Global solvability for the Kirchhoff equations in exterior domains of dimension three

Transcription

1 J. Differential Equations 21 ( Global solvability for the Kirchhoff equations in exterior domains of dimension three Taeko Yamazaki Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba, , Japan Received 29 January 24; revised 2 September 24 Available online 2 December 24 Abstract We consider the initial (boundary value problem for the Kirchhoff equations in exterior domains or in the whole space of dimension three, and show that these problems admit timeglobal solutions, provided the norms of the initial data in the usual Sobolev spaces of appropriate order are sufficiently small. We obtain uniform estimates of the L 1 (R norms with respect to time variable at each point in the domain, of solutions of initial (boundary value problem for the linear wave equations. We then show that the estimates above yield the unique global solvability for the Kirchhoff equations. 24 Elsevier Inc. All rights reserved. MSC: primary 35L7; secondary 35L2 Keywords: Kirchhoff equation; Wave equation; Quasilinear hyperbolic equation; Exterior domain. Introduction We consider the global solvability of the following initial boundary value problem for quasilinear hyperbolic equations of Kirchhoff type for initial data in the usual Sobolev spaces: 2 u ( 2 t 2 = m u 2 L Δu in [, Ω, (.1 2 address: yamazaki@ma.noda.tus.ac.jp /$ - see front matter 24 Elsevier Inc. All rights reserved. doi:1.116/j.jde

2 T. Yamazaki / J. Differential Equations 21 ( u u(,x= u (x, t (,x= u 1(x in Ω, (.2 u(t, x = on [, Ω, (.3 where Ω be an exterior domain in R n with smooth and compact boundary Ω, or Ω = R n. Throughout this paper we pose that the function m satisfies m(λ C 1 ([, and inf λ m(λ(= c >. (.4 Here we note that the decay of the L p (Ω norm (p >2 of the solution u as t does not directly imply that m( u 2 2 Δ is a small perturbation of m( u L Δ. L 2 This fact is used for proving the global solvability of semilinear wave equations 2 u t 2 Δu + F(x, u = f(t,x. In case Ω = R n, the global solvability was proved for small initial data satisfying some decay conditions as x. Greenberg Hu [7] first showed the unique global solvability for small initial data with some decay condition in case n = 1, by introducing a transformation from the solution into a pair of unknown functions. D Ancona and Spagnolo [3] generalized the result of [7] for arbitrary n and more general m. D Ancona and Spagnolo [5] proved the global solvability of (.1 (.2 with Ω = R 3 for initial data (u,u 1 with small u H 1,2 + u 1 H 1,2 norm. Here, H s,k (s R,k denotes the Hilbert space defined by H s,k ={f ; x k D s f L 2 (R n } with the norm f H s,k = x k D s f L 2, where x =(1 + x 2 1/2 and D s f = F 1 [ ξ s f ˆ]. In case n = 1, Rzymowski [16] relaxed the assumption of [7]. He showed the unique global solvability for initial data (u,u 1 C 3 (R C 2 (R such that x u, xx u, xxx u,u 1, x u 1, xx u 1 C L 1 (R, where C L 1 (R denotes the set of allintegrable continuous functions tending to as x, and that xx u x u L 1 + x u x u L x u 1 u L 1 1 is sufficiently small, where denotes convolution. In case Ω is an exterior domain, Racke [15] first showed the global solvability, and Heiming [8,9] improved Racke s result. They obtained a smallness condition on the generalized Fourier transform of the initial data sufficient for the unique global solvability. In a previous paper [19], we gave smallness conditions on the usual Sobolev norm of the initial data, sufficient for the unique global solvability, where Ω is an exterior domain in R n or the whole space R n, for dimension n 4. In [19], we derived the unique global solvability for the Kirchhoff equation from the fact that the L p -norm (p >2(n 1/(n 3 of the solution of the linear wave equation decays with respect to the time variable t of order (n 1(1/2 1/p > 1. We cannot use the method in [19] in case n = 3, since (n 1(1/2 1/p = 1 2/p 1 for every p 2, and moreover we cannot expect the decay of order higher than one even in the case Ω = R 3. In this paper, we first reduce the unique global solvability of the Cauchy problem for the abstract Kirchhoff equation for sufficiently small initial data in certain function

3 292 T. Yamazaki / J. Differential Equations 21 ( space to the integral estimate x Ω v(t, x dt C f Z, (.5 with respect to a certain norm Z of the unique solution v(t, x of the corresponding Cauchy problem for the linear abstract hyperbolic equation with initial value (v(, t v( = (f, i( Δ 1/2 f. Next we prove that the estimate (.5 is satisfied by the unique solution of the initial (boundary value problem for the wave equation in exterior domains in R 3 or whole space R 3. Then, combining these facts, we obtain some sufficient conditions on the usual Sobolev norm of the initial data, for the unique global solvability of the initial (boundary value problem. 1. Results First, we introduce notations used in this paper. For a closed operator B in a Banach space, let D(B and R(B denote the domain of B and the range of B, respectively. For 1 <p< and s, let W s p (R3 = { } f S (R 3 F 1 [ ξ s f ˆ] L p (R 3, with the norm f W s p = F 1 [ ξ s f ˆ ]. For a domain Ω in R3 with smooth boundary, L p let W s p (Ω = {f } g Wp s (R3 such that g Ω = f, { } with the norm f W s p (Ω = inf g W s p (R 3 g Ω = f, and let Wp, s (Ω be the completion of C (Ω with respect to the norm Wp s (Ω. In case p = 2, W2 s(r3, W2 s(ω and Wp, s (Ω are denoted by H s (R 3, H s (Ω and H s (Ω, respectively. For k N {}, let W1 k (Ω = f L1 (Ω D α f(x L 1 (Ω, α k with the norm f W k 1 = α k D α f(x L 1 (Ω.

4 T. Yamazaki / J. Differential Equations 21 ( Following Shibata Tsutsumi [18], let S(t,x; d = S(d denote a solution of the mixed problem: 2 u Δu = f in R Ω, t2 u(t, x = on R Ω, u u(,x= (x, (,x= ψ(x t in Ω for the data d = (, ψ,f. Let S (t, x; d = S (d (d= (, ψ,f denote a solution of the Cauchy problem, 2 v t 2 Δv = f in R R3, (1.1 v v(,x= (x, t (,x= ψ(x in R3. Definition 1. Let L 2 be an integer. Let (u,u 1 H L (Ω H L 1 (Ω, and let g 2 i= C i ([, ; H L 1 i (Ω. Define u j successively by u j := Δu j 2 + ( j 2 t g(,x (j 2. The data (u,u 1,g is said to satisfy the compatibility condition of order L 1 for u = g in Ω, if u j H 1 (Ω (j =, 1,...,L 1, u L L 2 (Ω. Hence, in case g =, the data (u,u 1, satisfy the compatibility condition of order L 1 especially if u H L(Ω and u 1 H L 1 (Ω H L 2 (Ω for odd L, and u H L (Ω H L 1 (Ω and u 1 H L 1 (Ω for even L. It is well known that there exists a unique solution S(t,x; d 2 i= C i (R; H L i (Ω if d = (u,u 1,g satisfy the compatibility condition of order L 1. (See Mizohata [13] and Ikawa [1]. Definition 2. An exterior domain Ω with smooth boundary is said to be non-trapping if the following is satisfied: Let G(t,x,y = S(t,x,d(y for d(y = (, δ( y,, where δ is the Dirac delta function and y is an arbitrary point in Ω. Let a and b be arbitrary positive constants with a b such that Ω {x R n ; x <a}. Then there exists a positive number T depending only on n, a, b and Ω such that ( G(t, x, yv(y dy C [T, Ω b Ω for every v such that p v Ω a, where Ω r ={x Ω x <r}.

5 294 T. Yamazaki / J. Differential Equations 21 ( It is known that if the complement of Ω is star-shaped, then Ω is non-trapping (see Lax and Phillips [12, Chapter V, Proposition 3.1]. Now we state our results. Theorem 1. Let n be a positive integer, and let Ω be a domain in R n with smooth boundary. Let A be a non-negative self-adjoint operator in H = L 2 (Ω. Let Z be a Banach space with norm Z contained in D (Ω such that Z L 2 (Ω is dense in L 2 (Ω. Let G be a subset in D(A Z. Let a 1, a 2, a 3 and a 4 be real numbers such that a 1 2 3, a 2 1, a 3 1 and a Assume that there exists a positive constant C such that for every f G, the following Cauchy problem for the linear abstract hyperbolic equation 2 v + Av =, t R, t2 v( = f, v (1.2 t ( = ia1/2 f has a unique solution v i=,1,2 ( C i R; D(A 1 i/2, satisfying the estimate v(t, x dt C f Z. (1.3 x Ω Then there exists a positive number δ such that the following holds: Assume that (u,u 1 D(A D(A 1/2, A a 1u,A a 2u,A a 3u 1,A a 4u 1 G, A 3/2 a 1u,A 1 a 3u, A 1 a 2u 1,A 1/2 a 4u 1 L 1 (Ω and A a 1 u ZA 3/2 a 1 L u + A a 2 u ZA 1 a 2 L u 1 1 (Ω 1 (Ω + A a 3 u 1 Z A 1 a 3 u L 1 (Ω + A a 4 u 1 Z A 1/2 a 4 u 1 L 1 (Ω δ, (1.4 then the following Cauchy problem for the abstract hyperbolic equation of Kirchhoff type 2 u 2 ( A t 2 + m 1/2 u 2 L 2 (Ω Au =, t R, u( = u, u (1.5 t ( = u 1,

6 has a unique global solution Furthermore, we have T. Yamazaki / J. Differential Equations 21 ( u i=,1,2 ( C i R; D(A 1 i/2. ( d A u(t dt m 1/2 2 dt <. (1.6 L (Ω 2 Next, we show that the unique solution v(t, x of initial (boundary value problem for the wave equation in R 3 satisfies (1.3. By using the well-known exact formula of the solution of the linear wave equation in R 3, we can show the following: Theorem 2. Let p, p, q and q be real numbers such that 1 < q <3/2 <q 2 p < 3 < p. Let M be a non-negative integer. Then there exists a constant C = C(M such that for every initial data (, ψ ( ( W M+1 q (R 3 H M+2 (R 3 W M q (R3 H M+1 (R 3, the unique global solution v(t, x = S (t,, d (d = (, ψ, satisfies the estimate x R 3 α M D α v(t, x dt C( W M + p W + W M + W + ψ M p q M q W M + ψ q W. (1.7 M q Combining Theorem 2 with M = and Theorem 1, we obtain the following sufficient condition for the unique global solvability of Kirchhoff equation in R 3. Theorem 3. Let p, p, q and q be the same constants in Theorem 2. Then there exists a positive constant δ such that the following holds: If the initial data (u,u 1 ( W 2 1 (R3 H 3 (R 3 2 (1.8 satisfies ( u W 1 p + u W 2q + u W 2 q + u 1 W 1 p + u 1 W 2q + u 1 W 2 q ( u W + u 1 21 W < δ, (1.9 21

7 296 T. Yamazaki / J. Differential Equations 21 ( then the Cauchy problem for the Kirchhoff equation (.1 (.2 with Ω = R 3 has a unique global solution u i=,1,2 ( C i R; H 3 i (R 3. Furthermore, we have t R d ( dt m u(t 2 L 2 dt <. Moreover, if (u,u 1 H L (R 3 H L 1 (R 3 for L 3, then u i=,1,2 ( C i [, ; H L i (R 3. Remark 1. D Ancona and Spagnolo [5] (see also [3,4] proved the global solvability of (.1 (.2 with Ω = R n (n N for initial data (u,u 1 with small u H 1,2 + u 1 H 1,2 norm, where f H 1,2 = x 2 D f L 2 (see Introduction of this paper. Our assumption on the initial data is different from that by D Ancona-Spagnolo [3 5] in the sense that we put the assumption of the smallness of the initial data of the usual Sobolev norm whereas D Ancona-Spagnolo put that of weighted Sobolev norm. And neither assumption implies the other. We obtain the following estimate for the unique solution of the linear wave equation in exterior domains. Theorem 4. Let Ω be a non-trapping exterior domain in R 3 with smooth boundary. Let p, p, q and q be the same constants in Theorem 2. Then there exists a constant C such that the following holds: Assume that the initial data (, ψ satisfy the following: W 7 p (Ω W 7 p (Ω H 1 (Ω, (W 7 q (Ω W 7 q (Ω3, ψ W 7 q (Ω W 7 q (Ω H 1 (Ω. Then v(t, x = S(t, ; d with d = (, ψ, satisfies the estimate v(t, x dt x Ω C( W 7 + p W + 7 p W 7 + q W + ψ 7 q W 7 + ψ q W. 7 q (1.1

8 T. Yamazaki / J. Differential Equations 21 ( Combining Theorems 1 and 4, we obtain the unique global solvability for the Kirchhoff equation in exterior domains: Theorem 5. Let Ω be a non-trapping exterior domain in R 3 with smooth boundary. Let p, p, q and q be the same constants as in Theorem 2. Then there exists a positive constant δ such that the following holds: If the initial data (u,u 1 ( W 8 p, (Ω W 9 q, (Ω W 9 q, (Ω W 2 1 (Ω 2 (1.11 satisfies ( u W 8 p + u W 9q + u W 9 q + u 1 W 7 p + u 1 W 8q + u 1 W 8 q ( u W + u 1 21 W < δ, ( then the mixed problem for the Kirchhoff equation (.1 (.3 has a unique global solution u i=,1,2 ( C i R; H 8 i (Ω. Furthermore, we have t R d ( dt m u(t 2 L 2 <. Moreover, if (u,u 1 H L (Ω H L 1 (Ω for L>8, then u i=,1,2 ( C i R; H L i (Ω. 2. Proof of Theorem 1 If u(t is a solution of (1.5, then w(t = u( t is a solution of (1.5 with u 1 replaced by u 1. Thus, it suffices to prove Theorem 1 for t. Local solvability: As is stated in [19], the local solvability of the Cauchy problem (1.5 is shown by Arosio Garavaldi [1] (see also Arosio Panizzi [2] as follows: For a solution of u of problem (1.5, define the energy of order α as ( A E α (u, t := m 1/2 2 A u(t L (Ω2 α/2 u(t 2 A + (α 1/2 u (t 2 L 2 (Ω 2 L 2 (Ω

9 298 T. Yamazaki / J. Differential Equations 21 ( and the initial energy as E ( α (u,u 1 := m ( A 1/2 u The Hamiltonian H is L 2 (Ω ( A H (u, t := M 1/2 u(t 2 2 A α/2 2 A u + (α 1/2 2 u 1 L 2 (Ω 2 L 2 (Ω + u (t 2 L 2 (Ω, L 2 (Ω. where ρ M(ρ = m(s 2 ds (ρ. Then we have H (u, t constant. Theorem A (Arosio-Garavaldi [1]. Let m Lip loc ([, + and m ν >. Then there exists T>which depends only on c in (.4, the Hamiltonian H and the initial energy E ( 3/2 (u,u 1, such that if α 3/2, u D(A α/2 and u 1 D(A (α 1/2, then the Cauchy problem (1.5 admits a unique solution in the space i=,1 Ci ([,T; D(A (α i/2. Moreover, the solution can be uniquely extended to a maximal solution uin i=,1 Ci ([,T u ; D(A (α i/2, and at least one of the following statements is valid: T u =+ E 3/2 (u, t + as t (T u. A priori Estimate: Let u i=,1 C i ([,T u ; D(A (2 i/2 be the unique maximal solution of (1.5 given by Theorem A. We use the same transformation as in [19], which is analogous to that in Greenberg Hu [7] and D Ancona- Spagnolo [3,5], where they used the transformation expressed by ξf(u or ξ F(u instead of A 1/2. Put ( A c(t = m 1/2 u(t 2 L 2 (Ω

10 T. Yamazaki / J. Differential Equations 21 ( for t <T u. Since H (u, is a continuous function of (u,u 1 with respect to the metric given by A 1/2 u L 2 (Ω and u 1 L 2 (Ω, there exists a constant e 1 depending only on e such that H (u, e 1 for every (u,u 1 D(A 1/2 H such that E ( 1 (u,u 1 e. We abbreviate E α (u, t and H (u, t to E α (t and H(t respectively. As is noted in [19], the identity H(t H( for all t [,T u and the inequality M(ρ c 2 ρ imply A 1/2 u(t 2 L 2 (Ω H( c 2 for all t [,T u. (2.1 Put c 1 = m(x, c 2 = m (x. (2.2 x H(/c x H(/c Define τ(t := t V(t := c(t 1/2 e iτ(ta1/2 ( u(t t W(t := c(t 1/2 e iτ(ta1/2 ( u(t t c(s ds, (2.3 ic(ta 1/2 u(t, + ic(ta 1/2 u(t, (2.4 ψ(r, t := (A 1/2 e 2irA1/2 W(t, V (t, ψ V (r, t := (A 1/2 e 2irA1/2 V (t, V (t, ψ W (r, t := (A 1/2 e 2irA1/2 W(t, W(t (2.5 for r R, t <T u. Here we note that d A 1/2 u(t dt Then we have the following lemma. 2 L 2 (Ω = Im ψ(τ(t, t. (2.6

11 3 T. Yamazaki / J. Differential Equations 21 ( Lemma 1. Assume that (u,u 1 satisfies 4 ψ(, L 1 + 2( ψ V (, L 1 + ψ W (, L 1 < c2 4c 2 (= δ 1, we put. (2.7 Here f(,t L 1 = f(r,t dr. Assume also that c(t satisfy T c (t c(t dt 1 2 (2.8 for some T (,T u. Then we have T c (t c(t dt 1 2. (2.9 Remark 2. In [19], we proved that if ψ(,, ψ V (, and ψ W (, are small with respect to suitable norms, and if [,T (1+ t d c (t /c(t δ 3 for some T (,T u, where δ 3 is a sufficiently small positive constant, then (1 + t d c (t < δ 3. [,T c(t 2 That is, this means polynomial decay property of c (t, whereas (2.8 means the integrability of c (t. In the proof of [7,3 5], D Ancona and Spagnolo proved polynomial decay property of c (t as t. On the other hand, in the proof of [15], Rzymowski proved the integrability of c (t in case n = 1, by using a representation expressed by forward waves and backward waves. Proof of Lemma 1. As is stated in the proof of Lemma 1 in [19], wehave ψ(r, t = ψ(r, 1 t c (s ( ψv (r τ(s, + ψ 2 c(s W (r τ(s, ds + 1 t c (s s c (σ ( ψ(r τ(s + τ(σ, σ 2 c(s c(σ + ψ( r + τ(s + τ(σ, σ dσ ds. (2.1

12 T. Yamazaki / J. Differential Equations 21 ( By (2.1 and Fubini s theorem, we have ψ(,t L 1 ψ(, L t 2 ( t + c (s c(s ds c (s for t <T. This inequality and assumption (2.8 yield c(s ds( ψ V (, L 1 + ψ W (, L 1 2 ψ(, σ L 1 (2.11 σ t ψ(, σ L 1 2 ψ(, L ( ψ V (, σ<t 2 L 1 + ψ W (, L 1. (2.12 From formula (2.1 with r = τ(t, wehave T ψ(τ(t, t dt T ψ(τ(t, dt T t T t c (s c(s c (s c(s ( ψ V (τ(t τ(s, + ψ W (τ(t τ(s, ds dt s c (σ ( ψ(τ(t τ(s + τ(σ, σ c(σ + ψ( τ(t + τ(s + τ(σ, σ dσ ds dt = I 1 + I 2 + I 3. (2.13 We change the variable η = τ(t in the above. Then, since c(t dt = dη and c(t c, we have I 1 = T ψ(τ(t, dt 1 T c(t ψ(τ(t, dt = 1 τ(t ψ(η, dη c c 1 c ψ(, L 1. (2.14 Applying Fubini s theorem and changing the variable η = τ(t τ(s, wehave T t c (s ψ c(s V (τ(t τ(s, ds dt

13 32 T. Yamazaki / J. Differential Equations 21 ( T T c (s = ψ s c(s V (τ(t τ(s, dt ds 1 T c (s c c(s ds ψ V (, L 1 1 ψ V (, L 1, 2c by assumption (2.8. We can estimate T t c (s ψ c(s W (τ(t τ(s, ds dt in the same way, and we obtain I c ( ψ V (, L 1 + ψ W (, L 1. (2.15 Applying Fubini s theorem, changing the variable η = τ(t τ(s+τ(σ and using (2.8 and (2.12, we have T t c (s s c (σ ψ(τ(t τ(s + τ(σ, σ dσ ds dt c(s c(σ T s T c (s c (σ = ψ(τ(t τ(s + τ(σ, σ dt dσ ds s c(s c(σ 1 T T c (s c (σ ψ(η, σ dη dσ ds c c(s c(σ We can estimate 1 c ( T c (s c(s ds 2 ψ(, σ L 1 σ T 1 ( ψ(, c L ( ψ V (, L 1 + ψ W (, L 1. T t c (s c(s s in the same way and we obtain c (σ ψ( τ(t + τ(s + τ(σ, σ dσ ds dt c(σ I 3 1 ( ψ(, c L ( ψ V (, L 1 + ψ W (, L 1. (2.16

14 T. Yamazaki / J. Differential Equations 21 ( Substituting (2.14 (2.16 into (2.13, and using assumption (2.7, we obtain T ψ(τ(t, t dt 1 ( 2 ψ(, c L ( ψ V (, 2 L 1 + ψ W (, L 1 < c 8c 2. (2.17 From formula (2.6, inequality (2.17 and the definitions of c and c 2 together with (2.1, it follows that T c (t T c(t dt = 4 m ( A 1/2 u(t 2 L 2 (Ω ψ(τ(t, t m ( A 1/2 u(t 2 L 2 (Ω dt 1 2, and the proof of Lemma 1 is complete. We now complete the proof of Theorem 1. (i First we show that assumption (1.4 implies (2.7 in Lemma 1 by taking δ = min{c 2, 1} c 4(2 + 2c 2 C. (2.18 By the definitions (see (2.3 (2.5, we have ψ(r, = (A 1/2 e 2irA1/2 W(, V ( = c((e 2irA1/2 Au,A 1/2 u + c( 1 (e 2irA1/2 A 1/2 u 1,u 1 + i(e 2irA1/2 Au,u 1 + i(e 2irA1/2 A 1/2 u 1,A 1/2 u, (2.19 ψ V (r, = (A 1/2 e 2irA1/2 V(, V ( = c((e 2irA1/2 Au,A 1/2 u + c( 1 (e 2irA1/2 A 1/2 u 1,u 1 i(e 2irA1/2 Au,u 1 + i(e 2irA1/2 A 1/2 u 1,A 1/2 u, ψ W (r, = (A 1/2 e 2irA1/2 W(, W ( = c((e 2irA1/2 Au,A 1/2 u + c( 1 (e 2irA1/2 A 1/2 u 1,u 1 +i(e 2irA1/2 Au,u 1 i(e 2irA1/2 A 1/2 u 1,A 1/2 u.

15 34 T. Yamazaki / J. Differential Equations 21 ( We shall estimate the L 1 -norm with respect to r R, of the first term of the right-hand side of (2.19. By Fubini s theorem we have (e 2irA1/2 Au,A 1/2 u L 1 = (e 2irA1/2 A a 1 u,a 3/2 a 1 u L 1 (e 2irA1/2 A a 1 u (x(a 3/2 a 1 u (x dx dr Ω = (e 2irA1/2 A a 1 u (x(a 3/2 a 1 u (x dr dx Ω = (A 3/2 a 1 u (x (e 2irA1/2 A a 1 u (x dr dx Ω (A 3/2 a 1 u (x dx (e 2irA1/2 A a 1 u (x dr. (2.2 Ω x Ω Note that (e 2irA1/2 A a 1u (x = v(2r, x, where v(t, x is the unique solution of the linear equation (1.2 with f = A a 1u in Theorem 1. Since v(t, x satisfies (1.3, it follows from (2.2 that (e 2irA1/2 Au,A 1/2 u C A 3/2 a 1 L u L 1 A a 1 u Z. ( (Ω The other terms in (2.19 are estimated in the same way, and we obtain ψ(, L 1 C ( c( A a 1 u ZA 3/2 a 1 L u 2 + A a 2 u ZA 1 a 2 L u 1 1 (Ω 1 (Ω + A a 3 u ZA 1 a 3 L 1 u + c( 1 A a 4 u ZA 1/2 a 4 L 1 u 1. 1 (Ω 1 (Ω The terms ψ V (, L 1 and ψ W (, L 1 are estimated by the same formula. Hence, if the initial data (u,u 1 satisfy assumption (1.4 of Theorem 1 with δ defined by (2.18, then the assumption (2.7 of Lemma 1 is satisfied. (ii By (i, we can apply Lemma 1. By using the continuity of f(t:= t c (s c(s ds

16 T. Yamazaki / J. Differential Equations 21 ( with respect to t and the fact that f( =, Lemma 1 yields Tu c (s c(s ds 1 2. (2.22 It is easy to see that ( t c (s E 3/2 (t E 3/2 ( exp 2 c(s ds. (2.23 Inequalities (2.22 and (2.23 yield t<tu E 3/2 (t <. Thus by Theorem A, we see that T u =, which means the global solvability. Inequality (2.22 with T u = means ( Proof of Theorem 2: an integral estimate for the linear wave equation in R 3 Without loss of generality, we can assume that q and q are the dual exponents of p and p, respectively, that is, real numbers such that 1/p + 1/q = 1 and 1/ p + 1/ q = 1. Since S ( t,x; (, ψ, = S (t, x; (, ψ,, it suffices to prove (1.7 with replaced by. The following formula of Kirchhoff is well known for the unique solution v(t, x of the initial value problem for the linear wave equation in R 3. (See [6, Section 2.4], for example. v(t, x = 1 4πt 2 (tψ(y + (y + (y (y xds(y B(x,t (x R 3, t >, (3.1 where B(x,t is the open ball with radius t centered at x. Thus, we have 4π = v(t, x dt B(x,t ( 1 ψ(y + 1 t t 2 (y + 1 t 2 (y (y x ds(y dt ( 1 ψ(y 1 + R 3 y x y x 2 (y + 1 y x 2 (y (y x dy. (3.2

17 36 T. Yamazaki / J. Differential Equations 21 ( By using Hölder s inequality, we have 1 ψ(y dy R 3 y x ( ψ(y 1/q ( q dy ( + y x 1 y x 1 ψ(y 1/ q ( q dy 1/p y x dy p. y x 1 1/ p y x dy p. (3.3 y x 1 Since p<3 < p, it follows from (3.3 that 1 ψ(y ( dy C ψ R 3 y x L q + ψ L q. (3.4 Since q <3/2 <q, we have in the same way that 1 R 3 y x 2 (y dy ( ( + y x 1 y x 1 (y 1/ p ( p dy (y 1/p ( p dy 1/ q y x 2 q dy y x 1 y x 2q dy y x 1 1/q C( L p + L p. (3.5 In the same way as in the proof of (3.4, we have 1 R 3 y x 2 (y (y x dy C( L q + L q. (3.6 Substituting (3.4 (3.6 into (3.2, we obtain the required estimate (1.7 in case M =. Since D α u(t, x is also a solution of the wave equation with initial value (D α u(x,, t D α u(x, = (D α,d α ψ, we obtain estimate (1.7 for general non-negative integer M. 4. Proof of Theorem 3: global solvability for the Kirchhoff equations in R 3 Let A = Δ with D(A = H 2 (R 3. Take Z = L p (R 3 W 1 q (R3 W 1 q (R3, G = Z H 2 (R 3

18 T. Yamazaki / J. Differential Equations 21 ( and a 1 = a 4 = 1/2, a 2 = a 3 = in Theorem 1. First we check that assumption (1.3 of Theorem 1 is satisfied. Let f G = Z D(A = W 1 q (R3 H 2 (R 3. Then (f, ia 1/2 f ( ( W 1 q (R3 H 2 (R 3 L q (R 3 H 1 (R 3. Thus, by Theorem 2 with M = and (, ψ = (f, ia 1/2 f, the unique solution v(t, x of (1.2 satisfies the following estimate: v(t, x dt C( L p + L p + L q + L q + ψ L q + ψ L q C ( f L p + f W + f 1q W. 1 q Hence assumption (1.3 is satisfied. Since A 1/2 g W l r (R 3 C g Wr l+1 (R 3 for every g Wr l+1 (R 3 (1 <r<, l N {}, and since Ag W l 1 (R 3 C g W1 l+2 (R 3 for every g Wr l+2 (R 3 (l N {}, Theorem 1 implies the unique existence of a global solution u ( i=,1,2 Ci R; H 2 i (R 3 of the Cauchy problem for the quasilinear wave equation of Kirchhoff type (.1 and (.2 for the initial data (u,u 1 belonging to (1.8 and satisfying (1.9 for sufficiently small δ. Since the regularity of the solution follows from that of the initial data, the proof of Theorem 3 is complete. 5. Proof of Theorem 4: an integral estimate for the linear wave equation in exterior domains Shibata Tsutsumi [18] showed a local energy decay estimate. In the case n = 3 and the forcing term equals, their estimate is stated as follows: Theorem B (Shibata and Tsutsumi [18, Lemma Ap.4 and Proof of Lemma 4.3]. (Local energy decay. Let n = 3. Let γ,a and b be any real numbers with < γ 2 and a,b r. Let M( 2 be an integer. Let u, u 1 be functions satisfying the conditions: (i u H M (Ω, u 1 H M 1 (Ω, (ii (u,u 1 satisfies the compatibility condition of order M 1 for u = in Ω, (iii p u i Ω a, i =, 1. Then there exist positive constants c = c(a, b, Ω and C = C(M,a,b,Ω such that S(t,, d H M (Ω b Ce ct ( u H M (Ω + u 1 H M 1 (Ω (5.1 for all t, d = (u,u 1,.

19 38 T. Yamazaki / J. Differential Equations 21 ( Shibata Tsutsumi [17,18] obtained the L p L q decay estimate of the solution of the linear wave equation in exterior domains Ω, by the combination of the local energy decay estimate and the decay estimate of the solution in R n through the cut-off argument. Here, using the integrability of the solution in R 3 (Theorem 2 instead of the L p L q decay estimate of the solution in R 3 in their argument, we prove Theorem 4. Proof of Theorem 4. Since d = (u,u 1, satisfies the compatibility condition of order 1, there exists a unique solution 2 S(t,x; d C i (R; H 2 i (Ω. i= Let r be a fixed positive constant such that Ω {x R 3 ; v r }. Choose ρ C (R3 such that ρ 1 and that ρ(x = { 1 when x Br +1, when x R 3 \ B r +2, respectively. Then by the uniqueness of solutions, we have S(t,x; d = S(t,x; d + S(t,x; d, (5.2 where d = ((1 ρ,(1 ρψ, d = (ρ, ρψ,. 1. Estimate of S(t,x; d Choose μ 1 C (R3 such that μ 1 1 and that { 1 when x R μ 1 (x = 3 \ B r +1, when x B r. Then we have S(t,x; d = μ 1 (xv(t, x + w(t, x, (5.3

20 T. Yamazaki / J. Differential Equations 21 ( where v(t, x = S (t, x; d, w(t, x = S(t,x; d 1, d 1 = (,,h 1, h 1 = 2 μ 1 v (Δμ 1 v. From Theorem 2, it follows that μ 1 (xv(t, x dt x R 3 ( C(M,μ 1 + Lp + L Lq + L + ψ + p q Lq ψ L. q (5.4 Next we estimate w(t, x = S(t,x; d 1 divided into inside and outside. (1 Inside estimate (x Ω r +1 ofw(t, x = S(t,x; d 1 By Sobolev s imbedding theorem, we have w(t, x dt w(t, L (Ω r +1 dt x Ω r +1 C w(t, H 2 (Ω r +1 dt. (5.5 By Duhamel s principle, we can write as where w(t, x = t S(t s, x,d 2 (s ds, (5.6 d 2 (s = (,h 1 (s,,. Since the port of h 1 is included in Ω r +1\Ω, d 2 (s trivially satisfies the compatibility condition for u =. Thus, we obtain by Theorem B that w(t, H 2 (Ω r +1 t C t S(t s, x,d 2 (s H 2 (Ω r +1 ds e β(t s h 1 (s H 1 (Ω r +1 ds. (5.7

21 31 T. Yamazaki / J. Differential Equations 21 ( Substituting (5.7 into (5.5, we obtain x Ω r +1 w(t, x dt C C = C C t t s e β(t s h 1 (s H 1 (Ω r +1 ds dt e β(t s v(s H 2 (Ω r +1 ds dt e β(t s v(s H 2 (Ω r +1 dt ds v(s H 2 (Ω r +1 ds. (5.8 Using Sobolev s imbedding theorem in the last integrand in (5.8, we obtain x Ω r +1 w(t, x dt C = C α 4 = C α 4 v(s W 4 1 (Ω r +1 ds Ω r +1 C(r Ω r +1 α 4 x Ω r +1 D α v(t, x dx ds D α v(t, x ds dx D α v(s, x ds. (5.9 Substituting (1.7 with M = 4 of Theorem 2 into the above, we obtain x Ω r +1 w(t, x dt C( W 4 + p W + 4 p W 4 + q W + ψ 4 q W 4 + ψ q W. (5.1 4 q Applying the same method to D α w(t, x ( α 3 instead of w(t, x, we obtain x Ω r +1 α 3 D α w(t, x dt C( W 7 + p W + W 7 + W + ψ 7 p q 7 q W 7 + ψ q W. ( q (2 Outside estimate (x R 3 \ B r +1 ofw = S(t,x; d 1

22 T. Yamazaki / J. Differential Equations 21 ( We extend w(t, x to the whole space by w(t, x := { S(t,x; d1 when x Ω when x/ Ω. Then, by the uniqueness of the solution of the wave equation in R 3,wehave where μ 1 (xs(t, x; d 1 = μ 1 (xw(t, x = S (t, x; d 2, d 2 = (,,h 2, h 2 = 2 μ 1 w (Δμ 1 w. By Duhamel s principle, we can write as μ 1 (xw(t, x = t where d 3 (s = (,h 2 (s,. Hence we have S (t s, x; d 3 (s ds for every x R 3, (5.12 μ 1 (xw(t, x dt = = t s S (t s, x; d 3 (s ds dt S (t s, x; d 3 (s dt ds S (r, x; d 3 (s dr ds (5.13 for every x R 3. By Theorem 2 with M =, by the fact that the port of h 2 (s is included in bounded domain Ω r +1, and by Sobolev s imbedding theorem, we have S (r, x; d 3 (s dr C ( h 2 (s Lq + h 2 (s L q C w(s W 3 1 (Ω r +1 for every x R 3. Substituting (5.14 into (5.13, we have μ 1 (xw(t, x dt C w(s W 3 1 (Ω r +1 ds = C D α w(s, x dx ds α 3 Ω r +1 (5.14

23 312 T. Yamazaki / J. Differential Equations 21 ( = C α 3 Ω r +1 C(r for every x R 3. This inequality and (5.11 yield x R 3 \B r +1 x R 3 w(t, x dt μ 1 (xw(t, x dt α 3 x Ω r +1 D α w(s, x ds dx D α w(s, x ds (5.15 C( W 7 + p W + W 7 + W + ψ 7 p q 7 q W 7 + ψ q W. ( q From (5.3, (5.4, (5.1 and (5.16, it follows that x Ω S(t,x; d dt C( W 7 + p W + W 7 + W + ψ 7 p q 7 q W 7 + ψ q W. ( q 2. Estimate of S(t,x; d, (1 Inside estimate (x Ω r +3 ofs(t,x; d Since d satisfies the compatibility condition of order 1 and its port is included in Ω r +2, we have by Theorem B that S(t,x; d H 2 (Ω r +3 Ce βt ( H 2 + ψ H 1. (5.18 With the aid of Sobolev s imbedding theorem, the above inequality yields x Ω r +3 S(t, ; d dt C C S(t, ; d L (Ω r +3 dt S(t, ; d H 2 (Ω r +3 dt e βt dt ( H 2 + ψ H 1 C ( H 2 + ψ H 1. (5.19

24 T. Yamazaki / J. Differential Equations 21 ( (2 Outside estimate (x R 3 \ B r +3 ofs(t,x; d Choose μ 2 C (R3 such that μ 2 1 and that Put μ 2 (x = { 1 when x R 3 \ B r +3 when x B r +2 w 2 (x := { S(t,x; d when x Ω when x/ Ω. Since μ 2 (x = on the port of the data d, we have by the uniqueness of the solution in R 3 that μ 2 (xw 2 (t, x = S (t, x; d 4, (5.2 where d 4 = (,,h 3, h 3 = 2 μ 2 w 2 (Δμ 2 w 2. By Duhamel s principle, we can write as μ 2 (xw 2 (t, x = t where d 5 (s = (,h 3 (s,. Hence we have S (t s, x; d 5 (s ds, (5.21 μ 2 (xw 2 (t, x dt = = t s S (t s, x; d 5 (s ds dt S (t s, x; d 5 (s dt ds S (r, x; d 5 (s dr ds (5.22 for every x R 3. By Theorem 2 together with the fact that the port of h 3 (s is included in a bounded domain Ω r +3 and that q <q 2, we have S (r, x; d 5 (s dr C ( h 3 (s Lq + h 3 (s L q C w 2 (s H 1 (Ω r +3 (5.23

25 314 T. Yamazaki / J. Differential Equations 21 ( for every x R 3. From this inequality and (5.18, it follows that S (r, x; d 5 (s dr Ce βs ( H 2 + ψ H 1 for every x R 3. (5.24 Substituting (5.24 into (5.22, we obtain x R 3 \B r +3 S(t,x; d dt x R 3 μ 2 (xw 2 (t, x dt C( H 2 + ψ H 1. (5.25 It follows from (5.19 and (5.25 that x Ω S(t,x; d dt C( H 2 + ψ H 1. (5.26 Inequalities (5.17 and (5.26 together with (5.2 imply (1.1 of Theorem Proof of Theorem 5: global solvability for the Kirchhoff equations in exterior domains By using Theorem 4, we apply Theorem 1 to the mixed problem for the quasi-linear hyperbolic equation of Kirchhoff type (.1 (.3 in an exterior domain. Let A = Δ with D(A = H 2 (Ω H 1 (Ω. Take Z = W 7 p (Ω W q 8 (Ω W 8 q (Ω, G = W 7 p, (Ω W q, 8 (Ω W 8 q, (Ω and a 1 = a 4 = 1/2, a 2 = a 3 = in Theorem 1. We shall check that the assumption of Theorem 1 are satisfied. Let 1 <r<. Let A r = Δ with domain D(A r = W 2 r (Ω W 1 r, (Ω. Then A r is a densely defined closed operator in L r (Ω. In case r = 2, A 2 equals A. Its fractional power A 1/2 r is determined by the closure of A 1/2 r x = 1 π λ 1/2 (λi + A r 1 A r xdλ

26 T. Yamazaki / J. Differential Equations 21 ( for x D(A r (see Komatsu [11]. Hence A 1/2 r x = A 1/2 r x for x D(A r D(A r. As is stated in the proof of Theorem 4 of [19], we have the following fact by using Prüss Sohr [14]; r g Wr l A (Ω and 1/2 r g C g W l r (Ω Wr l+1 (Ω (6.1 A 1/2 for every g Wr, l+1 (Ω, where l is an arbitrary non-negative integer. (In [19], the number r is assumed to satisfy 1 <r 2. But this assumption is not necessary. Let f G. Then, since f D(A p D(A q D(A q D(A, we have A 1/2 p f = A1/2 q f = A 1/2 q f = A 1/2 f. Thus, by (6.1, we have A 1/2 A f = 1/2 W 6 p (Ω p f W C f 6 p (Ω W 7 p (Ω, A 1/2 A f = 1/2 f C f W 7 r (Ω W 7 W r (Ω r 8(Ω (r = q, q. (6.2 r Hence (f, ia 1/2 f (W 7 p (Ω W q 8 (Ω W 8 q (Ω H 2 (Ω (W q 7 (Ω W 7 q (Ω H 1 (Ω. (6.3 Thus, by Theorem 4, the unique solution v(t of (1.2 satisfies the following estimate: x Ω v(t, x dt ( C f W 7 p + f W + f 7 p W 7 q + f W + ia 1/2 f + ia 1/2 f 7 q W 7 q W 7 q C ( f W + f 7 p W + f 8q W. 8 q Hence we see that the assumption of Theorem 1 is satisfied for sufficiently small δ >. By (6.1, initial condition (1.11 and (1.12 for sufficiently small δ > imply assumption (1.4 of Theorem 1. Thus, by Theorem 1, we obtain a unique global solution u i=,1,2 ( C i R; H 2 i (Ω of the mixed problem for the quasilinear wave equation of Kirchhoff type (.1 (.3.

27 316 T. Yamazaki / J. Differential Equations 21 ( Since H L D(AL/2, the regularity of the initial data yields the regularity of the solution by Theorem A, and the proof of Theorem 5 is complete. Acknowledgments The author expresses her sincere gratitude to the referee for valuable suggestions and comments. References [1] A. Arosio, G. Garavaldi, On the mildly degenerate Kirchhoff string, Math. Methods Appl. Sci. 14 ( [2] A. Arosio, S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 ( [3] P. D Ancona, S. Spagnolo, A class of nonlinear hyperbolic problems with global solutions, Arch. Rational Mech. Anal. 124 ( [4] P. D Ancona, S. Spagnolo, Nonlinear perturbations of the Kirchhoff equation, Comm. Pure Appl. Math. 47 ( [5] P. D Ancona, S. Spagnolo, Kirchhoff type equations depending on a small parameter, Chinese Ann. Math. 16B (4 ( [6] L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, [7] J.M. Greenberg, S.C. Hu, The initial-value problem for a stretched string, Quart. Appl. Math. 5 ( [8] C. Kerler ( = C. Heiming, Differenzierbarkeit im Bild und Abbildungseigenschaften verallgemeinerter Fouriertransformationen bei variablen Koeffizienten im Außengebiet und Anwendungen auf Gleichungen vom Kirchhoff-Typ, Ph.D. Thesis, Universität Konstanz, [9] C. Heiming, Mapping properties of generalized Fourier transforms and applications to Kirchhoff equations, Nonlinear Differential Equations Appl. 7 ( [1] M. Ikawa, Mixed problems for hyperbolic equations of second order, J. Math. Soc. Japan 2 ( [11] H. Komatsu, Fractional powers of operators, Pacific J. Math. 1 ( [12] P.D. Lax, R.S. Phillips, Scattering Theory, revised ed., Academic Press, San Diego, [13] S. Mizohata, Quelques problèmes au bord, du type mixte, pour d équations hyperboliques, Séminaire sur les équations aux dérivées partielles, Collège de France, 1966/67, pp [14] J. Prüss, H. Sohr, Imaginary powers of elliptic second order differential operators in L p -spaces, Hiroshima Math. J. 23 ( [15] R. Racke, Generalized Fourier transforms and global, small solutions to Kirchhoff equations, Appl. Anal. 58 ( [16] W. Rzymowski, One-dimensional Kirchhoff equation, Nonlinear Anal. 48 ( [17] Y. Shibata, Y. Tsutsumi, Global existence theorem of nonlinear wave equation in the exterior domain, in: M. Mimura, T. Nishida (Eds., Proceedings of the Conference on Recent Topics in Nonlinear Partial Differential Equations, Hiroshima, 1983, Lecture Note in Numerical and Applied Analysis, vol. 6, Kinokuniya/North-Holland, Tokyo/Amsterdam, 1984, pp [18] Y. Shibata, Y. Tsutsumi, On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain, Math. Z. 191 ( [19] T. Yamazaki, Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three, Math. Methods Appl. Sci. 27 (