Ω := R 2 \ O. Let ∂ ν denote the outer normal derivative on ∂Ω.
We consider the mixed problem for semilinear wave equations in Ω with the Neumann boundary condition: ( x ∈ Ω, ∂ t u(0, x) = ψ(x), x ∈ Ω, (1.1) where φ and ψ are C ∞ -functions compactly supported in Ω, and G : R 3 → R is a nonlinear function. We will study the case of the cubic nonlinearity with small initial data and obtain an estimate from below for the lifespan of the solution in terms of the size of the initial data. Here by the expression "small initial data" we mean that there exist m ∈ N, s ∈ R and a small number ε > 0 such that (1 + |x| 2 ) s |∂ α x ϕ(x)| 2 dx. (1.2) A large amount of works has been devoted to the study of the mixed problem for nonlinear wave equations in an exterior domain Ω ⊂ R n for n ≥ 3, mostly with the Dirichlet boundary condition. To our knowledge very few results deal with the global existence or the lifespan estimate for the exterior mixed problems of nonlinear wave equations in 2D; in [SSW11] the global existence for the case of the Dirichlet boundary condition and the nonlinear terms depending only on u is considered; in [K12] one of the authors obtained an almost global existence result for small initial data under the assumptions that |G(∂u)| ≃ (∂u) 3 , the obstacle is star-shaped and the boundary condition is of the Dirichlet type (see Remark 1.4 below for the detail).
Here we will treat the problem with the Neumann boundary condition in 2D and obtain an analogous result to [K12]. However, because we have a weaker decay property for the solution to the Neumann exterior problem of linear wave equations in 2D (see Secchi and Shibata [SS03]), we will obtain a slightly worse lifespan estimate than in the Dirichlet case.
Definition 1.1. To the mixed problem (1.1) we can associate the recurrence sequence {v j } j∈N * with v j : Ω → R such that where N * denotes the set of nonnegative integers and G (k) is defined as above (cf. (1.4)). We say that (φ, ψ, G) satisfies the compatibility condition of infinite order in Ω for (1.1) if φ, ψ ∈ C ∞ (Ω), and one has Our aim is to prove the following result.
Theorem 1.1. Let O be a convex obstacle. Consider the semilinear mixed problem (1.1) with given compactly supported initial data (φ, ψ) ∈ C ∞ (Ω)× C ∞ (Ω) and a given nonlinear term G(∂u) which is a homogeneous polynomial of cubic order as in (1.3). Assume that (φ, ψ, G) satisfies the compatibility condition of infinite order in Ω for (1.1). Under these assumptions, there exist ε 0 > 0, m ∈ N, s ∈ R such that, if ε ∈ (0, ε 0 ] and φ H m+1,s (Ω) + ψ H m,s (Ω) ≤ ε, (1.5) then the mixed problem (1.1) admits a unique solution u ∈ C ∞ ([0, T ε ) × Ω) with T ε ≥ exp(Cε −1 ), (1.6) where C > 0 is a suitable constant which is uniform with respect to ε ∈ (0, ε 0 ]. Remark 1.2. The only point where we require that the obstacle O is convex is to gain the local energy decay (see Lemma 7.5 below). In general one can treat the obstacles for which Lemma 7.5 holds. Unfortunately, for the Neumann problems in 2D, up to our knowledge it is not known if there exists non-convex obstacles satisfying such a local energy decay.
Remark 1.3. One can ask if it is possible to gain a global existence result maintaining our assumption on the growth of G.
In general the answer to this question is negative since the blow-up in finite time occurs for F = (∂ t u) 3 when n = 2. Indeed, it was proved in [G93] that for any R > 0 we can find initial data such that the blow-up for the corresponding Cauchy problem occurs in the region |x| > t + R. This result shows the blow-up for the exterior problem with any boundary condition if we choose sufficiently large R, because the solution in |x| > t + R is not affected by the obstacle and the boundary condition, thanks to the finite propagation property (see [KK12] for the corresponding discussion in 3D). In order to look for global solutions one could investigate the exterior problem with suitable nonlinearity satisfying the so-called null condition.
Remark 1.4. If we consider the Cauchy problem in R 2 , or the Dirichlet problem in a domain exterior to a star-shaped obstacle in 2D, an analogous result to Theorem 1.1 holds with (1.7) and this lifespan estimate is known to be sharp (see [G93] for the Cauchy problem and [K12] for the Dirichlet problem). One loss of the logarithmic factor in the decay estimates causes this difference between the lifespan estimates (1.6) and (1.7) (see Theorem 2.1 and Remark 7.1 below). It is an interesting problem whether our lower bound (1.6) is sharp or not for the Neumann problem.

Preliminaries
In this section we introduce some notation which will be used throughout this paper and some basic lemmas for the proof of Theorem 1.1.
Throughout the paper we shall assume 0 ∈ O so that we have |x| ≥ c 0 for x ∈ Ω for some positive constant c 0 . We shall also assume that O ⊂ B 1 , where B r stands for an open ball with radius r centered at the origin of R 2 . Thus a function v = v(x) on Ω vanishing for |x| ≤ 1 can be naturally regarded as a function on R 2 .
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 L 2 (Ω) norm is denoted by · L 2 Ω , while the norm · L 2 without any other index stands for · L 2 (R 2 ) . Similar notation will be used for the L ∞ norms.
• 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 ∞ Ω with dummy variables (s, y) for a function h on [0, t) × Ω, which means sup 0≤s<t h(s, ·) L ∞ Ω . • For m ∈ N and s ∈ R, by H m,s (Ω) we denote the weighted Sobolev space with norm defined by (1.2). Moreover H m (Ω) and H m (R 2 ) are the standard Sobolev spaces. • We denote by C ∞ 0 (Ω) the set of smooth functions defined on Ω which vanish outside B R for some R > 1.
Let ν ∈ R. We put This weight function w ν will be used repeatedly in the a priori estimates of the solution u to (1.1). We shall often use the following inequality Finally, for a ≥ 1 we set Since O ⊂ B 1 , we see that Ω a = ∅ for any a ≥ 1.

2.2.
Vector fields associated with the wave operator. We introduce the vector fields :

2)
and also Hence, for i, j = 0, 1, 2, 3, we have [Γ i , Γ j ] = 3 k=0 c k ij Γ k with suitable constants c k ij . Moreover, for i = 0, 1, 2 and j = 0, 1, 2, 3 we also have . The standard multi-index notation will be used for these sets of vector fields, such as These quantities will be used to control the influence of the initial data to the L ∞ norms of the solution.
Using the vector fields in Γ, we obtain the following Sobolev-type inequality.
Proof. It is well known that for w ∈ C 2 0 (R 2 ) we have Klainerman [Kl85] for the proof ). Let χ = χ(x) be a nonnegative smooth function satisfying χ(x) ≡ 0 for |x| ≤ 1 and χ(x) ≡ 1 for |x| ≥ 2. If we rewrite v as v = χv + (1 − χ)v, then we have χv ∈ C ∞ 0 (R 2 ) and (2.3) leads to By using the Sobolev embedding to estimate the last term, we arrive at This completes the proof.
2.3. Elliptic estimates. The following elliptic estimates will be used in the energy estimates.
If we put h = ∆v 1 , the function v 1 solves the elliptic problem Now we consider v 2 . Note that v 2 can be regarded as a function in R 2 and we can write . Let us recall that ∂ β x w L 2 (R n ) ∆w L 2 (R n ) for any w ∈ H 2 (R n ) and |β| = 2. Writing α = β + γ with |β| = 2 and |γ| = |α| − 2, we have Combining this inequality with the estimate for v 1 , we find (2.4).
(2.6) It is known that for u 0 ∈ H 2 (Ω), u 1 ∈ H 1 (Ω) and f ∈ C 1 [0, T ); L 2 (Ω) , the mixed problem (2.6) admits a unique solution provided that (u 0 , u 1 , f ) satisfies the compatibility condition of order 0, that is to say, (see [I68] for instance). Under these assumptions for u 0 := (u 0 , u 1 ), the solution u of (2.6) will be denoted by S[ u 0 , f ](t, x). We set K[ u 0 ](t, x) for the solution of (2.6) with f ≡ 0 and L[f ](t, x) for the solution of (2.6) with u 0 ≡ (0, 0); in other words we put so that we get where K[ u 0 ] and L[f ] are well defined because both of (u 0 , u 1 , 0) and (0, 0, f ) satisfy the compatibility condition of order 0. In order to obtain a smooth solution to (2.6), we need the compatibility condition of infinite order.
Definition 2.1. Suppose that u 0 , u 1 and f are smooth. Define u j for j ≥ 2 inductively by We say that (u 0 , u 1 , f ) satisfies the compatibility condition of infinite order in Ω for (2.6), if one has ∂ ν u j = 0 on ∂Ω for any nonnegative integer j.
We say that (u 0 , u 1 , f ) ∈ X(T ) if the following three conditions are satisfied: [I68] for instance).
The following decay estimates play important roles in our proof of the main theorem.
Theorem 2.1. Let O be a convex set and k be a nonnegative integer.
(ii) Let 0 < η < 1/2 and µ > 0. Then we have (iii) Let 0 < η < 1 and µ > 0. Then we have We will prove Theorem 2.1 in Section 7 below, by using the so-called cut-off method to combine the corresponding decay estimates for the Cauchy problem with the local energy decay.

The abstract argument for the proof of the main theorem
Since the local existence of smooth solutions for the mixed problem (1.1) has been shown by [SN89] (see also the Appendix), what we need to do for showing the large time existence of the solution is to derive suitable a priori estimates: following [SN89], we need the control of Let u be the local solution of (1.1), assuming (1.5) holds for large m ∈ N and s > 0. Let T * be the supremum of T such that (1.1) admits a (unique) classical solution in [0, T ) × Ω. For 0 < T ≤ T * , a small η > 0, and nonnegative integers H and K we define We neglect the first sum when H = 0. Similarly we neglect summations taken over the empty set as K varies. We also put Observe that E H,K (0) can be determined only by φ, ψ and G and that we have for suitably large m ∈ N and s > 0 depending on H and K. From (1.5) for such m ∈ N and s > 0, we see that E H,K (0) is finite. The previous inequality can be obtained combining the embedding H r (Ω) ֒→ L ∞ (Ω) for r > 1 with the trivial inequality and · s f H m (Ω) . In order to optimize m or s it is possible to use sharpest embedding theorem in weighted Sobolev spaces proved for example in [GL04].
Our goal is to show the following claim.
Claim 3.1. We can take suitable H and K and sufficiently large m and s, so that there exist positive numbers C 1 , P and Q and a strictly increasing continuous function R : provided that (1.5) holds with ε ≤ 1. Here C 1 , P , Q and R are independent of ε and T .
Let us explain how from (3.1) we can gain the lifespan estimate. Suppose that the above claim is true. If we assume (1.5) for some m and s which are sufficiently large, then, as we have mentioned, there exists C * > 0 such that E H,K (0) < 2C * ε. We may assume C * ≥ max{C 1 , 1}. We set ε 0 = min{(2C * ) −1 , 1} and suppose that 0 < ε ≤ ε 0 , so that we have ε ≤ 1 and 2C * ε ≤ 1. We put We are going to prove Since T * (ε) ≤ T * , and R is an increasing function, we obtain Therefore we get T * (ε) = T * , because otherwise the continuity of E H,K (T ) implies that there exists T > T * (ε) satisfying E H,K ( T ) ≤ 2C * ε, which contradicts the definition of T * (ε). However, if T * (ε) = T * , and H, K are sufficiently large, we can prove and we can extend the solution beyond the time T * by the local existence theorem, which contradicts the definition of T * . Therefore (3.3) is not true, and we obtain (3.2). This means that, for any ε ≤ ε 0 , there existsC > 0 such that It remains to show (3.4). It is evident that In order to estimate u L ∞ T * L 2 Ω we will use the expression which leads to As a conclusion, we obtain (1.6), once we can show that Claim 3.1 is true with P = Q = 1. This will be done in the next three sections.

Energy estimates for the standard derivatives
In this section we are going to estimate In the first subsection, we consider the case where j ≥ 0 and |α| = 1. This can be done directly through the standard energy inequalities. In the second subsection, the case where j ≥ 1 and |α| ≥ 2 will be treated with the help of the elliptic estimate, Lemma 2.2. In the third subsection, we consider the case where j = 0 and |α| ≥ 2. Lemma 2.2 will be used again, but this time we need the estimate of u L ∞ T L 2 (Ω R+1 ) for some R > 0, which is not included in the definition of E H,K (T ). Since we are considering the 2D Neumann problem, it seems difficult to use some embedding theorem to with some positive integer k. Instead, we will employ the L ∞ estimate, Theorem 2.1, for this purpose.
4.1. On the energy estimates for the derivatives in time. First we set Let j be a nonnegative integer. Since ∂ t commutes with the restriction of the function to ∂Ω, we have ∂ ν ∂ j t u(t, x) = 0 for all (t, x) ∈ (0, T ) × ∂Ω. Therefore, by the standard energy method, we find Recalling the definition of E H,K (T ), for j + |α| ≥ 1 we have Applying (4.1) and the Leibniz rule we find It is also clear that if j + |α| ≥ 1, one has As a trivial consequence of (2.1), we find w 1/2 (t, x) ≤ t −1/2 , so that After integration this gives for any integer s ≥ 0.

4.2.
On the energy estimates for the space-time derivatives. Since the spatial derivatives do not preserve the Neumann boundary condition, we need to use elliptic regularity results. We shall show that for j ≥ 1 and k ≥ 0 it holds It is clear that (4.3) follows from (4.2) when j ≥ 1 and k = 0, 1. Next we suppose that (4.3) holds for j ≥ 1 and k ≤ l with some positive integer l. Let |α| = l+1 and j ≥ 1. Since |α| ≥ 2, we apply to ∂ j t u the elliptic estimate (Lemma 2.2) and we obtain By (4.3) for k ≤ l, we see that the second term has the desired bound. On the other hand, using the fact that u is a solution to (1.1), for the first term we have which is the desired bound. Finally, observing that w 1/2 (t, x) ≤ 1, we get Combining these estimates, we obtain (4.3) for j ≥ 1 and k = l + 1. This completes the proof of (4.3) for j ≥ 1 and k ≥ 0.

4.4.
Conclusion for the energy estimates of the standard derivatives. If m and s are sufficiently large, (1.5) and the Sobolev embedding theorem lead to Summing up the estimates in this section, we get for each K ≥ 7.

On the energy estimates for the generalized derivatives
Throughout this section and the next one, we suppose that K is sufficiently large, and we assume that E K (T ) ≤ 1. 5.1. Direct energy estimates for the generalized derivatives. Let |δ| ≤ K − 2. Recalling (2.2), it follows that where ν = ν(x) is the unit outer normal vector at x ∈ ∂Ω and dS is the surface measure on ∂Ω.
Since G(∂u) is a homogeneous polynomial of order three, we can say that Applying the Hölder inequality and taking the L ∞ norm of the first factor, we arrive at since |δ| ≤ K − 2. Now we treat the boundary term, by means of the trace theorem. Since ∂Ω ⊂ B 1 , the norms of the generalized derivatives on ∂Ω are equivalent to the norms of the standard derivatives. Hence for all t ∈ (0, T ) we have Moreover, by the trace theorem and (4.6), we see that because of the assumption |δ| ≤ K − 2. Here we put R 0 (s) = s + s 2 .

From (5.5), it follows that
On the other hand, for |δ| ≤ K − 14 it holds Summing up these estimates and integrating (5.1), we get We repeat the above procedure once again with 1 ≤ |δ| ≤ K − 20. Being |δ| + 6 ≤ K − 14, from (5.6) we have A |δ|+4 (t) t 2η E 3 K (T ). In turn this implies In this case I δ (t) ≤ t −1 E 4 K (T ). After integration we get 1≤|δ|≤K−20 This estimate is the best we can obtain with our methods due to the estimate of I δ (t).

Proof of pointwise estimates
In this section, we go back to the Neumann problem (2.6) and will prove Theorem 2.1 by combining the decay estimates for the Cauchy problem in R 2 and the local energy decay estimate through the cut-off argument.

Then we have
For the proof, we refer to [K07].
Observe that the first term on the right-hands side of (7.3) can be evaluated by applying the decay estimates for the whole space case. In contrast, the local energy decay estimates for the mixed problem work well in estimating S j [ u 0 , f ] for 1 ≤ j ≤ 4, because we always have some localized factor in front of the operators L, S and in their arguments. 7.2. Known estimates for the 2D linear Cauchy problem. In this subsection we recall the decay estimates for solutions of homogeneous wave equation. Since ΛK 0 [v 0 , v 1 ] = K 0 [Λv 0 , Λv 1 ] by (2.2), we find that Proposition 2.1 of [Ku93] leads to the following.
The following two lemmas are proved for m = 0 in [D03]. For the general case, see [K12].
≡ 0 for |x| ≥ a and satisfying also the compatibility condition of order 0, that is to say, ∂ ν φ 0 (x) = 0 for x ∈ ∂Ω (see for instance Lemma 2.1 of [SS03]; see also Morawetz [M75] and Vainberg [V75]). Now let (u 0 , u 1 , f ) ∈ X a (T ) with some a > 1. Let u j for j ≥ 2 be defined as in Definition 2.1. Then, by Duhamel's principle, it follows that for any γ ∈ (0, 1]. In conclusion for any j ∈ N * , we have In order to evaluate ∂ α S[Ξ] for 2 ≤ |α| ≤ m, we have only to combine (7.19) with a variant of (2.4) : where 1 < b < b ′ and ϕ ∈ H m (Ω) with m ≥ 2; we can easily obtain (7.20) from (2.4) by cutting off ϕ for |x| ≥ b ′ . In order to complete the proof, one has to apply this inequality recalling the equation Invoking (7.19), we finally get the basic estimate (7.17).
7.4. Proof of Theorem 2.1. The following lemma is the main tool for the proof of Theorem 2.1.
and for any 0 ≤ η < ρ we have Proof. First we note that for any smooth function h : [0, T ) × Ω → R such that supp h(t, ·) ⊂ B R for any t ∈ [0, T ) and suitable R > 1, it holds that Clearly the same estimate holds for h : [0, T ) × R 2 → R.
Finally we prove (7.24) by using (7.8) and (7.11). It follows that Observe that the logarithmic term on the left-hand side is equivalent to a constant when x ∈ Ω b . Thus we get (7.24), because our assumption ensures that support of data and supp g(t, ·) are contained in Ω. This completes the proof. Now we are in a position to prove Theorem 2.1.
Remark 7.1. The main difference between the Dirichlet and the Neumann boundary cases is in the logarithmic loss in the local energy decay estimate (7.16). Due to this term, comparing our result with the one in [K12], we see that the estimates for S 2 [Ξ] and S 4 [Ξ] are worse in the Neumann case.

Appendix: A local existence theorem of smooth solutions
Here we sketch a proof of the following local existence theorem for the semilinear case (for the general case, see [SN89]). We underline that the convexity assumption for the obstacle is not necessary for the local existence result.  Let v j for j ≥ 0 be given as in Definition 1.1. First we show the following result.
for m ≥ 2. We show the existence of u by constructing an approximate sequence u (n) ⊂ Y m+2 T , and proving its convergence for suitably small T > 0. Throughout this proof, C M denotes a positive constant depending on M , but being independent of T . In order to keep the compatibility condition, we need to choose an appropriate function for the first step: for a moment, we suppose that we can choose a function u (0) ∈ Y m+2 Now we are in a position to show that u (n) converges to a local solution of (1.1) on [0, T ] with appropriately chosen T . For simplicity of description, we put for 0 ≤ k ≤ m + 2. Note that we have h Y m+2 T sup t∈[0,T ] m+2 k=0 |||h(t)||| k . We also set G n (t, x) = G ∂u (n) (t, x) for n ≥ 0. Combining the elementary inequality with the standard energy inequality for ∂ j t u (n) with 0 ≤ j ≤ m + 1, we get for a multi-index β and using the elliptic estimate, given in Lemma 2.2, we have for 2 ≤ k ≤ m + 2. By induction we get control of |||u (n) (t)||| k for 0 ≤ k ≤ m + 2, and obtain T for all n ≥ 1, provided that T ≤ T ′ M . Now we see that if T ≤ T ′ M , then {u (n) } is a Cauchy sequence in Y m+2 T , and there is u ∈ Y m+2 T such that lim n→∞ u (n) − u Y m+2 T = 0. It is not difficult to see that this u is the desired solution to (1.1).
Uniqueness can be easily obtained by the energy inequality.
Theorem A.1 is a corollary of Lemma A.1.

Proof of Theorem A.1.
The assumption on the initial data guarantees that for each m ≥ 3, there is a positive constant M m such that φ H m+2 (Ω) + ψ H m+1 (Ω) ≤ M m . Hence, by Lemma A.1, there is T m = T (m, M m ) > 0 such that (1.1) admits a unique solution u ∈ Y m+2 Tm . Note that we may take T 3 = T (3, R). We put (A.7) Our aim is to prove that (1.1) admits a solution u ∈ m≥3 Y m+2 T 3 . Then the Sobolev embedding theorem implies that u ∈ C ∞ [0, T 3 ] × Ω , which is the desired result. For this purpose, we are going to prove the following a priori estimate: for each m ≥ 3, if u ∈ Y m+2 T is a solution to (1.1) with some T ∈ (0, T 3 ], then there is a positive constant C m , which is independent of T , such that (A.8) Once we obtain this estimate, by applying Lemma A.1 repeatedly, we can see that u ∈ Y m+2 Now the Gronwall Lemma implies l+3 k=0 |||u(t)||| k ≤ C(1 + T 3 ) exp CC 2 l (1 + T 3 )T 3 =: C l+1 for 0 ≤ t ≤ T (≤ T 3 ), which implies u Y l+3 T ≤ C l+1 for 0 ≤ T ≤ T 3 . This completes the proof of (A.8).