GLOBAL AND ALMOST GLOBAL EXISTENCE OF SMALL SOLUTIONS TO A DISSIPATIVE WAVE EQUATION IN 3D WITH NEARLY NULL NONLINEAR TERMS

. The existence of global small O ( " ) solutions to quadratically non- linear wave equations in three space dimensions under the null condition is shown to be stable under the simultaneous addition of small O ( ⌫ ) viscous dis- sipation and O (  ) non-null quadratic nonlinearities, provided that " / ⌫ ⌧ 1 . When this condition is not met, small solutions exist “almost globally”, and in certain parameter ranges, the addition of dissipation enhances the lifespan.

In Theorem B of [9], Ponce proved that the initial value problem for (1.1) has a unique, global, strong solution for initial data that is sufficiently small. His argument is based on the dissipative properties of the linear equation, and although he does not quantify it, the size of the initial data must be small relative to the viscosity parameter. On the other hand, in the hyperbolic case ⌫ = 0, when the nonlinear terms satisfy the Klainerman null condition, there exist global small solutions, see [6,1]. One would expect this result to be stable under viscous perturbations, and moreover, based on Ponce's result one also expects that the introduction of small viscosity would allow a simultaneous nonlinear perturbation from the null condition. However, proving this requires the by no means routine adaptation of hyperbolic methods to dissipative equations such as (1.1).
The main result of this paper, Theorem 4.1, shows that these suppositions do indeed hold. We define a parameter which measures the deviation of the nonlinearity from being null, and we quantify the limitation on the size of the initial data, roughly ⌫/ , leading to global existence. We also obtain in Theorem 4.3 lower bounds for the lifespan of solutions in cases where our global existence result does not hold. In the hyperbolic case, it is well-known that the lifespan T (") of small solutions of size " satisfies an "almost global" lower bound of the form [T (")] " C > 1, see [3,5]. We show that dissipative effects can improve the almost global lifespan of the hyperbolic case if the viscosity ⌫ is large enough relative to the size of the data.
These results require decay estimates which are uniform with respect to ⌫, given in Sections 8 and 9. The derivation of the decay estimates extends the weighted L 2 approach, introduced in [7] and refined in [11], to the case of partial dissipation, at the expense of introducing space-time weighted norms. To implement the method of [11], it is helpful to reformulate the problem as a first order system, and this is done in the next section. The decay estimates are coupled with energy estimates based on the translational, rotational, and scaling vector fields, derived for two distinct energy levels in Sections 10 and 11. The energy estimates are also nonstandard insofar as occurrences of the scaling vector field are indexed separately because the linear equation is not scaling invariant. Hidano [2] has also used energy norms with a limitation on the use of scaling operator. A short discussion of local existence is provided in Section 12, for completeness.
There is an extensive literature on existence of solutions to dissipative wave equations in three space dimensions. However, the authors are aware of only one other work, [8], with uniformity in the viscosity parameter. There the underlying nonlinear hyperbolic system is Hamiltonian with a positive definite conserved energy, and the nonlinearity is bounded at infinity. It is possible to use the dissipation to establish local well-posedness in spaces of low regularity and to prove global existence for large data.
3. Notation. We shall employ the vector fields r, ⌦ = x^r, S = t@ t + r@ r , S 0 = r@ r . (3.1) We avoid the vector field @ t because in order to control @ k t u(0), the second order operator L forces the initial data to have 2k derivatives, which clashes with the hyperbolic case ⌫ = 0. The Lorentz rotations x@ t + tr are likewise unsuitable. Although the system (2.2a), (2.2b) is not scale invariant, it is possible to effectively use the scaling operator S. However, its usage will be indexed separately (see the space X p,q and the energy E p,q defined below). The rotational operators ⌦ are modified in the usual way when used with vector-valued functions, consistent with the rotational invariance of the linear system. Thus, we take e This definition is dictated by the fact that @(⌦ i ') = e ⌦ i @', for scalar functions '. We emphasize that these operators are vectorial, and we use the notation ( e ⌦ i u) ↵ to denote the ↵-th component of the vector e ⌦ i u 2 R 4 . However, notice that ( e ⌦ i u) 0 = ⌦ i u 0 , and so all of the vector fields act as scalars on the 0-th component of vectors u 2 R 4 . We shall frequently rely on the decomposition It will be convenient to use the abbreviation = {r, e ⌦}, however, the fields S 0 and S will always be tracked individually. Thus for example, given integers 0  q  p, we define the space This is a Hilbert space with the inner product

BOYAN JONOV AND THOMAS C. SIDERIS
The index p indicates the total number of allowable derivatives while the index q limits the number of occurrences of S 0 which, in practice, could be strictly less than the total p. These spaces characterize the initial data that we shall consider, and they may be used to establish local well-posedness for our system for appropriate pairs (p, q), see Section 12.
Given a solution of (2.2a)-(2.2d), our main objective will be to obtain a priori estimates for the energy for two sets of pairs (p, q), informally referred to as high and low. If u(0) = u 0 , it will be convenient to write (with a slight abuse of notation).
To get a priori bounds for the energy, it will be necessary to also obtain dispersive estimates. These will be derived using weighted L 2 -estimates in two space-time regions which we shall refer to as the interior and exterior regions.
We now define cut-off functions which determine these regions and the quantities which in Section 9 will be shown to have good decay properties. Given a function we define where we have used the common notation hti = (1 + t 2 ) 1/2 . The parameter ⌧ 1 will be chosen in Lemma 8.1. We have This is not a partition of unity. We can say that We shall frequently rely on the property that In the interior region, we shall derive estimates in for q < p, see Theorem 9.1.
In the exterior region, it is critical to decompose the solution into its orthogonal and tangential components along the light cone, Pu and Qu, respectively. These projections are defined as follows: Thanks to the fact that e ⌦ j ! = 0 and @ r ! = 0, we obtain The quantities to be bounded in Theorem 9.2 are again with q < p. Given a quadratic nonlinearity N of the form (2.2d), we associate to it a cubic polynomial P N (y) = C↵ , y ↵ y y`, y 2 R 4 . We say that N is null if P N (y) = 0, for all y 2 N = {y 2 R 4 : y 2 0 y 2 1 y 2 2 y 2 3 = 0}, where N is the collection of null vectors in R 4 .
As a final notational remark, we shall write A . B if there exists a generic constant C, independent of the initial data and the parameters ⌫, " 2 , (the latter two defined in Theorem 4.1) such that A  CB. Constants may depend on Theorem 4.1 (Global existence). Choose (p, q) such that p 11, and p q > p ⇤ , = max{|⌦ a P N (y)| : y 2 N , kyk = 1, |a|  p ⇤ } (4.1) and assume that  1.
There are positive constants C 0 , C 1 > 1 with the property that if the initial data for some " 2 ⌧ 1, and and Remarks.
• The assumption (4.2a) only requires the norm ku 0 k X p,q to be finite, but ku 0 k X p ⇤ ,p ⇤ must be correspondingly small. • The parameter measures the deviation of the nonlinearity from the null condition, see the remark following Lemma 5.3. If = 0, then (4.2a), (4.2b) hold for all 0 < ⌫  1, and the existence criterion is uniform in ⌫, consistent with the hyperbolic case ⌫ = 0. • The restriction p 11 is made so that p ⇤ + 3  p holds, see Proposition 11.1. • Successive restrictions on the size of the parameter " arise in Theorem 9.1 and in Propositions 10.1 and 11.1. Generic constants are not permitted to depend on ", so it will be possible to decrease the size of " when necessary. • As soon as the initial data u 0 meets the criterion (4.2a) for a single sufficiently small ", one can take the infimum over all such ". As a consequence, the bounds (4.3a), (4.3b) hold with " 2 replaced by Outline of Proof. Given data u 0 2 X p,q satisfying (2.2b), it can be shown using Picard iteration that the IVP for (2.2a)-(2.2d) has a local solution u 2 C([0, T ); X p,q ) where T depends only on ku 0 k X p,q , and the fixed constants ⌫ and C↵ , , see Section 12.
To establish global existence, it is enough to prove that ku(t)k X p,q remains finite. This norm can not be directly bounded by E 1/2 p,q [u](t) because the X p,q norm is based on S 0 while E 1/2 p,q [u](t) uses S. However, the norm ku(t)k X p,q can be controlled by a function which depends only on ku 0 k X p,q , ku(t)k X p,0 , and T , see Section 12. By definition, Thus, it is enough to show that the energy E p,q [u](t) remains finite.
If T 2 T , then by Proposition 10.1 and (P2) so by the local existence theorem, (P1) holds for some T 0 > T . Using the assumptions (4.2a), (4.2b), and (P2) with Proposition 11.1, we get and so we have by continuity that (P2) holds for 0  t < T 00 , with T < T 00  T 0 . This shows that (0, T 00 ) ⇢ T , and so T is open. The nonempty connected set T is both open and closed in (0, 1), and therefore equal to (0, 1).
Proof. The first inequality follows from Theorem 9.1 and Proposition 10.1, and the second follows from Theorem 9.2.

Remarks.
• The following table summarizes the results of the Theorems. The basic smallness restriction (4.2a) must always be enforced.
Almost global existence with diffusion enhanced lifespan (4.5b) Almost global existence with hyperbolic lifespan (4.5c) • The cases (4.5a), (4.5b) show that diffusive effects are important when ⌫ C 1 ". The constants C 0 , C 1 , depend on max |C↵ , |, and we have not verified that the parameter range in (4.5b) is nontrivial. In any case, it is clear from (4.5a)-(4.5c) that the quantity " /⌫ controls the transition from global to almost global existence.
For the remainder of the article, we assume that properties (P1) and (P2) hold. In the following sections, we are going to establish a series of a priori estimates culminating in Propositions 10.1, 11.1, and 11.2.

Commutation. Recall the linear operator defined in
For any multi-index a and any integer k 0, we have The appearance of additional Laplacian terms on the right-hand side of (5.1a) reflects the lack of scaling invariance for the operator L.
For the nonlinear form (2.2d), we define the commutators The higher order commutators [(S + 1) k a , N] are defined inductively, each being a nonlinear form of the type (2.2d). Of course, by Lemma 5.1 a nonlinear commutator could be nonzero only for pure e ⌦ derivatives. Using the higher order commutators, we have a Leibnitz-type formula: Remark. Here again we emphasize that [ a3 , N] = 0, unless the derivative a3 involves only e ⌦. Lemma 5.3. For any quadratic nonlinearity of the form (2.2d) Thus, from the chain rule and (5.2) we obtain the first statement: . Suppose that N is null. The one-parameter family of rotations U (s) = exp( sZ i ) leaves the set of null vectors N invariant. Thus, for any y 2 N , we have . This shows that P [⌦i,N ] is also null. Remark. If N is null, then since the operators ⌦ i act tangentially along N , we have ⌦ a P N (y) = 0, for all a, and the parameter defined in Theorem 4.1 vanishes.

BOYAN JONOV AND THOMAS C. SIDERIS
The key term is Notice that Lemma 5.3 gives and (5.3b) follows.
Proof. Using the cutoff function defined in (3.3a), apply (6.1a) to u(t). This produces We apply (6.1c) to the second integral Thus, we see that On the other hand, we have using (6.1c) again This shows that (Clearly, the same bound holds for khri 1/2 u(t)k L 1 , but we do not need it.) To prove (6.2a), apply (6.3) to the function ⇣u(t), and use (3.3d). Applying (6.1d) to ⇣ru(t) yields (6.2b). The inequality (6.2c) follows by applying (6.1b) to ⇣u(t). Since we can get (6.2d), by applying (6.1d) to ⌘u(t).
Finally, we prove (6.2e). By (6.1d) applied to ⌘Qu(t), we have Using (3.3d) and the commutation property (3.5), we see that X By linearity, we have we deduce from (6.4) , from which (6.2e) follows by Young's inequality.
provided the right-hand side is finite.
In the special case when provided the right-hand side is finite.
Proof. In the case k 1 + |a , using the Sobolev inequality (6.2a) we have the following bound: And in the case The second statement of the lemma follows similarly from the preceding arguments.
, provided the right-hand side is finite.
In the special case when provided the right-hand side is finite.
Proof. In the case k 1 + |a , using the Sobolev inequality (6.2d) we have the following bound: And in the case , we similarly have: The second statement of the lemma follows analogously.
Two slightly more specialized instances of this basic argument occur in the proof of Proposition 11.1.
Thanks to (8.1b), the cross term can be rewritten as which can be estimated below by using (3.3d) and Young's inequality. With this, (8.2) yields Choose 0  ✓  1, multiply the preceding inequality by hti ✓ 2 , integrate in time, and rearrange the result: We now focus on time derivative term on the right. A simple calculation reveals that and so, using integration by parts, we get Combining this with the inequality (8.3), we obtain Next, thanks to Lemma 8.2 (to follow), we gain control of the full gradient on the left: Then, inserting t 2 = hti 2 1 on the left, we may write According to definition (3.3b), we have r  hti on the support of ⇣, so for sufficiently small, the term can be absorbed on the left. This key step yields Finally, consider the first term on the right. We have (8.4). The desired estimate follows immediately since ⌫  1.
Remark. Notice that the time integration used in this lemma arises from the dissipative term in the equation.
In the proof of Lemma 8.1, we used the following simple coercivity estimate: ) and r^w = 0, then Using integration by parts and the property (3.3d), we may write The result follows by an application of Young's inequality.
We now establish a higher order version of Lemma 8.1. Proposition 8.3. Assume that in (3.3b) is sufficiently small and that ⌫  1. Fix 0  q < p. Suppose that S k G 2 L 2 ([0, T ]; X p k 1,0 ), k = 0, . . . , q for some 0 < T < 1. If u is a solution of (8.1a), (8.1b), (8.1c) such that then for any 0  ✓  1, Proof. We shall prove this by induction on q. Fix a multi-index |a|  p 1. By the commutation relations (5.1a), (5.1b), we have that a u solves (8.1a), (8.1b) with a G on the right. Apply Lemma 8.1 to get Summing over |a|  p 1 gives the result for q = 0. Now take any 1  r < p, and assume that the result holds when q = r 1. Choose a and k such that k 6 = 0, |a| + k  p 1, and k  r. By (5.1a), (5.1b), we have that S k a u solves (8.1a), (8.1b) with on the right. Apply Lemma 8.1 again: Notice that this inequality holds for k = 0, as well, by the result for q = 0. Perform a summation over |a| + k  p 1, k  r. This yields The result now follows by the induction hypothesis.
Next, we turn our attention to the exterior region.

BOYAN JONOV AND THOMAS C. SIDERIS
Insertion of the identity r@ r u 0 u 0 = r · ⇥ r@ r u 0 ru 0 1 2 x|ru 0 | 2 ⇤ + 1 2 |ru 0 | 2 , followed by integration by parts yields By (3.3d) and the fact that r & hti on the support of ⌘, we see that

Using integration by parts and (3.3d), we get
. The estimate (8.6) for I now follows by Young's inequality. ), k = 0, . . . , q 1, for some 0 < T < 1. If u = (u 0 ,ū) is a solution of (8.1a), (8.1b), (8.1c) such that Proof. First, we note that from (3.4a), and (3.2), we have for each j, , and by (8.1c), Therefore, since r & ht + ri ht ri on supp ⌘, we obtain A bit of algebraic manipulation produces the relation ⌘|r! · @ rū + t@ r u 0 |  ⌘|r@ rū + tru 0 | R 3 + O(|⌦u 0 |). Combining the preceding estimates gives us the bound . Now take any multi-index a with |a|  p 1. By (5.1a), we can apply the preceding inequality to a u to get which proves the result in the case q = 0. The result for 0 < q < p follows from (5.1b) and induction, as in the proof of Propostion 8.3.
Proof. Fix a pair (p,q) = (p,p 1) with 2 p  p. Choose a multi-index a and an integer k, with |a| + k p 1 and k q =p 1. Then, using Lemmas 5.2 and 5.1, we have that k⇣S k ⌦ a N (u, ru)k 2 L 2 is bounded by a sum of terms of the form k⇣S k1 a1 u S k2 a2+1 uk 2 L 2 , (9.3) with |a 1 | + |a 2 |  |a| and k 1 + k 2  k. Thus, k 1 + k 2 + |a 1 | + |a 2 |  k + |a| p 1 and k 1 + k 2 p 1. Lemma 7.1 implies that the terms in (9.3) can be bounded by (a multiple of) . Therefore, an application of Proposition 8.3 with G = N (u, ru) yields for any 0 < ✓  1. We are going to apply this for two pairs (p,q). First, let (p,q) = (p ⇤ , p ⇤ 1). Sincep = p ⇤ 5, we get In this case, (9.4) yields Choose 0 such that 0 < ✓ +  1. By (9.1), the right-hand side is bounded by For " 2 sufficiently small, the last term above can be absorbed on the left, and then the inequalities (9.2a) follow immediately. Next, we use the pair (p,q) = (p ⇤ + 1, p ⇤ ) in (9.4). Again since p ⇤ 5, we havē We obtain from (9.4) Choose > 0 such that 0 < ✓ + < 1. Since p ⇤ + 1  q  p, the right-hand side can be estimated above by By (9.2a) and (9.1), the last integral is bounded (by C" 2 ), and so the inequality (9.2b) now follows.
Proof. Choose a multi-index a and an integer k, such that |a| + k  p 1, k  q. Then, using Lemmas 5.2 and 5.1, we have that the quantity k⌘S k ⌦ a N (u, ru)k 2 L 2 is bounded by a sum of terms of the form k⌘S k1 a1 u S k2 a2+1 uk 2 L 2 , (9.6) with |a 1 | + |a 2 |  |a| and k 1 + k 2  k. Thus, k 1 + k 2 + |a 1 | + |a 2 |  k + |a|  p 1 and k 1 + k 2  p 1. Lemma 7.2 implies that the terms in (9.6) can be bounded by (a multiple of) (9.5). The result is now a consequence of Proposition 8.5. 10. High energy estimates.
Then there exists a constant C 1 > 1 such that Thanks to the symmetry of the coefficient matrices (2.2c), we obtain the basic energy identity For p q 0, we can combine this with (5.1a) to get For q > 0, using the definition B = e 0 ⌦ e 0 and integration by parts we get the bound . It follows from induction on q and Young's inequality that If we combine (10.2) with Lemma 5.2, we find The reader will note that we have deliberately left the absolute value signs on the "outside" of the integrals because we will later exploit cancellations of the coefficients of the nonlinear terms. Special care must be taken for the terms in the sum with |a 2 | + k 2 = |a| + k = p. To simplify the notation when analyzing these terms, set v = S k a u. Then using the fact that @ ↵ v = @ v ↵ , we may write (10.4) By (6.1a) and the assumption (10.1), we have It follows that the right-hand side of (10.4) is bounded by
Recalling the definition p ⇤ = ⇥ p+5 2 ⇤ , we obtain using Theorem 9.1 . Therefore, we also obtain the bound , after a possible increase in the size of the constant C 1 . Note however, that the choice of C 1 is independent of ". The size of C 1 may always be increased, while the size of " may always be decreased. The statement of Proposition 10.1 now follows.
Proof. This is simply (10.3) from the proof of Proposition 10.1 in the case where (p, q) = (p ⇤ , p ⇤ ).
There exists a constant C 0 > 1 such that if Proof. We continue from the inequality of Corollary 10.2. Using the cut-off functions defined in (3.3b) and recalling (3.3c), we can write: with Interior Low Energy. The first integral I 1 on the right of (11.4) is bounded by To estimate this, we follow the same strategy as in Lemma 7.1.
In the case k 1 + |a , we have using (6.2a) [u](t) Next, we consider the terms in . By Hardy's inequality (6.2c) and the Sobolev inequality (6.2b), we can NONLINEAR DISSIPATIVE WAVE EQUATIONS 1433 write: [u](t) Thanks to the assumption p 11, we have h p ⇤ +5 2 i  p ⇤ , so altogether for the interior low energy we have: [u](t) By Theorem 9.1 and Proposition 10.1, we can estimate these three integrals as follows. Note that " in (11.1) must again be taken small enough, 2C 1 " < 1 is sufficient, in addition to our earlier restrictions.
First integral: Third integral: . Combining these estimates, we have Exterior Low Energy. Thanks to Lemma 5.4, (5.3a), the second integral I 2 on the right of (11.4) is estimated by ⇥ h!, @ r S k2 a2 ui R 4 (S k a u) 0 dxdt (11.6b) and (11.6c) Before estimating the main term I 0 2 above, we dispatch the easiest terms I 00 2 . We claim that within the the range of indices of the sum, where k 2 + |b|  p ⇤ . Thus, from (11.9), we have shown that Inserting the estimates (11.11) and (11.8) into (11.6a), we find that By Proposition 10.1, we get provided C 1 " < 1/3. We deduce from (11.4), (11.5), (11.12) that for every 0  T < T 0 . Thus, there exists a constant C 0 > 4 such that for every 0  T < T 0 . In other words, the function .
We now conclude the proof with a standard argument, using (11.13) to show that (11.2) implies (11.3). Suppose that S(T ) < 4A 0 . Then by (11.13), we have Remark. The reader will note that (11.12) is the only point in this paper where ⌫ > 0 is relied upon. In particular, we never use the estimates for ⌫ 2 Z int and ⌫ 2 Z ext given in Corollary 4.2.
We now consider the situation when the condition (11.2) does not hold. 14) Remark. The constant C 0 may be assumed to be the same in Propositions 11.1 and 11.2. The constant C 1 is the one given by Proposition 10.1. Proof. We continue with the same notation as used in the proof of Proposition 11.1. All of the estimates derived there up to and including (11.13) are valid under the current hypotheses, insofar as the assumption (11.2) is used only in the final paragraph of the proof.
Using (11.13) with Proposition 10.1, we have Alternatively, we may avoid using the dissipation when estimating I 0 2 in (11.6b). Consider the terms in the sum for I 0 2 with k 2 + |a 2 | 6 = p ⇤ . By Lemma 7.2, these can be estimated by The remaining terms have the form ⌘P N (!)! ! µ!j u @ j (S k a u) µ (S k a u) 0 dxdt.
12. Remarks on local existence.
Proof. This can be shown using the Fourier transform, energy estimates, and induction on q.
Theorem 12.3. If u 0 2 X p,q , p 4, satisfies (2.2b), then there exists a T > 0 depending only on ku(0)k X p,q , ⌫, and max ↵, ,`| C↵ , | such that the IVP for (2.2a), (2.2b) has a unique solution u 2 C([0, T ], X p,q ). Proof. For |a| + k  p, k  q, and u 2 X p,q , we may write (as in Proposition 10.1) S k 0 a N (u, ru) = f 0 + r ·f, with f = (f 0 ,f ) 2 X p,q . Thanks to the energy estimate (12.1), the map is a contraction on C([0, T ], B 1 (u(0))), provided T is sufficiently small, where B 1 (u 0 ) denotes the closed ball of radius one with center u 0 in X p,q . Proposition 12.4. There is a continuous function : R 3 ! R + such that the local solution of Theorem 12.3 satisfies ku(t)k X p,q  (ku 0 k X p,q , t, sup 0st ku(s)k X p,0 ), 0  t  T.
Proof. This is proven by induction on q. For |a| + k  q, the equation satisfied by v = S k 0 a u is linear in v.