Solutions with Prescribed Local Blow-up Surface for the Nonlinear Wave Equation
-
Thierry Cazenave
,
Yvan Martel
Abstract
We prove that any sufficiently differentiable space-like hypersurface of
1 Introduction
1.1 Main Result
We consider the nonlinear energy-subcritical or -critical wave equation
for
When a solution u with initial data at
We prove that any sufficiently differentiable space-like hypersurface of
Theorem 1.1.
Let
Let
There exist
It follows from (1.4) that
Compared to previous results (see Section 1.3), Theorem 1.1 applies to any space dimension
Remark 1.2.
In the definition of
1.2 Definition of the Maximal Influence Domain
We adapt the presentation of [1, Chapter III] (see also [24]) to the framework of
For any
Definition 1.3.
An open set Ω of
For Ω an influence domain containing
From the above definition, either ϕ is identically
Recall that by the Cauchy theory in the energy space
From the local Cauchy theory, it is standard to define the notion of maximal solution and maximal time of existence
To define the notion of maximal influence domain corresponding to an initial data, we first extend the Cauchy theory of
Suppose that
Next, consider the solution
By time-translation invariance of the equation and considering the map
Now, we define the notion of solution in an influence domain.
Definition 1.4.
Let
For any
moreover,
For any
Definition 1.5.
For any
It follows that, for any initial data
1.3 Previous Results
Under certain assumptions, it is known that the upper boundary of the maximal influence domain is a blow-up surface in the sense that the solution blows up (at the same rate as the ODE) on the surface, and the blow-up surface is
Constructing solutions of the wave equation (1.1) with prescribed blow-up surface is a classical question. Results similar to Theorem 1.2 have been proved in several cases. For the wave equation with cubic nonlinearity, it is proved in [18, Theorem 10.14, p. 192] that there exist solutions (locally defined around the blow-up surface) blowing up exactly on a prescribed surface of class
A related question is the study of the blow-up set, which is the intersection of the blow-up surface with the hyperplane
1.4 Strategy of the Proof of Theorem 1.1
We follow closely the strategy of [6] (see also [7]). It is based on the construction of an appropriate approximate solution which blows up at
in the dual variables
1.5 Notation
We fix a smooth, even function
Let
In the present article, we use multi-variate notation and results from [8]. For
For
Recall that for two functions
We write
We recall the Faa di Bruno formula (see in [8, Corollary 2.10]). Let
where
We will also need to differentiate in space and time, so we define multi-index notation in space-time:
For
Then, for two functions
We write
Last, we write in this context the Faa di Bruno formula. Let
where
2 Blow-up Ansatz
2.1 Change of Variables
Let
We perform a change of variable related to ψ
so that
Therefore, equation (1.1) on
In this section, we focus on finding ansatz for this equation under assumption (2.1).
2.2 First Blow-up Ansatz
Let
where
be an integer.
We consider the function
It follows that A is of class
We define a basic blow-up ansatz
where
which satisfies
Since the functions ψ and A are of class
In view of (2.2), it is natural to set
We gather in the next lemma the properties of
Lemma 2.1.
The function
Moreover, for any
If
(2.10)If
(2.11)If
(2.12)(2.13)
Furthermore, if
Proof.
First, we observe that the function κ is constant for
Proof of (2.9). This follows from direct computations.
Proof of (2.10). For
In particular,
Let
For
For
where
As before, we use for
Since
We obtain, for all
which proves (2.10) for
We use the notation
where
Using (2.15) and
Proof of (2.11). We estimate the three terms in (2.8). It follows from Leibniz’s formula (1.14), the properties of ψ,
Using once more that
Proof of (2.12). It follows from the properties of the functions ψ and A that
Estimate (2.12) follows immediately. Then we have, for any
which implies (2.13).
Proof of (2.14). Since
2.3 Refined Blow-up Ansatz
Starting from
Let
where
Lemma 2.2.
There exist
If
(2.16)If
(2.17)(2.18)If
(2.19)If
(2.20)(2.21)
Remark 2.3.
To complete the energy control in Section 3, we need an error estimate of the form
as in [6] (see (3.28)), as well as an estimate of the form
Proof of Lemma 2.2.
We observe that (2.19), (2.20) and (2.21) for
Proof of (2.16). Let
Indeed, we have by Leibniz’s formula
and thus using (2.10) and (2.19) for
For
where
This means that we can integrate this term on
For
which proves (2.22). Similarly, using Leibniz’s formula, we check the estimate
In particular, for
where, using
Thus, by integration on
For
which proves (2.23).
Using estimates (2.10), (2.22), (2.23) and again Leibniz’s formula, we obtain, for all
These estimates imply (2.16) for
Proof of (2.17)–(2.18). For
For
For
Choosing
In the case
which is (2.17).
To prove (2.18), we note that by (2.16), and using
Proof of (2.19). Note that (2.19) for
By direct computations, we briefly check that the function
Indeed, we have
and thus, using (2.9),
Differentiating in s again, and using (2.9), we obtain
which is (2.24).
Using (2.24),
We estimate each term of the right-hand side above for
For the first term, recall that for
Now, we treat the next three terms in the expression of
we see that these three terms are estimated by
Finally, we estimate
and
These estimates imply
For
Thus,
Now, we consider the case
Therefore, using the notation
and, by the Faa di Bruno formula (1.15), for
To estimate the term
Thus, for
Second, for
Thus, we have proved
and so by integration in
We now estimate
and thus
Moreover, for
To estimate the term
For
For
Second, for
Thus, we obtain
Integrating in
Combining (2.26) and (2.27), we have proved for
In conclusion, we have estimated all terms in the expression of
Proof of (2.20)–(2.21). For
3 Construction of a Solution of the Transformed Equation (2.2)
Let the function χ be given by (1.7), let
and impose the following additional condition on k:
Recall that
Our main result of this section is the following.
Proposition 3.1.
Assume that
There exist
which is a solution of (2.2) in
with λ given by (3.1). In addition, there exist a constant C and a function
a.e. on
We construct the solution v of Proposition 3.1 by a compactness argument. For any n large, let
We let n be sufficiently large so that
Let
We define the sequence of solution
The nonlinearity
We set, for all
thus
Proposition 3.2.
There exist
for all
Proof.
We fix
We define the auxiliary function Q as follows:
where, by abuse of notation, we denote
Thus, setting
(by the definition of Q and
The nonlinear term
We define the following energy functional related to the above formulation of the equation of w:
We also define a weighted norm related to the above functional
Since we may be dealing with
and
For future reference, we establish two estimates on
Thus, since
Similarly, by (2.10),
and
Step 1: Coercivity. We claim the following estimates.
Lemma 3.3.
It holds
For
Proof.
First, we prove the following estimates. For any
We have, using (3.13),
This proves (3.17) and the proof of (3.18) is similar. Moreover, using (3.12),
which proves (3.19); the proof of (3.20) is similar.
We prove (3.15). The inequality
Since
Last, we prove (3.16). Let
The triangle inequality and the Taylor inequality (1.8) yield
where
From (2.17),
and so
For the first term, we prove the following general estimate: for any
Indeed, using Hölder’s inequality and the embedding
In particular, from (3.24), it holds
In the case
and so, using (3.24)
Last, since
In conclusion, we have obtained, for
which, combined with (3.15), implies that for
Step 2: Energy Control. We claim that there exist
provided
Proof of (3.26). We compute
First, we remark the negative contribution of
Second, we compute
For
Second, by integration by parts,
By the definition of
Using (3.13), we also have
Now, by the expressions of Q and
and thus
Similarly, using (3.12), (3.13), (3.14)
and
Using the same estimates and then (3.19), we finish estimating
Thus, for some constant
Next, integrating by parts, using the identities
and integrating again by parts, we find
By (3.13) and the definition of
Similarly, using (3.13) and (3.14), we have
For
Thus,
For
For
and we need only estimate
Since
Recall that by (3.1),
and that (3.2) is equivalent to
Thus, for
It follows that
For this term, we have obtained
Finally, by the Cauchy–Schwarz inequality,
Using (3.27), we obtain
In conclusion for
To continue with the proof of (3.26), we estimate the term
Using (3.24), the first term is controlled as follows:
In the case
where
In conclusion, we have proved
We proceed similarly for
by (1.9) and Taylor’s inequality,
Using (3.29), we conclude that
Now, we estimate
By the triangle inequality, Taylor inequality (1.11),
Using (3.29), we conclude that
Finally, we estimate
By the triangle inequality and Taylor’s inequality (1.11)
Using (2.18),
Using (3.29) and
Choosing
Step 3: Higher-Order Energy Terms. We claim that for any
Differentiating (3.9) with respect to s, setting
Differentiating
First, by the Cauchy–Schwarz inequality
From
and then (3.19) with
Thus, using also
Second, using (2.10) and (2.18),
so that Taylor’s inequality (1.10) yields
We have
Moreover, since
Thus,
Third, from (2.21), for
and thus, following the proof of (3.28), we have
The above estimates prove (3.30) for
We now prove (3.30) for
(3.32)
Differentiating
The term
so that
Moreover, from (2.21), for
and thus, following the proof of (3.28), we have
Finally,
so that
Step 4: Conclusion. Since
and by continuity,
for some constant
for some constant
This gives, for all
By integration, using
Thus, from (3.16), it holds, for
It follows from (3.15) and the definition of
This completes the proof of the proposition. ∎
Proof of Proposition 3.1.
We set
From Proposition 3.2, there exist
for all
It follows from estimate (3.33) that there exist a subsequence of
It is then easy to pass to the limit in (3.34), and it follows that
in
it holds
and, using the definition of
Finally, we prove (3.6). We write
On the other hand,
Since
Next, we write
It follows from (2.20) that
by (2.10). Using again (3.41), we conclude that
Since
4 Proof of Theorem 1.1
In this section, we use the following notation. We let
4.1 Cut-Off of the Local Hypersurface
Let φ be a function satisfying (1.3) (see statement of Theorem 1.1). Without loss of generality, by the invariance by rotation of equation (1.1), we assume that
(For dimension 1, the reduction is done by possibly changing
On the one hand, from this definition and the properties of χ, it holds
On the other hand, from
it holds on
We fix
The first constraint on
4.2 Construction of the Function ψ
We claim that for any
(As observed before, we ignore
and we compute, using (4.3),
and
Thus, for fixed
so that by (4.5)
for all
on
Moreover, using (4.9), we see that
For all
Equivalently, the functions ψ and
We check that ψ is of class
First, since φ is of class
Second, from (4.1), it follows that
Last, we estimate
and for
It follows from (4.3) that
In particular, we see that
so that (3.3) is proved.
4.3 Definition of an Appropriate Solution of the Transformed Equation
We assume (2.3), (2.4), (3.1), (3.2) and we consider the function ψ defined in (4.12)–(4.13). Note that ψ is of class
4.4 Returning to the Original Variable
Let
Recall (see (4.1) and (4.3)) that
It follows that the space-time region
is an influence domain in the sense of Section 1.2. (See Figure 1.) Moreover, let
The set
Given
It is well known that
Since ψ is of class
The map Λ has the expression
and it follows that
We prove that
and that
In the case where
Property (4.20) follows. Moreover,
by (4.16). Thus (4.21) is proved in this case.
In the case where
Using (4.10), we replace
Recall that by (4.8), we have
which means that
hence, using (4.9),
on
Using (4.16), we see that
We now set
We refer to [18, Exercise 10.7.c] for a similar use of the Lorentz transform. Note that by (4.21), u is well defined.
Let ω be an open subset of
Since
Next, it follows from (4.25) that
and
where the argument of ψ is y and the argument of
Similar formulas hold for all first and second space-time derivatives of
Since
in
Set
and
so that
4.5 Choice of a Solution of the Nonlinear Wave Equation
We apply Section 1.2 to extend u, which is a solution of (1.1) on
and that
4.6 Blowup on the Local Hypersurface and End of the Proof
We show blowup on the local hypersurface by proving (1.4). For this, we further restrict the size of the hypersurface. Arguing as in the proof of (4.18), we see that
Thus we see that if
We fix
Let
Given
and, given
We claim that there exist
where
Assuming (4.31)–(4.32), we conclude the proof of (1.4). Given
so that, using
Therefore,
Applying (3.6) and (4.31), we deduce that
It follows from (4.32), (2.14) and (4.30) that
Furthermore,
Next,
Estimate (1.4) follows from (4.33)–(4.36).
It remains to prove the claim (4.31)–(4.32). Let
In the case
Using the right-hand side inequality in (4.9), and then (4.3), we deduce
and (4.37) by using again (4.3).
Next we claim that
Indeed, by (4.19) for
so that
Estimate (4.38) follows by using the triangle inequality
Estimating
Since
It now remains to prove that if
Using (4.39), we obtain
which proves the claim for
Finally, we prove that the hypersurface
This is absurd, since by (1.4), given
This completes the proof of the theorem, where
Dedicated to Laurent Véron on the occasion of his 70th birthday
Funding statement: Lifeng Zhao was partially supported by the NSFC Grant of China No. 11771415.
A The Wave Equation (2.2)
Let
A.1 The Associated Semigroup
Let X be the Hilbert space
and consider the linear operator
with domain
which proves that
It is easy to solve the second equation by the Lax–Milgram theorem, and we obtain a solution
A.2 The Nonlinear Equation
Using the notation
where
A.3 Regularity
Suppose
A.4 The Case of Equation (3.7)
Equation (3.7) is equation (A.1), where f is replaced by
B Uniqueness on Light Cones
We state and prove a uniqueness property for solutions of the nonlinear wave equation on light cones (Proposition B.2), for which we could not find a reference. We first recall in the following remark the relevant results concerning the local well-posedness of the Cauchy problem.
Remark B.1 (Local Well-Posedness).
Let
of the wave equation
We also discuss the property
in the case
(i) Case
(ii) Case
(iii) Case
(iv) Case
Proposition B.2 (Uniqueness on Light Cones).
Let
be two solutions of the wave equation
The proof of Proposition B.2 relies on the following local estimates.
Lemma B.3.
Let
satisfy
for all
In (B.4) and (B.5), the constant C independent of h, R, τ and t.
Proof.
We define
We let
Therefore, estimate (B.4) with z replaced by
To conclude the proof, we show that z and
satisfies
By finite speed of propagation, it follows that
for all
On the other hand, it follows from (B.6) that
Proof of Proposition B.2.
We need only prove the result for τ small, the general case follows by iteration.
The Case
Since
The Case
By Hölder’s inequality, it follows that
where all the integrals are on
where all the integrals are on
the conclusion follows by choosing τ sufficiently small. ∎
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Articles in the same Issue
- Frontmatter
- Solutions with Prescribed Local Blow-up Surface for the Nonlinear Wave Equation
- On Nodal Solutions of the Nonlinear Choquard Equation
- Limit Configurations of Schrödinger Systems Versus Optimal Partition for the Principal Eigenvalue of Elliptic Systems
- Fractional Powers of Monotone Operators in Hilbert Spaces
- Nonradial Solutions for the Hénon Equation Close to the Threshold
- 𝜔+-Type Index of GL+(2)-Paths
- The Concentration Behavior of Ground States for a Class of Kirchhoff-type Problems with Hartree-type Nonlinearity
- p-Harmonic Functions in ℝN+ with Nonlinear Neumann Boundary Conditions and Measure Data
Articles in the same Issue
- Frontmatter
- Solutions with Prescribed Local Blow-up Surface for the Nonlinear Wave Equation
- On Nodal Solutions of the Nonlinear Choquard Equation
- Limit Configurations of Schrödinger Systems Versus Optimal Partition for the Principal Eigenvalue of Elliptic Systems
- Fractional Powers of Monotone Operators in Hilbert Spaces
- Nonradial Solutions for the Hénon Equation Close to the Threshold
- 𝜔+-Type Index of GL+(2)-Paths
- The Concentration Behavior of Ground States for a Class of Kirchhoff-type Problems with Hartree-type Nonlinearity
- p-Harmonic Functions in ℝN+ with Nonlinear Neumann Boundary Conditions and Measure Data
