Solutions to semilinear wave equations of very low regularity

This paper finds solutions to semilinear wave equations with strongly anomalous propagation of singularities. For very low Sobolev regularity we obtain solutions whose singular support propagates along any ray inside or outside the light cone. In one dimension these solutions exist for any Sobolev exponent $s<\frac{1}{2}$ in space, while classical results show that the singular support of solutions with higher regularity is contained in the light cone. The spatial Fourier transform of these anomalous solutions is supported in a half-line. We obtain wellposedness results in such function spaces when the problem is ill-posed for Sobolev data without the support condition and, in some cases, obtain wellposedness below $L^2(\mathbb{R})$. The results are based on new multiplier theorems for Sobolev spaces satisfying the support condition. Extensions to higher space dimensions are given.


Introduction
This paper observes new phenomena for the wellposedness and propagation of singularities for semilinear wave equations with initial data of very low Sobolev-regularity.We address the problem ∂ 2 t u − ∆u = ±u p , u(x, 0) = u 0 (x), ∂ t u(x, 0) = u 1 (x), (1) in space dimension n, where p ≥ 2 is assumed to be a positive integer.The specialization to this case is needed for two reasons.First, we wish to study propagation of singularities from the initial data, and hence we need a smooth nonlinearity in order to avoid the occurrence of additional singularities.Second, we will use the Hörmander product of distributions, so only integer powers are amenable.
It is a general principle in linear wave propagation that sharp wave crests (singularities) propagate along light cones, or more precisely, along the bicharacteristics of the linear wave operator ∂ 2 t u − ∆u.For semilinear wave equations, singularities of sufficiently smooth solutions are known to propagate along unions of bicharacteristics, where in space dimension n > 1 new -but weaker -singularities may arise at points of intersection of incoming wave crests.In dimension n = 1, a fundamental result for problem (1) assures that the singularities of any solution in L ∞ loc (R 2 ) propagate only along light cones [34,37].
In this article we obtain solutions in C([−T, T ] : H s loc (R)), for any s < 1 2 , whose singular support lies inside or outside the light cone.As C([−T, T ] : H s loc (R)) ⊂ L ∞ loc (R 2 ) when s > 1 2 , they establish a sharp threshold s = 1  2 for the Sobolev exponent between solutions with expected, respectively anomalous, propagation of singularities.Note that the maximal regularity of solutions with unexpected properties has attracted significant recent interest for equations from continuum mechanics and geometry [10].
For the anomalous solutions u presented here, at fixed time t the Fourier transform u with respect to x is supported in a half-line.Motivated by this fact, we also extend the range of wellposedness for problem (1) to data and solutions u with this property.The paper addresses two different, yet related issues: (a) anomalous propagation of singularities for solutions to (1) of low Sobolev regularity in one and higher space dimension and (b) wellposedness for a certain class of initial data of low Sobolev regularity in one space dimension.The main results of this article can be summarized as follows: (a) (Anomalous propagation) For any c = ±1 there exist solutions to (1) with singular support along the line {x + ct = 0, t ∈ R}, i.e., along any ray off the light cone.More precisely, for any c = ±1 and any s < is the content of Theorem 10.Note that it improves the wellposedness results for data in H s (R) without the support condition, where problem (1) The solutions constructed in (a) do not fall into the known H s -wellposedness regimes.
Section 7 addresses the extension of this Theorem to higher space dimensions n > 1.We give examples of anomalous solutions to (1) which belong to C([−T, T ] : Here the nonlinearity is even defined classically as the p-th power of an L p loc -function.A similar extension of the wellposedness theory remains open.
The remainder of this introduction is devoted to a literature review of propagation of singularities and of critical Sobolev exponents.
The investigation of propagation of singularities in semilinear hyperbolic equations and systems started with the discovery of Jeffrey Rauch and Michael Reed [34,36] that -unlike in the linear case -singularities may arise that cannot be traced back via bicharacteristics to singularities in the initial data, but may be produced at later times by the interaction of singularity bearing bicharacteristics.For a survey of the huge number of results up to around 1990 we refer to the monograph [3].Rauch and Reed coined the term anomalous singularities for this phenomenon.However, these "anomalous singularities" still propagated along characteristics/bicharacteristics, as opposed to the noncharacteristic singularities in the present paper, which are even more anomalous.
There is one exception, namely the wave equation with smooth nonlinearity f (•) in one space dimension (actually any (2 × 2)-first order system in n = 1) where the propagation is as in the linear case.This is due to the fact that there are only two characteristic directions, thereby avoiding nonlinear interaction at later times.Here the results of [34,37] say that for distributional solutions to the semilinear wave equation which belong to L ∞ loc (R 2 ) no anomalous singularities arise.(This applies, in particular, to solutions which belong to C([−T, T ] : H s (R)) with s > 1 2 .)For example, if the singular support of the initial data is {x = 0} then the solution is smooth except possibly along the light cone {|x| = |t|}.
In higher space dimensions, the first and prototypical result is due to Rauch [32].It says the following: Suppose that u is a distributional solution to (1) (even with a polynomial nonlinearity) which belongs to H s loc (R n × R) with s > (n + 1)/2 and let the initial data belong to C ∞ (R n \ {0}).Then u is C ∞ on {|x| > |t|}, and it belongs to H s+1+σ loc (R n × R) on {|x| < |t|} for all σ < s − (n + 1)/2.It is also known that the singular support of the solution may contain the solid cone {|x| ≤ |t|}, see [1], where an example is given with Sobolev regularity just above 3s − n + 2 in {|x| < |t|}.
These results date back to a time when the investigation of critical exponents had not yet been picked up.Accordingly, the usual setting was in H s loc (R n × R) with s > (n + 1)/2, in which case H s loc is an algebra.The methods were commonly based on a microlocal analysis of the nonlinear action [4,35], as well as on paradifferential calculus [6].Few papers addressed propagation of local regularity in lower Sobolev regularity, as the paper [13] which went as low as s > 0 (but still requiring L ∞ loc ); see also the early counterexamples of anomalous bicharacteristic behavior in low regularity in the second part of [33].
In the meantime the wellposedness of problem (1) for data of low regularity has been clarified.Recall that problem (1) is locally wellposed in H s if, for every Further, u is required to belong to a space on which the p-th power is welldefined (usually L p loc (R n+1 )), and the map (u 0 , u 1 ) → u should be continuous.The p-th power here may also be understood as a Fourier product, see Section 3.
As summarized in [8,11], the critical regularity for local H s -wellposedness of problem ( 1) is Essentially, wellposedness has been established for s ≥ s crit , possibly with additional constraints in certain ranges of p and n, while illposedness has been proven for s < s crit , again with certain gaps in the ranges.Relevant literature is [16,18,19,20,21,40], as well as recent directions for the probabilistic wellposedness [7,24,25,26,41].For more details, the reader is referred to the summaries in [8,11].
The case n = 1 deserves special attention.The critical exponent is The stronger bound is needed in order to have H s (R) ⊂ L p (R).It was shown in [8] that problem (1) is H s -illposed for s < s sob .In addition, it was shown there that norm inflation takes place for 1 2 − 1 p−1 < s < s sob and for s ≤ − 1 2 .Further, the authors also showed that the solution map is discontinuous at (0, 0) for s ≤ 1 2 − 1 p−1 .These results were complemented by [11] which proved norm inflation also in the range s < 0. It is also noted in [8] that problem ( 1) is locally H s -wellposed when n = 1 and s ≥ s sob .

Notation
The notation generally follows [39].In particular, the Fourier transform is used in the form The Sobolev spaces and local Sobolev spaces, respectively, are defined by The distribution (x + i0) λ ∈ S ′ (R) is defined as It is an entire function of λ ∈ C, see e.g.[12,Section I.3.6].Its Fourier transform is given by Remark 1.The following properties are easy to show.We assume here that λ < 0 so that the pseudofunction ξ −λ−1 + is locally integrable.Then, for λ < 0, the following equivalences hold:

Multiplication of distributions
This section serves to recall the products of distributions which will be used to define the integer powers in the semilinear wave equation (1).
Let S, T ∈ S ′ (R n ).The S ′ -convolution of S and T is said to exist, if where S − (x) = S(−x).In this case, the convolution is defined by S * T, ϕ = (ϕ * S − )T, 1 , and If the S ′ -convolution of Fu and Fv exists, one may define the Fourier product The definition can be localized as follows.Assume that for every x ∈ R n there is a neighborhood Ω x and χ x ∈ D(R n ), χ x ≡ 1 on Ω x , such that the S ′ -convolution of F(χ x u) and F(χ x v) exists.Locally near x, the product u • v is defined to be F −1 (F(χ x u) * F(χ x v)).Globally, it is defined by a partition of unity argument.
Remark 2.Here are some special cases in which the Fourier product exists.
(a) The existence of the S ′ -convolution of S, T ∈ S ′ (R n ) is guaranteed if both S and T have their support in a closed, acute and convex cone Γ in R n .Further, S * T is also supported in Γ, and the map (S, T ) → S * T is separately continuous in S ′ (R n ) [42, I.5.6,I.4.5].Let For u, v ∈ S ′ Γ (R n ), the product u • v is thus definable by (4) and belongs to S ′ Γ (R n ).Thus S ′ Γ (R n ) forms an algebra with respect to multiplication, and the multiplication map is separately continuous.
A simple calculation shows that the S ′ -convolution u * v exists and coincides with the ordinary convolution.By the exchange formula for L 2 -functions, the Fourier product u • v coincides with the ordinary product of two L 2functions.Using localization as indicated above, the same holds for the product of two L 2 loc -functions.In particular, for u, v ∈ H s loc (R n ) with s > n/2, the Fourier product exists and coincides with the product in the algebra whose Fourier product does not exist, as shown in the recent paper [28].However, if both the ordinary product and the Fourier product exist, they necessarily coincide.
(c) The product defined by Hörmander's wave front set criterion [14], requiring that for every , is also a special case.This can be seen by localizing the arguments establishing case (a), see e.g.[22,Proposition 6.3].
The remark shows that all products of functions and distributions occurring in this paper can be subsumed under the framework of the Fourier product.Further details and a discussion of different products of distributions can be found in [22].
Therefore, all integer powers exist both in the sense of the Fourier product and using Hörmander's wave front set criterion.Their Sobolev regularity is summarized in Remark 1.

Anomalous propagation of singularities to 1D-semilinear wave equations
In this section, we consider the propagation of singularities for the semilinear wave equation ( 1) in one dimension.The following proposition exhibits explicit solutions with stationary singular support.They are used below to construct solutions whose singular support propagates in arbitrary noncharacteristic directions.
Proposition 4. For every s < 1 2 there are λ < 0 and p ∈ N such that (a) the distribution u 0 (x) = (x + i0) λ belongs to H s loc (R), singsupp u 0 = {0}, and (b) u(x, t) ≡ u 0 (x) is a distributional solution to the semilinear wave equation where the nonlinear term is understood in the sense of the Fourier product.Its singular support is the noncharacteristic line {(0, t) : t ∈ R}.
Similarly, for any 1 = p ∈ N there are λ and s < 1 2 such that u(x, t) ≡ u 0 (x) is a distributional solution of (1).
Multiplying the solution u of ( 5) by a constant, we obtain a corresponding solution of (1).
The special solutions exhibited here are self-similar solutions to the semilinear wave equation.However, they do not belong to the classes of functions considered e.g. in [5,17,29,30,38].
Proof of Proposition 4. The function C → S ′ (R), λ → (x + i0) λ is analytic and it is well-known that The support of its Fourier transform is [0, ∞), so all integer powers make sense by means of the Fourier product.Further, Remark 5. (a) Anomalous propagation of singularities.When s > 1 2 , equation (5) with initial data in H s (R) × H s−1 (R) would have a unique solution which belongs to C 0 ([−T, T ] : ), so the anomalous singular support of the solution from Proposition 4 would be ruled out by the results in [34,35].
Using certain Lorentz transformations, it is possible to transform the stationary solutions u 0 (x) = (x + i0) λ from Proposition 4 to time dependent solutions with singular support on noncharacteristic rays.Starting with the case n = 1, the transformation x t → L x t , L = cosh θ sinh θ sinh θ cosh θ keeps the quadratic form x 2 − t 2 invariant, while the transformation respectively.In particular, if u(x, t) ≡ u 0 (x) is a stationary solution to (6) (with ∂ t u(x, 0) = 0), then v(x, t) = u 0 (x cosh θ + t sinh θ) solves the same equation with a sign change and with initial data while w(x, t) = u 0 (x sinh θ + t cosh θ) solves ( 6) with initial data w(x, 0) = u 0 (x sinh θ), ∂ t v(x, 0) = cosh θ u ′ 0 (x sinh θ).
Suppose now that u 0 (x) has singular support equal to x = 0.The reparametrization which has its singular support along the line {x + ct = 0, t ∈ R}, that is, inside the light cone, while the singular support of the initial data is still {x = 0}.Similarly, the reparametrization which has its singular support along the line {x ± ct = 0, t ∈ R}, that is, outside the light cone.In conclusion, the stationary solutions from Proposition 4 can be transformed to nonstationary solutions with singular support on any ray off the light cone.

Special products of distributions
The subsequent analysis requires refined estimates for products of Sobolev functions whose Fourier transform is supported in Γ = [0, ∞) ⊂ R. For later reference we present results for R n .
Let Γ be a closed, acute, convex cone in R n and s ∈ R. Notation: Note that the solutions given in Section 4 locally belong to H s Γ (R) at any fixed time t, for Γ = [0, ∞).The product of two members f ∈ H s 1 Γ (R n ) and g ∈ H s 2 Γ (R n ) is understood in the sense of the Fourier product.
Write ξ 0 for the n-dimensional integral ξ 1 . . .ξn etc.The proof of (a) starts with Minkowski's inequality for integrals and the observation that holds for the characteristic functions of the indicated n-dimensional intervals.Thus For ξ ≥ 0, η ≥ 0 and s 1 ≤ 0, σ − s 1 ≤ 0 (which holds for the σ under consideration provided s 2 ≤ n 2 ) one has .
Remark 7. The estimates in (a) are sharp.For example, when n = 1, s ≤ 0 and The estimates in (b) are not sharp.For example, if On the other hand, if f and g merely belong to H s 1 (R n ) and H s 2 (R n ), then f g are only known to belong to H σ (R n ) for σ < − n 2 + s 1 + s 2 under the condition that s 1 + s 2 ≥ 0 [15, Theorem 8.2.1], see also [2,Lemma 1.3] and [3, formula (1.5)].
In view of the intended application to the semilinear wave equation ( 1) we now assume that u ∈ H s Γ (R n ) where Γ is a cone as above.Let p be a positive integer.We wish to determine ranges for σ such that u p ∈ H σ Γ (R n ).Remark 8. (Sobolev properties of integer powers) The case s ≤ 0.Here The case 0 < s n 2 .Here Proposition 6(a) immediately gives that This follows by induction, using Proposition 6(a) and (b).Indeed, , and so on.
Note that in particular for s = n 2 , u p ∈ H σ Γ (R n ) for all p and σ < 0. The case s > n 2 .Here u p ∈ H s Γ (R n ) for all p since the latter space is an algebra.In all cases, u p H σ ≤ C u p H s for some constant C > 0.
The application of Proposition 6 does not produce new estimates for the power function on H s Γ (R n ) for n ≥ 2 and s ≤ n 2 .We henceforth concentrate on the case n = 1.In the context of the semilinear wave equation, the question arises whether u ∈ H s Γ (R) implies u p ∈ H s−1 Γ (R), that is, whether the σ in Remark 8 can attain a value ≥ s − 1.The answer is summarized in the following remark, which is of interest when s ≤ 1 2 .Remark 9. Suppose that u ∈ H s Γ (R), where Γ is a closed half-ray.Then u p ∈ H s−1 Γ (R) in the following cases: In all cases, u p H s−1 ≤ C u p H s for some constant C > 0.
6 Application to 1D-semilinear wave equations In this section, we address wellposedness in H s Γ of the Cauchy problem for the semilinear wave equation in one space dimension; here p ≥ 2 is a positive integer.We denote by Γ ⊂ R a closed half-ray which may be assumed to be the half-line [0, ∞).
Theorem 10.Assume that p and s are in the range given by (7).
Then there is T > 0 such that problem (8) has a unique distributional solution in C([−T, T ] : The goal is to construct a fixed point u = Mu in the for small T .Note that (for t ≥ 0) . Indeed, apply the estimate above to v = u(t + h), w = u(t).
(b) M : B T → B T .This follows from the estimate (c) M is a contraction on B T for small T .This follows from the similar estimate Existence and uniqueness follow.Also, u = Mu and u ∈ C([−T, T ] : . Finally, Gronwall's inequality gives local Lipschitz continuity.For p ≥ 3 a similar argument works using factorization of v p − w p and Proposition 6. Remark 11.Note that the lower bound in (7) is smaller than s sob in (2) only for p = 2 and p = 3, so in these cases Theorem 10 improves the results of [8,11].

Extensions to semilinear wave equations in higher dimensions
The results of the previous sections focused on new phenomena for semilinear wave equations on the real line.We now discuss their extension to higher dimensions.We shall again give explicit solutions to (1) exhibiting anomalous propagation of singularities.In these examples, the nonlinear term u p even exists in the classical sense of the p-th power of an p loc -function, which in this case coincides with the p-th power taken in the sense of the Fourier product.
In addition to (x + i0) λ , which was considered above, we introduce the radially symmetric pseudofunction r λ ∈ S ′ (R n ), defined by Remark 12.The product of the pseudofunctions r λ is defined as a Fourier product: First, one may extend the definition to λ ∈ C by taking finite parts in the poles.It was proved in [27,Satz 5] that the S ′ -convolution of Pf r α and Pf r β exists if and only if Re(α + β) < −n.This can be used to characterize the range of exponents for which the Fourier product exists.However, for the present paper, only the range λ ∈ R, −n < λ < 0 will be needed, in which case both r λ and F(r λ ) are locally integrable functions.We show that -in the indicated range of exponents -the Fourier product of r λ and r µ exists if λ + µ > −n.Up to constant factors, the respective Fourier transforms are ρ −λ−n and ρ −µ−n .Take ϕ ∈ S(R n ).Then (ϕ This is exactly the case when λ + µ > −n.Thus the Fourier product of r λ and r µ exists in this range.A proof that r λ •r µ = r λ+µ can be found, e.g., in [22,Example 5.4].Incidentally, r λ , r µ and r λ+µ are locally integrable functions in the range 0 > λ, µ > −n, λ + µ > −n, and the usual product equals r λ r µ = r λ+µ , thus coincides with the Fourier product.The same holds for integer powers (r λ ) p = r λp when λp > −n.
In particular, when λ > 2 − n and p = 1 − 2/λ, r λ belongs to L p loc (R n ), (r λ ) p = r λp and it satisfies the elliptic equation ∆r λ = λ(λ + n − 2)(r λ ) p , where the derivatives are understood in the weak sense and the pth power as the evaluation of the Nemytskii operator L p loc (R n ) → L 1 loc (R n ).
Remark 13.The following properties are easy to show.We assume here that λ < 0 so that the pseudofunction ρ −λ−n is locally integrable.