EXISTENCE QUESTIONS FOR NON-LINEAR WAVE EQUATIONS

TERENCE TAO

1. Introduction

In this note we consider the various possible existence results in for non-linear wave equations of the form

□u = F (u, ∇u)
u(x, 0) = f(x)
u_{t}(x, 0) = 0

where F (0, 0) = 0, and the data^{1} f assumed to be in the homogeneous Sobolev space^{2} Ḣ^{s}. Because we want these spaces to have reasonable propertes (e.g. to be localizable to compact sets, and to imbed in some L^{p} space) we will assume that 0 ≤ s < n/2. We consider the question of whether solutions u exist in some solution space X. We shall assume that X is a Banach space with the same scaling as Ḣ^{s}.

For a given f ∈ Ḣ^{s}, we define T^{∗}(f) to be the supremum of all the times T such that a (weak) solution u to the above problem exists in S_{T} = {(x, t) : 0 ≤ t ≤ T }.

We will suppose that F is essentially homogeneous in the sense that it scales like

F (u, ∇u) ∼ |u|^{p}|∇u|^{q}

for some p, q such that p + q − 1 > 0. there exists α, β ∈ R such that the class of non-linearities under consideration is invariant under the scalings

F (a, b) ← λ^{2−α}F (λ^{α}a, λ^{α−1}b).
F (a, b) ← λ^{−β}F (λa, λb).

Thus, the scaling

f_{λ}(x) = λ^{\frac{q−2}{p+q−1}} f(x/λ)
u_{λ}(x, t) = λ^{\frac{q−2}{p+q−1}} u(x/λ, t/λ)

leaves the problem essentially invariant.

We will call the quantity

κ = q − 2 + (p + q − 1)(\frac{n}{2} − s)

1991 Mathematics Subject Classification. 35L05.

.

1Unsurprisingly, all of these results for (f, 0) data generalize to (f, g) data in the usual fashion.

2We will remark on other data spaces at the end of this note.

the criticality of the problem. We say that the problem is sub-critical, critical, or super-critical if κ < 0, κ = 0, or κ > 0 respectively. Note that

∥f_{λ}∥Ḣ^{s} = λ^{\frac{κ}{p+q−1}} ∥f∥Ḣ^{s}
∥u_{λ}∥_{X_{λT}} = λ^{\frac{κ}{p+q−1}} ∥f∥_{X_{T}}

We now consider existence results for various types of data^{3}. We make three possible restrictions on the data f:

1. (Large data) ∥f∥_{X} < ∞.

2. (Small-norm data) ∥f∥_{X} < ε for some fixed ε > 0.

3. (Small data) f = εf_{0} for any f_{0} ∈ X and any 0 ≤ ε < ε(f_{0}).

Trivially any result for large data implies the corresponding result for small-norm data, which in turn implies the corresponding result for small data. We also consider the following distinction:

1. (Non-compact support) No restrictions on the support of f.

2. (compact support) f is supported on B(0, 1).

By translation and scale invariance, any result for data supported in the unit ball implies a corresponding result for data supported on other compact sets. Trivially, any result for non-compactly supported data implies the corresponding result for compactly supported data.

Finally, we consider three possible conclusions about the existence time T^{∗} = T^{∗}(f):

1. (Global existence) T^{∗} = ∞.

2. (Strong local existence) T^{∗} ≥ T (∥f∥_{X}) > 0 for some function T.

3. (Weak local existence) T^{∗} > 0.

Trivially, global existence implies strong local existence, which implies weak local existence.

2. Equivalences between existence results

A priori, we have eighteen possible existence results in the above framework:

Global  Global, small norm  Global, small
Global (compact) Global, small norm (compact) Global, small (compact)
Strong Strong, small norm Strong, small
Strong (compact) Strong, small norm (compact) Strong, small (compact)
Weak Weak, small norm Weak, small
Weak (compact) Weak, small norm (compact) Weak, small (compact)

3We will assume that whenever an existence result is proved, the corresponding uniqueness result is also proved.

However, besides the trivial relationships, we also have the following implications:

1. Global (compact) ⇒ Global

2. Strong (compact) ⇒ Strong

3. Global, small ⇒ Global, small norm

4. Weak, small (compact) ⇒ Strong, small-norm

5. Weak, small-norm ⇒ Weak

6. (κ ≠ 0) Global, small norm ⇒ Global

7. (κ ≠ 0) Strong, small norm ⇒ Strong

8. (κ ≥ 0) Strong ⇒ Global

9. (κ ≥ 0) Strong, small norm ⇒ Global, small norm

(1) follows immediately from finite speed of propogation (see e.g. [2], page 130) as does (2).

(3) follows from the following “cut and paste” argument. Suppose “Global, small norm” failed. Then for every n > 0 there exists data f_{n} of norm O(4^{−n}) which blows up at time T_{n}. By finite speed of propogation we may localize each f_{n} to a ball of radius T_{n}, and by translation invariance we may make these balls disjoint. Then, if we let f_{0} = ∑_{n} 2^{n}f_{n}, then f_{0} ∈ Ḣ^{s} and the data 2^{−n}f_{0} has no global solution for any n > 0, so “Global, small data” fails.

(4) is proven similarly (see e.g. [2], page 99). Suppose “Strong, small norm“ failed. Then for every n, m > 0 there exists data f_{n,m} of norm O(4^{−n−m}) which blows up at time O(4^{−n−m}). As before, we localize each f_{n,m} to a ball of radius 4^{−n−m}, and make these balls all disjoint and contained a ball of radius O(1). Then, if we let f_{0} = ∑_{n} ∑_{m} 2^{n+m}f_{n,m}, then f_{0} is compactly supported in X and the data 2^{−n}f_{0} has no solution for any T > 2^{−m}. Thus “weak, small (compact)” fails.

To prove (5), we use the cutoff argument (see e.g. ??) Suppose f ∈ Ḣ^{s}, and let ε be such that one has weak local existence for data with norms O(ε). Then one can find a sufficiently large R and balls B_{i} that cover B(0, R) such that such that f − fχ̃_{B(0,R)} and fχ̃_{B_{i}} have norms less than ε, and therefore have solutions for some non-zero time. Combining these solutions together using uniqueness and finite speed of propogation we obtain the result.

If κ ≠ 0, (6) and (7) is immediate from the fact that large-norm data can be always be scaled to small-norm data.

Finally, if κ ≥ 0, then (8) and (9) follow from the fact that any failure of global existence can be scaled to failure of strong local existence without increasing the norm of the data.

By combining these implications together into classes of logically equivalent statements, we obtain the following conclusions.

In the super-critical case κ > 0, there is only one equivalence class^{4}:

1. (WP) “Global” ⟺ … ⟺ “Weak, small (compact)”

There is no distinction between global and local, small and large, or compact and non-compact. This should not be interpreted as a statement that the global problems are as easy as the local problems, but rather that the local problems are at least as difficult as the global problems. In particular, if there is global blow-up for large data, then there is blow-up for arbitrarily small times even for very small, compact data.

In the critical case κ = 0, there are two equivalence classes:

1. (Global WP) “Global” ⟺ … ⟺ “Strong (compact)”

2. (WP in the critical sense) “Weak” ⟺ “Global, small-norm” ⟺ … ⟺ “Weak, small (compact)”.

Thus there is only one possible result which is easier than the global WP problem, which is well-posedness in the critical sense as formulated in [1]. Note that the two traditional criteria for WP in the critical sense, “Weak” and ”Global, small-norm”, are actually equivalent under our assumption of finite speed of propogation. In fact, to show WP in the critical sense it suffices to show weak local existence for very small data.

The two statements are distinct: a critical semi-linear equation with an indefinite conserved energy can blow up in finite time for large data, but still be well-posed in the critical sense for small data.

In the sub-critical case κ < 0, there are four equivalence classes, which we arrange from strongest to weakest:

1. (Global WP) “Global” ⟺ ”Global (compact)” ⟺ … ⟺ “Global, small”

2. (Global WP for small-norm compact data) “Global, small-norm (compact)”

3. (Global WP for small compact data) “Global, small (compact)”

4. (Local WP) “Strong” ⟺ … ⟺ “Weak, small (compact)”

Thus, although there is only one local well-posedness statement, there are additional possibilities for global existence when one looks at small or small-norm compact data. For example, for the semi-linear wave equation for very smooth data one always has local well-posedness (by e.g. the energy method), but one only has global WP for small-norm or small compact data for p less than Strauss’s power (see [2], [3]). As before, the global WP problem depends whether the energy is positive definite.

4For brevity we shall only list the strongest and weakest of the equivalent statements, as measured by the trivial implications

Many of these equivalences extend to spaces other than Ḣ^{s}, though the scaling arguments (6-9) are unlikely to work for inhomogeneous spaces. Also, the cutoff argument (5) does not work for smooth function spaces such as C^{k}, because one cannot make the norm of the data small by truncation. The finite speed of propogation is also crucial in many of these arguments.

References

[1] G. Ponce, Private communication.

[2] C. D. Sogge, Lectures on Nonlinear Wave Equations, Monographs in Analysis II, International Press, 1995.

[3] V. Georgiev and H. Lindblad, C. D. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations, to appear.

Department of Mathematics, UCLA, Los Angeles CA 90095-1555

E-mail address: keelmath.ucla.edu, taomath.ucla.edu