Global regularity for supercritical nonlinear dissipative wave equations in 3D

Nonlinear Analysis 152 (2017) 183–195

Contents lists available at ScienceDirect

Nonlinear Analysis www.elsevier.com/locate/na

Global regularity for supercritical nonlinear dissipative wave equations in 3D Kyouhei Wakasa a,∗ , Borislav Yordanov b,c a

College of Liberal Arts, Mathematical Science Research Unit, Muroran Institute of Technology, 27-1, Mizumoto-cho, Muroran, Hokkaido 050-8585, Japan b Office of International Affairs, Hokkaido University, Kita 15, Nishi 8, Kita-ku, Sapporo, Hokkaido 060-0815, Japan c Institute of Mathematics, Sofia, Bulgaria

article

info

Article history: Received 27 June 2016 Accepted 22 December 2016 Communicated by Enzo Mitidieri MSC: primary 35L70 secondary 35A05 Keywords: Wave equation Nonlinear damping Supercritical Regularity

abstract The nonlinear wave equation utt − ∆u + |ut |p−1 ut = 0 is shown to be globally wellk (R3 ) × H k−1 (R3 ) posed in the Sobolev spaces of radially symmetric functions Hrad rad for all p ≥ 3 and k ≥ 3. Moreover, global C ∞ solutions are obtained when the initial data are C0∞ and exponent p is an odd integer. The radial symmetry allows a reduction to the one-dimensional case where an important observation of Haraux (2009) can be applied, i.e., dissipative nonlinear wave equations contract initial data in W k,q (R) × W k−1,q (R) for all k ∈ [1, 2] and q ∈ [1, ∞]. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Dissipative nonlinear wave equations are well-posed in H k (Rn ) × H k−1 (Rn ) for all k ∈ [1, 2]and n ≥ 1 under the monotonicity condition of Lions and Strauss [14]. This global result makes no exception for nonlinear dissipations of supercritical power, as determined by invariant scaling, and gets around the stringent conditions for well-posedness of general nonlinear wave equations; see Ponce and Sideris [17], Lindblad [13], Wang and Fang [26] and the references therein. The monotonicity method has been less effective in studying higher regularity. It is still an open question whether supercritical problems are globally well-posed in Sobolev spaces with index k > 2. The purpose of this paper is to give an affirmative answer when n = 3 and initial data have radial symmetry. To state the result, let u = utt − ∆u be the d’Alembertian in R3+1 and Du = (∇u, ∂t u) be the space-time gradient of u. ∗ Corresponding author. E-mail addresses: [email protected] (K. Wakasa), [email protected] (B. Yordanov).

http://dx.doi.org/10.1016/j.na.2016.12.013 0362-546X/© 2017 Elsevier Ltd. All rights reserved.

K. Wakasa, B. Yordanov / Nonlinear Analysis 152 (2017) 183–195

184

We use ∥ · ∥q , for the norm in Lq (R3 ) with q ∈ [1, ∞], and Dα , for the partial derivative of integer order α = (α1 , α2 , α3 , α4 ). We consider the dissipative nonlinear wave equation u + |ut |p−1 ut = 0,

x ∈ R3 , t > 0,

(1.1)

with the Cauchy data u|t=0 = u0 ,

ut |t=0 = u1 ,

x ∈ R3 .

(1.2)

k−1 k Our main assumptions are p ≥ 3 and (u0 , u1 ) ∈ Hrad (R3 ) × Hrad (R3 ), where the radially symmetric Sobolev spaces are defined as k Hrad (R3 ) = {u ∈ H k (R3 ) : (xi ∂xj − xj ∂xi )u(x) = 0 for 1 ≤ i < j ≤ 3}.

Clearly, such u(x) depend only on |x| = (x21 + x22 + x23 )1/2 . The radially symmetric spaces are invariant under the evolution determined by (1.1), (1.2); see [22,20]. 3 2 Theorem 1.1. Assume that p ≥ 3 and (u0 , u1 ) ∈ Hrad (R3 ) × Hrad (R3 ). Then problem (1.1), (1.2) admits a unique global solution u, such that 3−|α|

Dα u ∈ C([0, ∞), Hrad (R3 )),

|α| ≤ 3.

Corollary 1.2. The global solution of problem (1.1), (1.2), given by Theorem 1.1, satisfies the following uniform estimates: for t ≥ 0,   t ∥Dα u(s)∥2∞ ds ≤ C1 (u0 , u1 ), 0

1≤|α|≤2



∥DDα u(t)∥2 ≤ C2 (u0 , u1 ),

|α|≤2

where Cj (u0 , u1 ), j = 1, 2, are finite whenever ∥u0 ∥H 3 + ∥u1 ∥H 2 is finite. It is easy to derive the propagation of higher regularity from Theorem 1.1 and Sobolev embedding inequalities, if the nonlinearity allows further differentiation. k−1 k Theorem 1.3. Let p be an odd integer and (u0 , u1 ) ∈ Hrad (R3 ) × Hrad (R3 ) with an integer k ≥ 4. Problem (1.1), (1.2) admits a unique global solution u, such that k−|α|

Dα u ∈ C([0, ∞), Hrad

(R3 )),

|α| ≤ k.

In addition, the following estimates hold uniformly in t ≥ 0:   t (k) ∥Dα u(s)∥2∞ ds ≤ C1 (u0 , u1 ), 1≤|α|≤k−1

0



(k)

∥DDα u(t)∥2 ≤ C2 (u0 , u1 ),

|α|≤k−1 (k)

where Cj (u0 , u1 ), j = 1, 2, are finite whenever ∥u0 ∥H k + ∥u1 ∥H k−1 is finite. ∞ ∞ If (u0 , u1 ) ∈ Crad (R3 ) × Crad (R3 ) and (u0 , u1 ) are compactly supported, then problem (1.1), (1.2) admits a unique global solution ∞ u ∈ C ∞ ([0, ∞), Crad (R3 )),

such that u(·, t) is also compactly supported for all t ≥ 0.

K. Wakasa, B. Yordanov / Nonlinear Analysis 152 (2017) 183–195

185

The above results rely heavily on the dissipative nonlinearity and radial symmetry. In less favorable circumstances, global regularity is well beyond the reach of existing methods which require the nonlinearity be slightly weaker than the linear differential operator. Nonlinear conservative wave equations and Schr¨odinger equations, for example, are well-understood only in subcritical and critical cases; see Struwe [23], Grillakis [4], Shatah and Struwe [19], Nakanishi [16] for the former and Bourgain [1], Grillakis [5], Colliander, Keel, Staffilani, Takaoka and Tao [3] for the latter. The energy supercritical nonlinearities allow mostly conditional results about the equivalence of certain norms and nature of blow up, such as Kenig and Merle [9] and Killip and Visan [10]. Exceptions are the well-posedness for log-supercritical nonlinearities and radial data by Tao [24] and the well-posedness and scattering for log–log-supercritical nonlinearities by Roy [18]. Dissipative and conservative nonlinear wave equations behave very differently even in Sobolev spaces with k ≤ 2 derivatives. This is evident from the invariant scaling of (1.1): u(x, t) → uλ (x, t) = λ(2−p)/(p−1) u(λx, λt), with λ > 0. Then ∥uλ (t)∥H˙ k = λ(2−p)/(p−1)+k−3/2 ∥u(λt)∥H˙ k ,

(1.3)

so the critical space for well-posedness is H˙ kc (R3 ) × H˙ kc −1 (R3 ) with kc = 3/2 + (p − 2)/(p − 1). However, the result of [14] shows that k ∈ [1, 2] is sufficient for any p > 1. Here the monotone dissipation plays a decisive role, since (1.1) remains dissipative after applying Dα with |α| = 1: Dα u + p|ut |p−1 Dα ut = 0.

(1.4)

We can use ut and Dα ut as multipliers for (1.1) and (1.4), respectively, to obtain  t 2 ∥Du(t)∥22 + 2 ∥us (s)∥p+1 p+1 ds = ∥Du(0)∥2 , 0  t ∥DDα u(t)∥22 + 2p ∥|us (s)|(p−1)/2 Dα us (s)∥22 ds = ∥DDα u(0)∥22 . 0

As derivatives of order k ≤ 2 turn out to be a priori bounded, the corresponding well-posedness results readily follow from the monotonicity method of [14]. Two differentiations of (1.1) yield an equation that is no longer dissipative. It becomes a nontrivial task to derive uniform estimates for derivatives of order three and higher. Some information is provided by the invariant scaling, which predicts the non-concentration of second-order norms if k = 2 and (2 − p)/(p − 1) + 1/2 > 0 in (1.3). Thus, the range of subcritical exponents is p < 3. For the complementary range p ≥ 3 and k > 2, the global well-posedness of (1.1) in H k (R3 ) × H k−1 (R3 ) requires a new idea. Until recently, the proof has been known only in the critical case p = 3 with radially symmetric data [25]. This paper shows that no critical exponent exists for the regularity of dissipative nonlinear wave equations with radial symmetry. The development of singularities is prevented by the monotonicity of seminorms involving second-order derivatives. We actually obtain, after setting r = |x| and X± (r, t) = (∂t ± ∂r )(r∂t u), that max sup |X± (r, t)| ≤ max sup |X± (r, 0)|, ±

r>0

±

t ≥ 0.

r>0

Such decreasing quantities are known only for linear and dissipative nonlinear wave equations in R3 with radial data. The proof is based on differentiating (1.1) in t and rewriting the equation for ∂t u as p (∂t ∓ ∂r )X± + |∂t u|p−1 (X+ + X− ) = 0. 2 A. Haraux [6] has applied the same idea to derive W 2,∞ (R) × W 1,∞ (R) estimates in the one-dimensional case. In higher dimensions with radial symmetry, similar W 1,∞ (R) × L∞ (R) estimates are derived by Joly,

K. Wakasa, B. Yordanov / Nonlinear Analysis 152 (2017) 183–195

186

Metivier and Rauch [8], Liang [12] and Carles and Rauch [2]. Since the one-dimensional reduction does not work for general data, the global well-posedness in H k (R3 ) × H k−1 (R3 ), with k > 2, remains completely open. The rest of this paper is organized as follows. Section 2 contains several basic facts and estimates for the 3 2 wave equation with radially symmetric data in R3 . The regularity problem in Hrad (R3 )×Hrad (R3 ) is studied k−1 k 3 in Section 3. In the final Section 4, we establish Theorem 1.3 about well-posedness in Hrad (R ) × Hrad (R3 ) ∞ ∞ with k ≥ 4 and Crad (R3 ) × Crad (R3 ) with compact support. 2. Basic estimates First of all, we state the energy estimates and Strichartz estimates for the radial wave equation in R3 × R. Lemma 2.1. Let u be a solution of the Cauchy problem in R3 × R u = F,

ut |t=0 = u1 .

u|t=0 = u0 ,

(a) For any source and initial data, u satisfies the energy estimate  ∥Du(t)∥2 ≤ C(∥∇u0 ∥2 + ∥u1 ∥2 ) + C

t

∥F (s)∥2 ds 0

with an absolute constant C for all t ≥ 0. (b) For radial source and initial data, u is also a radial function which satisfies 1/2

t



∥u(s)∥2∞ ds

t

 ≤ C(∥∇u0 ∥2 + ∥u1 ∥2 ) + C

∥F (s)∥2 ds

0

0

for all t ≥ 0. More generally, if |α| = k and k ≥ 1, then 

1/2

t

∥D

α

u(s)∥2∞ ds

0

≤C

 



β

∥DD u(0)∥2 +

t

 ∥D F (s)∥2 ds . β

0

|β|≤k

Part (a) can be found in Strauss [22], H¨ ormander [7], and Shatah and Struwe [20]. The first estimate (b) is the so-called “radial Strichartz estimate” for the 3D wave equation. Klainerman and Machedon [11] have found the homogeneous version of (b) which readily implies the non-homogeneous estimate stated here. The second part, concerning Dα u, needs verification since the latter function is no longer radial. We will use the idea of [11] and the Hardy inequality for radial functions. In fact, it is sufficient to consider the Cauchy problem u = 0,

u|t=0 = 0,

ut |t=0 = g,

with a radially symmetric g, and show that if |α| = k and k ≥ 1, then  t 1/2  α 2 ∥∇ u(s)∥∞ ds ≤C ∥∇β g∥2 . 0

|β|=k

We exclude time derivatives as these preserve radial symmetry. Recall the formula  t ∇α g(x + tω)dSω ∇α u(x, t) = 4π |ω|=1

(2.1)

K. Wakasa, B. Yordanov / Nonlinear Analysis 152 (2017) 183–195

187

and representations ∇α g(x) =

k 

g (l) (|x|)ak,l (x),

|ak,l (x)| ≤ Ck,l |x|l−k .

l=1

These imply  t+|x| k  1 λl−k g (l) (λ)λdλ, |∇ u(x, t)| ≤ Ck 2|x| |t−|x|| α

l=1

so we see that ∥∇α u(·, t)∥∞ ≤ Ck

k 

M [| · |l−k+1 g (l) ](t),

l=1

where M stands for the Hardy–Littlewood maximal function. Thus, the estimate  t 1/2 k  α 2 ≤C ∥| · |l−k g (l) ∥2 ∥∇ u(s)∥∞ ds 0

l=1

2

follows from the L (R+ ) boundedness of M . To complete the proof of (2.1), we will apply the Hardy inequality for radial functions [15] to the right hand side: ∥| · |l−k g (l) ∥2 ≤ Ck−l ∥∇k−l g (l) ∥2 .  The following are useful facts concerning the local well-posedness of problem (1.1), (1.2). These results can be found in [22,7,20], Remark 5.3. Lemma 2.2. Let k ≥ 3, p > k − 1 and (u0 , u1 ) ∈ H k (R3 ) × H k−1 (R3 ). (a) There exists T > 0, such that problem (1.1), (1.2) has a unique solution u satisfying Dα u ∈ C([0, T ], H k−|α| (R3 )),

|α| ≤ k.

Moreover, we have sup ∥Dα u(t)∥2 ≤ Ck , t∈[0,T ]

where T and Ck can be chosen to depend continuously on ∥u0 ∥H k + ∥u1 ∥H k−1 . (b) The continuation principle holds: if T∗ = T∗ (u0 , u1 ) is the supremum of all numbers T for which (a) holds, then either T∗ = ∞ or sup ∥Dα u(t)∥2 = ∞ t∈[0,T∗ )

for some α with |α| ≤ k. (c) If the data (u0 , u1 ) are radially symmetric, the solution of (1.1), (1.2) is also radially symmetric. Next, we state two preliminary estimates for problem (1.1), (1.2). These results, called the energy dissipation laws, are already discussed in the introduction. Lemma 2.3. Assume that (u0 , u1 ) ∈ H 3 (R3 ) × H 2 (R3 ) and |α| = 1. Let u be the solution of problem (1.1), (1.2) for t ∈ [0, T ], given by Lemma 2.2. Then

K. Wakasa, B. Yordanov / Nonlinear Analysis 152 (2017) 183–195

188

 t 1 2 ∥Du(t)∥2 + ∥us (s)∥p+1 p+1 ds = 2 0  t 1 ∥|us (s)|(p−1)/2 Dα us (s)∥22 ds = ∥DDα u(t)∥22 + p 2 0

1 ∥Du(0)∥22 , 2 1 ∥DDα u(0)∥22 , 2

for all t ∈ [0, T ]. Thus, the following norms of u are uniformly bounded: ∥Du(t)∥2 ≤ ∥Du(0)∥2 , ∥DDα u(t)∥2 ≤ ∥DDα u(0)∥2 , |α| ≤ 1,  T (p−1)/2 α (∥us (s)∥p+1 D us (s)∥22 )ds ≤ ∥Du(0)∥22 + ∥DDα u(0)∥22 . p+1 + ∥|us (s)| 0

Proof. Multiplying Eq. (1.1) by ut , we get 0 = (u + |ut |

p−1

 ut )ut =

|Du|2 2



− div(ut ∇u) + |ut |p+1 . t

The first-order energy identity follows from the integration on R3 and divergence theorem if u(x, t) has compact support with respect to x. More generally, we can approximate (u0 (x), u1 (x)) with compactly supported C ∞ functions and use the finite propagation speed to show that the boundary integral of div(ut ∇u) is zero. Property (a) in Lemma 2.2 implies that the approximations will converge to the actual solution. Similarly, we can differentiate Eq. (1.1) and multiply with Dα ut (x, t) to show the second-order energy identity. Let us recall that we work with solutions whose third-order derivatives belong to L2 (R3 ).  Finally, we state the strong version of Strauss inequality. The original version can be found as Radial Lemma 1 of Strauss [21]. 1 Lemma 2.4 (Strong Version of Strauss Inequality [21]). Let U ∈ Hrad (R3 ). There exists a constant C > 0, such that for every R > 0

R|U (R)| ≤ C∥U ∥H 1 (|x|>R) . Therefore lim R|U (R)| = 0.

R→∞

3. Global existence of radial solutions in H 3 × H 2 The following lemma is essential for the proof of Theorem 1.1. This approach to L∞ estimates of secondorder derivatives is borrowed from Haraux [6]. 3 2 Lemma 3.1. Let (u0 , u1 ) ∈ Hrad (R3 ) × Hrad (R3 ) and let u be the local solution of problem (1.1), (1.2) constructed in Lemma 2.2. There exists a positive constant C = C(∥u0 ∥H 3 , ∥u1 ∥H 2 ), such that

sup r|∂t2 u(r, t)| ≤ C,

(3.1)

r∈[0,∞)

sup |∂t u(r, t)| ≤ C, r∈[0,∞)

for all t ∈ [0, T∗ ).

sup r|∂r ∂t u(r, t)| ≤ C, r∈[0,∞)

(3.2)

K. Wakasa, B. Yordanov / Nonlinear Analysis 152 (2017) 183–195

189

Proof. Since the solution of (1.1) is radially symmetric, the equation becomes (∂t2 − ∂r2 )(ru) + r|∂t u|p−1 ∂t u = 0,

(3.3)

where r = |x|. Let us define X± (r, t) = (∂t ± ∂r )(r∂t u). 2k−1 We differentiate (3.3) in t to obtain an equation for ∂t u. Then we multiply through by X± with k ∈ N to obtain

(∂t − ∂r ) {X+ (r, t)}2k + pr|∂t u|p−1 ∂t2 u{X+ (r, t)}2k−1 = 0 2k and (∂t + ∂r ) {X− (r, t)}2k + pr|∂t u|p−1 ∂t2 u{X− (r, t)}2k−1 = 0. 2k Adding the above two equations and using X+ (r, t) + X− (r, t) = 2r∂t2 u, we have (∂t − ∂r ) 2k (∂t + ∂r ) 2k X+ + X− 2k−1 2k−1 ) = 0. + X− X+ + X− + p|∂t u|p−1 (X+ 2k 2k 2 Here we remark that the following inequality holds for a, b ∈ R and k ∈ N: (a + b)(a2k−1 + b2k−1 ) ≥ 0. From this inequality with a = X+ and b = X− , we derive 2k 2k 2k 2k ∂t (X+ + X− ) + ∂r (X− − X+ ) ≤ 0.

It is easy to see that integration on [0, R] × [0, t] implies  R  R  t 2k 2k 2k 2k 2k 2k {X+ (r, t) + X− (r, t)}dr ≤ {X+ (r, 0) + X− (r, 0)}dr + {X+ (R, s) − X− (R, s)}ds, 0

0

2k (0, t) X−

0

2k (0, t) X+

= 0. We also notice that |X± (R, t)| → 0 as R → ∞, which is a consequence of − since Lemma 2.4. Thus, we get 2k 2k 2k ∥X+ (·, t)∥2k L2k (R+ ) + ∥X− (·, t)∥L2k (R+ ) ≤ ∥X+ (·, 0)∥L2k (R+ ) + ∥X− (·, 0)∥L2k (R+ ) .

The two terms on the right hand side satisfy 2k−2 2 ∥X± (·, 0)∥2k L2k (R+ ) ≤ ∥X± (·, 0)∥L∞ (R+ ) ∥X± (·, 0)∥L2 (R+ ) , 3 2 where ∥X± (·, 0)∥L2 (R+ ) < ∞ by Lemma 2.4 and (u0 , u1 ) ∈ Hrad (R3 ) × Hrad (R3 ). This observation yields a more convenient estimate: 2k−2 2k 2 ∥X+ (·, t)∥2k L2k (R+ ) + ∥X− (·, t)∥L2k (R+ ) ≤ ∥X+ (·, 0)∥L∞ (R+ ) ∥X+ (·, 0)∥L2 (R+ ) 2 + ∥X− (·, 0)∥2k−2 L∞ (R+ ) ∥X− (·, 0)∥L2 (R+ ) .

Letting k → ∞, we have that max{∥X+ (·, t)∥L∞ (R+ ) , ∥X− (·, t)∥L∞ (R+ ) } ≤ max{∥X+ (·, 0)∥L∞ (R+ ) , ∥X− (·, 0)∥L∞ (R+ ) }.

(3.4)

The remaining part of the proof is standard. Since X+ (r, t) + X− (r, t) = 2r∂t2 u, claim (3.1) follows from (3.4). We can write 2∂r (r∂t u) = X+ (r, t) − X− (r, t) and  r 2r|∂t u(r, t)| ≤ |X+ (ρ, t) − X− (ρ, t)|dρ ≤ Cr. 0

K. Wakasa, B. Yordanov / Nonlinear Analysis 152 (2017) 183–195

190

Hence ∥∂t u(·, t)∥L∞ (R+ ) ≤ C/2 < ∞, which is the first claim in (3.2). We finally observe that 2r∂r ∂t u(r, t) = X+ (r, t) − X− (r, t) − 2∂t u(r, t), so the second claim in (3.2) follows from the first one and (3.4). The proof is complete.  Proof of Theorem 1.1. It is sufficient to show that ∥u(t)∥H 3 + ∥ut (t)∥H 2 does not blow up as t → T∗ , where u is the local solution of (1.1) given by Lemma 2.2. The first and second order norms of u are a priori bounded from Lemma 2.3, so the global existence of u is guaranteed by the next result about third order norms. 3 2 Proposition 3.2. Assume that (u0 , u1 ) ∈ Hrad (R3 ) × Hrad (R3 ) and let u be the local solution of problem (1.1), (1.2) constructed in Lemma 2.2. There exists a positive constant C = C(u0 , u1 , T∗ ) such that  ∥Dα u(t)∥2 ≤ C(u0 , u1 , T∗ ) |α|=3

for all t ∈ [0, T∗ ). Proof. Differentiating (1.1) twice, we find that Dα u is a weak solution of Dα u + p|ut |p−1 Dα ut + p(p − 1)|ut |p−3 ut Dα1 ut Dα2 ut = 0, where α = α1 + α2 with |α1 | = 1 and |α2 | = 1. Thus, Dα u satisfies the energy estimate in Lemma 2.1(a):  t  t α ∥DDα u(t)∥2 ≤ C∥DDα u(0)∥2 + C ∥up−1 (s)D u (s)∥ ds + C ∥up−2 (s)Dα1 us (s)Dα2 us (s)∥2 ds s 2 s s 0

0

for t ∈ [0, T∗ ). We notice that ∥up−1 (s)Dα us (s)∥2 ≤ C∥Dα us (s)∥2 , s by (3.1) in Lemma 3.1, and ∥up−2 (s)Dα1 us (s)Dα2 us (s)∥2 s

  α  D 1 us (s)   ,  ≤C  |·| 2

by (3.1) and (3.2) in Lemma 3.1. Thus, α



α

∥DD u(t)∥2 ≤ C∥DD u(0)∥2 + C 0

t

  t  α1  D us (s)   ds.  ∥D us (s)∥2 ds + C   |·| α

0

2

The third term of the right hand can be estimated by Hardy’s inequality:  t  t α α α ∥DD u(t)∥2 ≤ C∥DD u(0)∥2 + C ∥D us (s)∥2 ds + C ∥DDα1 us (s)∥2 ds. 0

0

We add these estimates for all |α| = 2 to get 

∥Dα u(t)∥2 ≤ C

|α|=3



∥Dα u(0)∥ + C



t



∥Dα u(s)∥2 ds.

0 |α|=3

|α|=3

Making use of Gronwall’s inequality, we finally have   ∥Dα u(t)∥2 ≤ C ∥Dα u(0)∥eCt . |α|=3

This completes the proof of global existence.

|α|=3



K. Wakasa, B. Yordanov / Nonlinear Analysis 152 (2017) 183–195

191

Proof of Corollary 1.2. We can now verify the uniform estimates of L2 norms and square integrability of L∞ norms on [0, ∞). Notice that inequality (b) in Lemma 2.1 holds on any interval [t0 , t] for all |α| = 1. Hence we get   |α|=1

t

1/2    t ∥up−1 (s)Dα us (s)∥2 ds. ∥Dα u(s)∥2∞ ds ≤C ∥DDα u(t0 )∥2 + C s

t0

|α|=1

t0

|α|=1

It is convenient to abbreviate the left hand side as 1/2   t α 2 ∥D u(s)∥∞ ds , N1 (t) =

t ≥ t0 ,

t0

|α|=1

and estimate the integrand on the right hand side as ∥up−1 (s)Dα us (s)∥2 ≤ C∥Dα u(s)∥∞ ∥us(p−1)/2 (s)Dα us (s)∥2 , s from (3.2). Applying the Cauchy inequality on [t0 , t] to the initial estimate, N1 (t) ≤ C





α

1/2

t

∥DD u(t0 )∥2 + CN1 (t)

∥u(p−1)/2 (s)Dus (s)∥22 ds s

.

t0

|α|=1

The above integral converges on [0, ∞) by Lemma 2.3. We can find t0 , such that 

t

C t0

1/2 1 2 ≤ , ∥u(p−1)/2 (s)Du (s)∥ ds s s 2 2

t ≥ t0 .

(3.5)

 Thus, N1 (t) ≤ 2C |α|=1 ∥DDα u(t0 )∥2 for all t ≥ t0 . If t < t0 , a similar estimate for N1 (t) follows from Theorem 1.1. Hence ∥Dα u(·)∥∞ ∈ L2 ([0, ∞)) for |α| = 1. Next, we show that the second order norms in Corollary 1.2 are also uniformly bounded. Set N2 (t) =

 

1/2

t

∥D

α

u(s)∥2∞ ds

t ≥ t0 ,

,

t0

|α|=2

and apply estimate (b) in Lemma 2.1 to obtain 

t

1/2  t ∥Dα u(s)∥2∞ ds ≤ C∥DDα u(t0 )∥2 + C ∥usp−1 (s)Dα us (s)∥2 ds

t0

t0



t

+C

∥usp−2 (s)(Dus (s))2 ∥2 ds,

(3.6)

t0

where |α| = 2. Noticing that (3.2) gives ∥up−1 (s)Dα us (s)∥2 ≤ C∥us (s)∥2∞ ∥Dα us (s)∥2 , s ∥up−2 (s)(Dus (s))2 ∥2 s

≤ C∥D

α

(3.7)

u(s)∥∞ ∥u(p−1)/2 (s)Dus (s)∥2 s

(3.8)

and applying the Cauchy inequality on [t0 , t], we get N2 (t) ≤ C



∥DDα u(t0 )∥2 + C



t

t0

|α|=2



t

+ CN2 (t) t0

 ∥us (s)∥2∞ 

 

|α|=2

1/2 2 ∥u(p−1)/2 (s)Du (s)∥ ds . s s 2

∥Dα us (s)∥2  ds

K. Wakasa, B. Yordanov / Nonlinear Analysis 152 (2017) 183–195

192

We use again (3.5) to derive N2 (t) ≤ 2C



∥DDα u(t0 )∥2 + 2C





t

∥us (s)∥2∞ 

t0

|α|=2

∥Dα us (s)∥2  ds

|α|=2

for sufficiently large t ≥ t0 . Let us introduce  N3 (t) = sup ∥DDα u(s)∥2 , |α|=2

 

t ≥ t0 .

s∈[t0 ,t]

Then we can write 

N2 (t) ≤ 2C

∥DDα u(t0 )∥2 + 2CN3 (t)



t

 ∥Du(s)∥2∞ ds .

(3.9)

t0

|α|=2

We will combine the above estimate with Lemma 2.1(a) for the interval [t0 , t]:  t  t  α ∥DDα u(t)∥2 ≤ C ∥DDα u(t0 )∥2 + C ∥up−1 (s)D u (s)∥ ds + C ∥up−2 (s)(Dus (s))2 ∥2 ds, s 2 s s t0

|α|=2

t0

where t ≥ t0 . It follows from (3.7) and (3.8) that  t  ∥DDα u(t)∥2 ≤ C ∥DDα u(t0 )∥2 + C ∥Du(s)∥2∞ ∥Dα us (s)∥2 ds t0

|α|=2



t

+C

1/2  t 1/2 2 . ∥Dus (s)∥2∞ ds ∥u(p−1)/2 (s)Du (s)∥ ds s s 2

t0

t0

Summing over |α| = 2 and applying Lemma 2.3, we simplify the estimate to    t    N3 (t) ≤ C ∥DDα u(t0 )∥2 + CN3 (t) ∥Du(s)∥2∞ ds + CN2 (t)  ∥DDα u(0)∥2  . t0

|α|=2

|α|=1

Since the integral 

∞

∥Du(s)∥2∞ ds

0

is convergent, we choose a sufficiently large t0 to finally get  

N3 (t) ≤ C

∥DDα u(t0 )∥2 + CN2 (t) 

|α|=2

 

∥DDα u(0)∥2 

(3.10)

|α|=1

for t ≥ t0 . From (3.10) and (3.9), we obtain  N3 (t) ≤ C



∥DDα u(t0 )∥2 + C 

|α|=2

∥DDα u(0)∥2  

|α|=1

 +C 

 

  |α|=1

∥DDα u(0)∥2 



 

∥DDα u(t0 )∥2 

|α|=2

 t 2 ∥Dus (s)∥∞ ds N3 (t).

t0

Choosing a sufficiently large t0 , we can bound N3 (t) in terms of the initial data. From inequality (3.9), we also have a uniform estimate of N2 (t) for t ≥ t0 . The estimates for t ∈ [0, t0 ) are simple. We notice that  E3 (t) = ∥DDα u(t)∥2 |α|=2

K. Wakasa, B. Yordanov / Nonlinear Analysis 152 (2017) 183–195

193

is bounded in Proposition 3.2. This gives the second claim in Corollary 1.2. For the first claim, we combine (3.6) with t0 = 0, Lemma 3.1 and Hardy’s inequality: 1/2  t  t α 2 α E3 (s)ds.  ∥D u(s)∥∞ ds ≤ C∥DD u(0)∥2 + C 0

0

4. Global existence of radial solutions in H k × H k−1 with k > 3 Here we consider Eq. (1.1) with a supercritical integer p = 2m + 1, m ∈ N. For k ≥ 3 and t ∈ [0, T∗ ), let us define 1/2    t β 2 α ∥D u(s)∥∞ ds , Ek (t) = ∥DD u(t)∥2 , Nk (t) = |α|=k−1

1≤|β|≤k−1

0

where u is the local solution of (1.1), (1.2) constructed in Lemma 2.2. Proof of Theorem 1.3. We use induction in k. From Theorem 1.1, we know that the following hold when k = 3: T∗ = ∞ and sup Ek (t) ≤ Ak (∥u0 ∥H k , ∥u1 ∥H k−1 ),

(4.1)

t∈[0,∞)

sup Nk (t) ≤ Bk (∥u0 ∥H k , ∥u1 ∥H k−1 ).

(4.2)

t∈[0,∞)

Assuming these claims for k, we will verify them for k + 1. The proof starts from applying Dα to (1.1) with p = 2m + 1 and solving for the highest derivatives:   2m−1 α Dα u = cm,1 u2m Dα1 ut Dα2 ut + cm,3 u2m−2 Dα1 ut Dα2 ut Dα3 ut t t D ut + cm,2 ut α1 +α2 =α



+ · · · + cm,l ut2m+1−l

α1 +α2 +α3 =α

Dα1 ut · · · Dαl ut = 0,

(4.3)

α1 +α2 +···+αl =α

where l = min{2m + 1, k}, cm,i are constants and |αi | ≥ 1 for i = 1, . . . , l. Making use of Lemma 2.1(a), we obtain  t  t  α α 2m α ∥DD u(t)∥2 ≤ C∥DD (0)∥2 + C ∥us D us ∥2 ds + C ∥u2m−1 Dα1 us Dα2 us ∥2 ds s 0

+





C

α1 +α2 +α3 =α

+ ··· +

α1 +α2 =α

0

t

∥u2m−2 Dα1 us Dα2 us Dα3 us ∥2 ds s

0

 α1 +α2 +···+αl =α

 C

t

∥us2m+1−l Dα1 us · · · Dαl us ∥2 ds.

(4.4)

0

Only the first integrand involves a derivatives of order k + 1. From (3.2), we get α 2 ∥u2m s D us ∥2 ≤ C∥us ∥∞ Ek+1 (s).

(4.5)

In the second integrand, there are two cases: max{|α1 |, |α2 |} = k − 1 and max{|α1 |, |α2 |} ≤ k − 2. We rely on (3.2) to derive ∥u2m−1 Dα1 us Dα2 us ∥2 ≤ ∥us ∥2m−2 ∥us ∥∞ ∥Dus ∥∞ Ek (s) s ∞    ≤ C ∥Dβ u(s)∥2∞  Ek (s) 1≤|β|≤2

(4.6)

K. Wakasa, B. Yordanov / Nonlinear Analysis 152 (2017) 183–195

194

and ∥u2m−1 Dα1 us Dα2 us ∥2 ≤ ∥us ∥2m−2 ∥us ∥∞ ∥Dγ us ∥∞ Ek−|γ|+1 (s) s ∞    ≤ C ∥Dβ u(s)∥2∞  Ek−|γ|+1 (s),

(4.7)

1≤|β|≤k−1

respectively, where γ is such that |γ| = max{|α1 |, |α2 |}. The third integrand also admits max{|α1 |, |α2 |, |α3 |} = k−2. Let γ be the multiindex satisfying |γ| = k−2. Applying (3.2), we have ∥Dγ us ∥∞ ∥Dus ∥∞ E2 (s) ∥u2m−2 Dα1 us Dα2 us Dα3 us ∥2 ≤ ∥us ∥2m−2 ∞ s    ∥Dβ u(s)∥2∞  . ≤ C

(4.8)

1≤|β|≤k−1

Finally, we consider all integrands with max{|αj | : j = 1, 2, . . . , l} ≤ k − 3. Since k ≥ 4, we can use the Sobolev embedding and (3.2) to obtain   l−2 k   2m+1−l α1 α2 αj 2m+1−l  β 2  ∥us D us D us · · · D us ∥2 ≤ ∥us ∥∞ ∥D u(s)∥∞ Ej (s) j=1

1≤|β|≤k−2

 ≤ C

 

∥Dβ u(s)∥2∞  .

(4.9)

1≤|β|≤k−2

We combine the basic estimate (4.4) with estimates (4.5)–(4.9). This yields  t ∥DDα u(t)∥2 ≤ C∥DDα u(0)∥2 + C ∥us (s)∥2∞ Ek+1 (s) + CNk2 (t) + Ck 0

for all t ∈ [0, T∗ ), where the constant Ck = Ck (∥u0 ∥H k , ∥u1 ∥H k−1 ). Adding all such estimates with |α| = k and using (4.1), (4.2), we have  t Ek+1 (t) ≤ Ck+1 + C ∥us (s)∥2∞ Ek+1 (s)ds, 0

for some constant Ck+1 = Ck+1 (∥u0 ∥H k+1 , ∥u1 ∥H k ). From the Gronwall inequality,  t  2 Ek+1 (t) ≤ Ck+1 exp ∥us (s)∥∞ ds . 0

This shows that Ek+1 (t) is uniformly bounded on every finite subinterval of [0, T∗ ). Hence, u can be continued to a global solution, such that supt∈[0,∞) Ek+1 (t) ≤ Ak+1 . It remains to check estimate (4.2). We can apply Lemma 2.1(b) to the solution of (4.3). The calculations are very similar to the above ones, so we give only the final estimate: 1/2  t   t α 2 ∥D u(s)∥∞ ds ≤C ∥us (s)∥2∞ Ek+1 (s)ds + Ck+1 . |α|=k

0

0

Noticing that supt∈[0,∞) Ek+1 (t) ≤ Ak+1 and supt∈[0,∞) N1 (t) ≤ B1 , we obtain the final estimate in (4.1) with k + 1, i.e., supt∈[0,∞) Nk+1 (t) ≤ Bk+1 . The proof of higher regularity by induction is complete. Similarly, we can show that C ∞ regularity is preserved during the evolution of compactly supported radial data. 

K. Wakasa, B. Yordanov / Nonlinear Analysis 152 (2017) 183–195

195

Acknowledgment The authors are very grateful to Professor Hideo Kubo for his useful comments. References [1] J. Bourgain, Global wellposedness of defocusing critical nonlinear Schr¨ odinger equation in the radial case, J. Amer. Math. Soc. 12 (1) (1999) 145–171. [2] R. Carles, J. Rauch, Focusing of sherical nonlinear pulses in R3+1 , III. Sub and supercritical cases, Tohoku Math. J. 56 (2004) 393–410. [3] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Global well-posedness and scattering in the energy space for the critical nonlinear Schr¨ odinger equation in R3 , Ann. of Math. 167 (2008) 767–865. [4] M. Grillakis, Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Ann. of Math. (2) 132 (3) (1990) 485–509. [5] M. Grillakis, On nonlinear Schr¨ odinger equations, Comm. Partial Differential Equations 25 (910) (2000) 1827–1844. [6] A. Haraux, Lp estimates of solutions to some non-linear wave equations in one space dimension, Int. J. Math. Model. Numer. Optim. 1 (1–2) (2009) 146–154. [7] L. H¨ ormander, Lectures on Nonlinear Hyperbolic Differential Equations, in: Math´ ematiques & Applications, vol. 26, Springer Verlag, Berlin, 1997. [8] J.-L. Joly, G. Metivier, J. Rauch, Nonlinear hyperbolic smoothing at a focal point, Michigan Math. J. 47 (2) (2000) 295–312. [9] C. Kenig, F. Merle, Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications, Amer. J. Math. 133 (4) (2011) 1029–1065. [10] R. Killip, M. Visan, The defocusing energy-supercritical nonlinear wave equation in three space dimensions, Trans. Amer. Math. Soc. 363 (7) (2011) 3893–3934. [11] S. Klainerman, M. Machedon, Space–time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (9) (1993) 1221–1268. [12] J. Liang, Nonlinear Hyperbolic Smoothing at a Focal Point (Ph.D. dissertation), 2002. [13] H. Lindblad, A sharp counterexample to the local existence of low-regularity solutions to nonlinear wave equations, Duke Math. J. 72 (1993) 503–539. [14] J.-L. Lions, W.A. Strauss, Some non-linear evolution equations, Bull. Soc. Math. France 93 (1965) 43–96. [15] R. Musina, Weighted Sobolev spaces of radially symmetric functions, Ann. Mat. Pura Appl. (1923-) 193 (6) (2014) 1629–1659. [16] K. Nakanishi, Scattering theory for nonlinear Klein-Gordon equation with Sobolev critical power, Int. Math. Res. Not. 31 (1999) 31–60. [17] G. Ponce, T.C. Sideris, Local regularity of nonlinear wave equations in three space dimensions, Comm. Partial Differential Equations 18 (1993) 169–177. [18] T. Roy, Global existence of smooth solutions of a 3D loglog energy-supercritical wave equation, Anal. PDE 2 (3) (2009) 261–280. [19] J. Shatah, M. Struwe, Regularity results for nonlinear wave equations, Ann. of Math. (2) 138 (3) (1993) 503–518. [20] J. Shatah, M. Struwe, Geometric Wave Equations, in: Courant Lecture Notes in Mathematics, vol. 2, American Mathematical Society, Providence, RI, 1998, New York University, Courant Institute of Mathematical Sciences, New York. [21] W.A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (2) (1977) 149–162. [22] W.A. Strauss, Nonlinear Wave Equations, in: CBMS Regional Conference Series in Mathematics, vol. 73, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1989, by the AMS, Providence, RI. [23] M. Struwe, Globally regular solutions to the u5 -Klein-Gordon equations, Ann. Sc. Norm. Super. Pisa (Ser. 4) 15 (1988) 495–513. [24] T. Tao, Global regularity for a logarithmically supercritical defocusing nonlinear wave equation for spherically symmetric data, J. Hyperbolic Differ. Equ. 4 (2) (2007) 259–265. [25] G. Todorova, D. Ugurlu, B. Yordanov, Regularity and scattering for the wave equation with a critical nonlinear damping, J. Math. Soc. Japan 61 (2) (2009) 625–649. [26] C. Wang, D. Fang, Ill-posedness for semilinear wave equations with very low regularity, Math. Z. 259 (2) (2008) 343–353.