# Anomalous solutions to nonlinear hyperbolic equations

Michael Oberguggenberger Unit of Engineering Mathematics, University of
Innsbruck, Technikerstraße 13, 6020 Innsbruck, Austria,
<EMAIL_ADDRESS>
###### Abstract

The behavior of sufficiently regular solutions to semilinear hyperbolic
equations has attracted a great deal of attention in the past decades,
concerning local/global existence, finite time blow-up, critical exponents,
and propagation of singularities. Solutions of lower regularity may exhibit
unexpected (anomalous) propagation of singularities. The purpose of this paper
is to present various striking examples that seemingly have not been addressed
in the literature so far. The key issue is the interpretation of the nonlinear
operations.

##  1 Introduction

This paper serves to display various unusual, or _anomalous_ solutions to
semilinear wave equations

$\frac{1}{c^{2}}\partial_{t}^{2}u-\Delta u=f(x,t,u),\quad u(x,0)=u_{0}(x),\
\partial_{t}u(x,0)=u_{1}(x)$ (1)

in space dimension $n\geq 1$, and to advection-reaction equations

$\frac{1}{c}\partial_{t}u+\partial_{x}u=f(x,t,u),\quad u(x,0)=u_{0}(x)$ (2)

in one space dimension as prototypical hyperbolic partial differential
equations. The nonlinear function $f$ is assumed to be smooth. Here typically
$f(x,t,u)=\pm|u|^{p}$ or $|u|^{p-1}u$. The main research direction in the past
decades has been to find bounds on the exponent $p$ and the regularity of the
initial data, asking about the existence of global solutions with small or
large initial data, local solutions, self-similar solutions, blow-up in finite
time or stability of blow-up. The reader is referred to the discussion in the
monograph [13], the survey article from the 1990s [35], a collection of
currently known critical exponents [23] and some of the papers discussing the
development of the field [16, 36]. Relevant literature on self-similar
solutions and stationary solutions as building blocks will be quoted at the
appropriate place in Section 4.

In the 1980s and 1990s, a central question has been propagation of
singularities, which started with the discovery of Jeffrey Rauch and Michael
Reed [28, 29] that in semilinear hyperbolic equations and systems,
singularities do not only propagate out from initial singularities along
characteristics or bicharacteristics as in the linear case, but may be created
at later times by the interaction of previous singularity bearing
(bi-)characteristics. For example, an initial singularity at the origin in
problem (1) may lead to singularities in the solution that fill up the solid
light cone [2]. For a survey of the vast literature up to around 1990 we refer
to the monograph [3]. Rauch and Reed coined the term _anomalous singularities_
for this phenomenon.

The results on anomalous singularities required sufficient overall regularity
of the solution, for example $H^{s}_{\rm loc}$-regularity with $s>(n+1)/2$,
and the mechanism for creating the anomalous singularities was still based on
characteristics, bicharacteristics and their interaction.

The anomalous solutions presented in this paper are distinguished by (a) lower
regularity than in the previous literature and (b) propagation along non-
characteristic curves. The majority of examples is based on non-regular
solutions to the corresponding stationary elliptic equation. Derivatives are
always understood in the sense of distributions. In an attempt to categorize
the solutions, four types will be singled out:

_Type I:_ | products defined by Hörmander’s wave front set criterion;
---|---
_Type II:_ | products and powers evaluated by Nemytskii operators;
_Type III:_ | limits of weak asymptotic solutions;
_Type IV:_ | sequential solutions, especially very weak solutions in the
| sense of Ruzhansky.

It is worth noting that all constructed solutions come with a certain
assertion of uniqueness.

The plan of the paper is as follows. Section 2 serves to recall results on
anomalous propagation of singularities for sufficiently regular solutions, for
reasons of comparison. Section 3 addresses Type I solutions, introducing the
employed multiplication of distributions and discussing the question of
regularization. Section 4 will exhibit seemingly harmless solutions lying in
an $L^{p}$-space on which the nonlinear operations are defined and continuous
(Type II). In Section 5 it will be shown that the solutions from Section 4
arise as limits of nets of asymptotic solutions (satisfying the equations up
to an error term converging weakly to zero, Type III). In Section 6 nets of
smooth functions $(u_{\varepsilon})_{\varepsilon>0}$ will be constructed that
solve the equations at each fixed $\varepsilon>0$, but need not necessarily
converge as $\varepsilon\to 0$ (Type IV). Nevertheless, their regularity
properties can be characterized by suitable estimates on their growth in terms
of negative powers of $\varepsilon$ as $\varepsilon\to 0$. The appendix serves
to recall some notions required to define the products arising in Type I
solutions.

The author has been aware of the existence of these anomalous solutions since
the early 1980s, but due to a lack of explanation, hesitated to publish them
so far. It is hoped that this publication will arouse interest in these types
of solutions among the community. Many more examples of similar nature are
known, collected by the author and in joint work with Hideo Deguchi [9].

What concerns notation, $H^{s}$ denotes the usual Sobolev space based on
$L^{2}$; ${\mathcal{C}}^{k}$ denotes the space of $k$-times differentiable
functions, ${\mathcal{C}}^{k}_{b}$ the subspace of functions with bounded
derivatives up to order $k$. The notation for spaces of test functions and
distributions follows [34]. The Fourier transform is defined as
${\mathcal{F}}\varphi(\xi)=\int{\rm e}\hskip 1.0pt^{-2\pi{\rm i}\hskip
1.0ptx\xi}\varphi(x)\hskip 1.0pt{\rm d}\hskip 0.5ptx$.

##  2 Propagation of singularities for regular solutions

This section serves to recall results from the 1980s on propagation of
singularities for solutions to semilinear hyperbolic systems. These results
hold for sufficiently regular solutions ($L^{\infty}_{\rm loc}$ in one space
dimension, $H^{s}_{\rm loc}$ for $s>(n+1)/2$ in space dimension $n$). We do
not strive for full generality – the quoted results will be contrasted with
the much less regular solutions to be constructed in the following sections.

We start with $(m\times m)$-systems of first order hyperbolic equations in one
space dimension, considering the initial value problem

$\begin{array}[]{l}(\partial_{t}+\Lambda\partial_{x})u(x,t)=f(x,t,u(x,t)),\
(x,t)\in R\\\\[4.0pt] u(x,0)=u_{0}(x),\ x\in R_{0}\end{array}$ (3)

where $R_{0}\subset\mathbb{R}$ is an interval and
$R\subset\mathbb{R}\times[0,\infty)$ is its domain of determinacy. Here
$u=(u_{1},\ldots,u_{m})$, $\Lambda={\rm diag}(\lambda_{1},\ldots,\lambda_{m})$
with real and constant entries $\lambda_{i}$, and $f=(f_{1},\ldots,f_{m})$ is
smooth. Let $x_{1},\ldots,x_{k}\in R_{0}$ and denote by $S_{0}$ the union of
characteristic lines emanating from $x_{1},\ldots,x_{k}$. Following [29],
construct the forward characteristic lines starting at the intersection points
of $S_{0}$ and call this set $S_{1}$. Let $S_{2}$ be the set of forward
characteristic lines starting from the intersection points of $S_{1}$.
Continue recursively to construct a sequence of sets $S_{j}$. Let $S$ be the
closure of $\bigcup_{j=0}^{\infty}S_{j}$ intersected with $R$.

######  Proposition 2.1.

Let $u\in(L^{\infty}(R))^{m}$ satisfy (3) in the sense of distributions and
take on the initial data $u_{0}\in(L^{\infty}(R_{0}))^{m}$. Suppose that
$u_{0}$ is ${\mathcal{C}}^{\infty}$ with each derivative uniformly bounded on
the complement of the finitely many points $x_{1},\ldots,x_{k}$. Then $u$ is
${\mathcal{C}}^{\infty}$ on $R\setminus S$ and all derivatives of $u$ have
continuous extensions from each connected component of $R\setminus S$ to its
closure.

###### Proof.

This is Theorem 1 from [29]. ∎

######  Remark 2.2.

(a) If the function $f$ is linear, then the solution $u$ is in
${\mathcal{C}}^{\infty}$ on $R\setminus S_{0}$ – singularities can only lie on
characteristic curves tracing back to the singularities of the initial data.
In the nonlinear case, the solution is not ${\mathcal{C}}^{\infty}$ on
$S\setminus S_{0}$, in general. The singularities belonging to $S\setminus
S_{0}$ in the nonlinear case have been termed _anomalous singularities_ by the
authors.

(b) In the scalar case and in the case of $(2\times 2)$-systems (thus $m=1$ or
$m=2$), $S=S_{0}$, so no anomalous singularities arise.

Next we recall a result of [27] on propagation of singularities for semilinear
wave equations. Consider the initial value problem

$\begin{array}[]{l}(\partial^{2}_{t}-\Delta)v(x,t)=f(v(x,t)),\
(x,t)\in\mathbb{R}^{n}\times\mathbb{R},\\\\[4.0pt] v(x,0)=v_{0}(x),\
\partial_{t}v(x,0)=v_{1}(x),\ x\in\mathbb{R}^{n},\end{array}$ (4)

where $f$ is a polynomial with $f(0)=0$, $\Delta$ denotes the $n$-dimensional
Laplace operator, and $u_{0}\in H^{s}_{\rm loc}(\mathbb{R}^{n})$, $u_{1}\in
H^{s-1}_{\rm loc}(\mathbb{R}^{n})$ with $s>(n+1)/2$. Note that $H^{s}_{\rm
loc}(\mathbb{R}^{n}\times\mathbb{R})$ is an algebra in this case, even
contained in the space of continuous functions, so $f(u)$ is classically
defined.

######  Proposition 2.3.

Let $s>(n+1)/2$ and $v\in H^{s}_{\rm loc}(\mathbb{R}^{n}\times\mathbb{R})$
satisfy (4) in the sense of distributions. Suppose that $v_{0}$ and $v_{1}$
belong to ${\mathcal{C}}^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$. Then $v$
is ${\mathcal{C}}^{\infty}$ on $\\{|x|>|t|\\}$, and it belongs to
$H^{s+1+\sigma}_{\rm loc}(\mathbb{R}^{n}\times\mathbb{R})$ on $\\{|x|<|t|\\}$
for all $\sigma<s-(n+1)/2$.

###### Proof.

This follows from Theorem 3.1, together with Theorem 1.1 of [27]. ∎

######  Remark 2.4.

In space dimension $n=1$, the solution $v$ is actually
${\mathcal{C}}^{\infty}$ in $\\{|x|<|t|\\}$, as follows from the Corollary to
Theorem 2 in [28] as well as the earlier paper [30].

It is known that the solution is not necessarily better than $H^{s+1+\sigma}$
in $\\{|x|<|t|\\}$ in space dimension $n\geq 2$. For a survey of the state of
the art around 1990, see [3].

##  3 Type I solutions – multiplication of distributions

In this section, we address weak solutions to nonlinear equations where the
involved products or powers exist in the sense of Hörmander’s wave front set
criterion [18]. The examples will be based on the one-dimensional distribution

$u_{0}(x)=\frac{1}{x+{\rm i}\hskip 1.0pt0}=\lim_{\varepsilon\to
0}\frac{1}{x+{\rm i}\hskip 1.0pt\varepsilon}={\rm vp}\frac{1}{x}-{\rm i}\hskip
1.0pt\pi\delta(x)$ (5)

also denoted by $\delta_{+}(x)$ in the physics literature. Here ${\rm
vp}\frac{1}{x}$ denotes the principal value distribution ${\rm
vp}\frac{1}{x}=\partial_{x}\log|x|$ and $\delta(x)$ is the Dirac measure. The
Fourier transform of $u_{0}(x)$ and its auto-convolution are

$({\mathcal{F}}u_{0})(\xi)=-2\pi{\rm i}\hskip 1.0ptH(\xi)\quad{\rm
and}\quad({\mathcal{F}}u_{0}\ast{\mathcal{F}}u_{0})(\xi)=-4\pi^{2}\xi H(\xi)$

where $H$ denotes the Heaviside function. In particular, the wavefront set of
$u_{0}$ is $\\{(0,\xi):\xi>0\\}$, thus $u_{0}^{2}$ exists according to
Hörmander’s criterion. Actually, it can simply be computed as Fourier product
(see Appendix),

$u_{0}^{2}={\mathcal{F}}^{-1}({\mathcal{F}}u_{0}\ast{\mathcal{F}}u_{0}),$

as well as all its powers. It holds that

$u_{0}^{2}(x)=\Big{(}\frac{1}{x+{\rm i}\hskip
1.0pt0}\Big{)}^{2}=-\Big{(}\frac{1}{x+{\rm i}\hskip
1.0pt0}\Big{)}^{\prime}={\rm Pf}\frac{1}{x^{2}}+{\rm i}\hskip
1.0pt\pi\delta^{\prime}(x)=-u_{0}^{\prime}(x)$ (6)

where ${\rm Pf}\frac{1}{x^{2}}$ is the Hadamard finite part distribution, and

$2u_{0}^{3}(x)=2\Big{(}\frac{1}{x+{\rm i}\hskip
1.0pt0}\Big{)}^{3}=\Big{(}\frac{1}{x+{\rm i}\hskip
1.0pt0}\Big{)}^{\prime\prime}=u_{0}^{\prime\prime}(x).$ (7)

###  3.1 A nonlinear advection-reaction equation

######  Proposition 3.1.

The distribution $u(x,t)\equiv u_{0}(x)$ given by (5) is a weak solution to
the initial value problem

$\frac{1}{c}\partial_{t}u+\partial_{x}u+u^{2}=0,\quad u(x,0)=u_{0}(x)$ (8)

for whatever $c\in\mathbb{R},c\neq 0$, where the square is understood in the
sense of Hörmander’s product.

###### Proof.

It is clear from (6) that $\partial_{x}u+u^{2}=0$ and that $\partial_{t}u=0$.
∎

Clearly, the mechanism producing this result is that the stationary solution
satisfies the nonlinear differential relation $u_{0}^{\prime}=-u_{0}^{2}$.
Further reasons why a genuine distribution can satisfy such a relation will be
discussed below. At first we wish to point out that the solution given in
Proposition 3.1 exhibits anomalous propagation of singularities. Indeed,

$\mathop{\mathrm{singsupp}}u=\\{(x,t):x=0,t\geq 0\\}$

while the expected singular support from Proposition 2.1 or Remark 2.2(b)
should be $\\{(x,t):x=ct\\}$. To be sure, $u_{0}$ does not belong to
$L^{\infty}$ as required in Proposition 2.1.

######  Remark 3.2.

It should be noted that anomalous propagation of singularities is not confined
to stationary solutions. The following example, due to Deguchi [9], shows that
any anomalous propagation speed is possible. Indeed,

$u(x,t)=\frac{1}{ax+bct+{\rm i}\hskip 1.0pt0}\quad{\rm with}\ a+b=1$

solves equation (8) with initial data $u_{0}(x)=(ax+{\rm i}\hskip
1.0pt0)^{-1}$.

######  Remark 3.3.

One possible explanation why the mentioned nonlinear differential relation, as
well as similar relations for the higher derivatives, hold for the specific
distribution (5) can be obtained by studying its representation as a boundary
value of an analytic function. Indeed, every distribution
$v\in{\mathcal{D}}^{\prime}(\mathbb{R})$ can be represented as the boundary
value of a function $\widehat{v}(z)$, analytic in
$\mathbb{C}\setminus\mathop{\mathrm{supp}}(u)$, in the sense

$v(x)=\lim_{\varepsilon\to 0}(\widehat{v}(x+{\rm i}\hskip
1.0pt\varepsilon)-\widehat{v}(x-{\rm i}\hskip 1.0pt\varepsilon)),$ (9)

see e.g. [37]. If $v$ is a distribution of compact support, $\widehat{v}(z)$
is given by the Fantappiè indicatrix

$\widehat{v}(z)=\frac{1}{2\pi{\rm i}\hskip
1.0pt}\big{\langle}v(x),\frac{1}{x-z}\big{\rangle}$

and in general by a partition of unity procedure. Further, $|\widehat{v}(z)|$
grows at most like a negative power of $|\mathop{\mathrm{Im}}z|$ as
$\mathop{\mathrm{Im}}z\to 0$, locally uniformly in $\mathop{\mathrm{Re}}z$.
The representation $\widehat{v}(z)$ is unique up to a function analytic on
$\mathbb{C}$. Further, every function $\widehat{v}(z)$, analytic in
$\mathbb{C}\setminus\mathbb{R}$ and satisfying the growth condition has a
distributional boundary value in the sense of (9).

If the support of $\widehat{v}(z)$ is contained in
$\\{\mathop{\mathrm{Im}}z>0\\}$, the representation is unique. Thus the space
of distributions ${\mathcal{H}}_{+}(\mathbb{R})$ whose Fantappiè parametrix
has support in the upper complex half plane is isomorphic to the space of
analytic functions in the upper complex half plane satisfying the mentioned
growth condition. However, the latter space is a differential algebra, the
differential-algebraic structure of which can be transported to
${\mathcal{H}}_{+}(\mathbb{R})$, rendering it a differential algebra [37].

This is exactly the case with $u_{0}(x)$ given by (5) for which

$\widehat{u}_{0}(z)=\left\\{\begin{array}[]{ll}\frac{1}{z},&\mathop{\mathrm{Im}}z>0,\\\\[2.0pt]
0,&\mathop{\mathrm{Im}}z<0.\end{array}\right.$

In the algebra of analytic functions in the upper half plane, the functional
relation

$\frac{\hskip 1.0pt{\rm d}\hskip 0.5pt^{k}}{\hskip 1.0pt{\rm d}\hskip
0.5ptz^{k}}\Big{(}\frac{1}{z}\Big{)}=(-1)^{k}k!\Big{(}\frac{1}{z}\Big{)}^{k+1},\quad
z\neq 0$

obtains. In this way, formulas (6) and (7) are explained. The differential-
algebraic relations persist in the boundary values.

####  3.1.1 Analytic regularization

It will be instructive to study the behavior of approximate solutions when the
initial data are regularized. The first obvious possibility is to consider the
analytic regularization defining the distribution $u_{0}(x)=1/(1+{\rm i}\hskip
1.0pt0)$. We wish to solve the regularized problem

$\frac{1}{c}\partial_{t}u_{\varepsilon}+\partial_{x}u_{\varepsilon}+u_{\varepsilon}^{2}=0,\quad
u_{\varepsilon}(x,0)=u_{0\varepsilon}(x)=\frac{1}{x+{\rm i}\hskip
1.0pt\varepsilon}.$ (10)

Solving (10) by the method of characteristics results in the unique classical
solution

$u_{\varepsilon}(x,t)=\frac{u_{0\varepsilon}(x-ct)}{1+ctu_{0\varepsilon}(x-ct)}=\frac{\frac{1}{x-ct+{\rm
i}\hskip 1.0pt\varepsilon}}{1+ct\frac{1}{x-ct+{\rm i}\hskip
1.0pt\varepsilon}}=\frac{1}{x+{\rm i}\hskip 1.0pt\varepsilon}.$

Thus, by simple arithmetic, $u_{\varepsilon}(x,t)\equiv u_{0\varepsilon}(x)$
and so the solution given in Proposition 3.1 coincides with the weak limit of
approximate solutions when the initial data are replaced by their analytic
regularization.

####  3.1.2 Regularization by convolution with a mollifier

The purpose of this subsection is to show that the convergence of the
approximate solution is a peculiarity of the analytic regularization and does
not hold if the initial data are regularized by convolution with a standard
Friedrichs mollifier
$\varphi_{\varepsilon}(x)=\varepsilon^{-1}\varphi(x/\varepsilon)$ with
$\varphi\in{\mathcal{D}}(\mathbb{R})$, $\int\varphi(x)\hskip 1.0pt{\rm
d}\hskip 0.5ptx=1$. For the sake of the argument, we take $\varphi\geq 0$
symmetric, $\mathop{\mathrm{supp}}\varphi\subset(-1,1)$. Thus let

$U_{0\varepsilon}(x)=(u_{0}\ast\varphi_{\varepsilon})(x)$

and let $U_{\varepsilon}(x,t)$ be the corresponding classical solution to (10)
with initial data $U_{0\varepsilon}$. By the method of characteristics,

$U_{\varepsilon}(x,t)=\frac{({\rm
vp\frac{1}{x}}\ast\varphi_{\varepsilon})(x-ct)-{\rm i}\hskip
1.0pt\pi\varphi_{\varepsilon}(x-ct)}{1+ct\big{(}({\rm
vp\frac{1}{x}}\ast\varphi_{\varepsilon})(x-ct)-{\rm i}\hskip
1.0pt\pi\varphi_{\varepsilon}(x-ct)\big{)}}.$

In particular,

$U_{\varepsilon}(ct-\varepsilon,t)=\frac{({\rm
vp\frac{1}{x}}\ast\varphi_{\varepsilon})(-\varepsilon)}{1+ct({\rm
vp\frac{1}{x}}\ast\varphi_{\varepsilon})(-\varepsilon)}$ (11)

We show that the solution $U_{\varepsilon}(x,t)$ blows up at latest at

$t_{\varepsilon}=\frac{-1/c}{({\rm
vp\frac{1}{x}}\ast\varphi_{\varepsilon})(-\varepsilon)}=\frac{1/c}{({\rm
vp\frac{1}{x}}\ast\varphi_{\varepsilon})(\varepsilon)}$

and that this number is of order $\varepsilon$ as $\varepsilon\to 0$. Thus
there is no global solution, when Friedrichs regularization is used.

Indeed, starting from the defining formula

$({\rm vp\frac{1}{x}}\ast\varphi_{\varepsilon})(x)=\lim_{\eta\to
0}\int_{|x-y|\geq\eta}\frac{\varphi_{\varepsilon}(y)}{x-y}\hskip 1.0pt{\rm
d}\hskip 0.5pty,$

some simple manipulations using the support properties of $\varphi$ lead to

$({\rm vp\frac{1}{x}}\ast\varphi_{\varepsilon})(-\varepsilon)=\lim_{\eta\to
0}\int_{-1+\eta/\varepsilon}^{\infty}\frac{\varphi(y)}{-\varepsilon(1+y)}\hskip
1.0pt{\rm d}\hskip
0.5pty=-\frac{1}{\varepsilon}\int_{\mathop{\mathrm{supp}}\varphi}\frac{\varphi(y)}{1+y}\hskip
1.0pt{\rm d}\hskip 0.5pty=-\frac{1}{\varepsilon}C_{\varphi}$

where $C_{\varphi}$ is a positive constant. This shows that the denominator in
(11) is indeed zero at $t_{\varepsilon}=\varepsilon/cC_{\varphi}$, while the
numerator is nonzero.

####  3.1.3 Separation in real and imaginary part

One might argue that the complex valued initial value problem (10) is actually
a real valued, nonstrictly hyperbolic system. This is indeed the case; the
real and imaginary part of the analytically regularized solution are

$u_{\varepsilon}(x,t)=\frac{1}{x+{\rm i}\hskip
1.0pt\varepsilon}=v_{\varepsilon}(x,t)+{\rm i}\hskip
1.0ptw_{\varepsilon}(x,t)=\frac{x}{x^{2}+\varepsilon^{2}}-{\rm i}\hskip
1.0pt\frac{\varepsilon}{x^{2}+\varepsilon^{2}}.$

The hyperbolic system for the real and imaginary part is

$\begin{array}[]{lcl}\partial_{t}v_{\varepsilon}+\partial_{x}v_{\varepsilon}&=&-v_{\varepsilon}^{2}+w_{\varepsilon}^{2},\\\\[4.0pt]
\partial_{t}w_{\varepsilon}+\partial_{x}w_{\varepsilon}&=&-2v_{\varepsilon}w_{\varepsilon}.\end{array}$

Here $v_{\varepsilon}(x,t)\to{\rm vp}\,\frac{1}{x}$ and
$w_{\varepsilon}(x,t)\to-\pi\delta(x)$ as $\varepsilon\to 0$. However, it is
well-known (and rather immediate) that $v_{\varepsilon}^{2}$ and
$w_{\varepsilon}^{2}$ do not converge in ${\mathcal{D}}^{\prime}(\mathbb{R})$
as $\varepsilon\to 0$. Thus the individual terms in the first line make no
sense in the limit. (By purely arithmetic manipulations involving $1/(x+{\rm
i}\hskip 1.0pt\varepsilon)$ and $1/(x-{\rm i}\hskip 1.0pt\varepsilon)$ and
their limits, the limit in the right-hand side of the second line is seen to
exist and to equal $-\pi\delta^{\prime}(x)$.)

###  3.2 A nonlinear wave equation

In the same vein, the distribution $u_{0}(x)$ can serve to produce a solution
to a semilinear wave equation in one space dimension.

######  Proposition 3.4.

The distribution $u(x,t)=u_{0}(x)$ given by (5) is a weak solution to the
initial value problem

$\frac{1}{c^{2}}\partial_{t}^{2}u-\partial_{x}^{2}u+2u^{3}=0,\quad
u(x,0)=u_{0}(x),\ \partial_{t}u(x,0)=0$ (12)

for whatever $c>0$, where the cubic term is understood in the sense of
Hörmander’s product.

###### Proof.

It is clear from (7) that $-\partial_{x}^{2}u+2u^{3}=0$ and that
$\partial_{t}u=0$. ∎

In the real-valued case, the wave equation (12) has a so-called defocusing
nonlinearity. For initial data $(u_{0},u_{1})$ in $H^{1}(\mathbb{R})\times
L^{2}(\mathbb{R})$, it would have a unique global finite energy solution [35],
belonging to
${\mathcal{C}}([0,\infty):H^{1}(\mathbb{R}))\cap{\mathcal{C}}^{1}([0,\infty):L^{2}(\mathbb{R}))$.
By local existence theory, it could also be extended to small negative times,
and hence would belong to $L^{\infty}_{\rm loc}$ in an open neighborhood of
the half plane. As in Remark 2.4, the Corollary to Theorem 2 in [28] would
imply that a singularity in the initial data at $x=0$ can only spread along
the characteristic lines $x=\pm ct$. Clearly, the solution given in
Proposition 12 neither has the required regularity properties nor does it show
the expected singularity propagation.

######  Remark 3.5.

The distribution $u_{0}(x)$ is homogeneous of degree $-1$. Thus
$u(x,t)=u_{0}(x)$ is a self-similar solution to (12), satisfying $\mu u(\mu
x,\mu t)=u(x,t)$ for all $\mu>0$.

##  4 Type II solutions – Nemytskii operators

This section addresses weak solutions, whereby the nonlinear terms are defined
by Nemytskii operators. We recall the pseudofunctions $R_{\lambda}$,
meromorphic functions of $\lambda\in\mathbb{C}$ with values in the space of
tempered distributions ${\mathcal{S}}^{\prime}(\mathbb{R}^{n})$ [11, Chapter
17]. For ${\rm Re\,}\lambda>-n$ they are given by

$\langle R_{\lambda},\varphi\rangle=\int|x|^{\lambda}\varphi(x)\hskip
1.0pt{\rm d}\hskip 0.5ptx$

and can be analytically continued to
$\mathbb{C}\setminus\\{-n-2k:k\in\mathbb{N}\\}$. Outside the poles, they
satisfy

$\Delta R_{\lambda}=\lambda(\lambda+n-2)R_{\lambda-2}$

In particular, when $\lambda>2-n$ and $p=1-2/{\lambda}$, $R_{\lambda}$ belongs
to $L^{p}_{\rm loc}(\mathbb{R}^{n})$, $(R_{\lambda})^{p}=R_{\lambda p}$ and it
satisfies the elliptic equation

$\Delta R_{\lambda}=\lambda(\lambda+n-2)(R_{\lambda})^{p},$

where the derivatives are understood in the weak sense and the $p$th power as
the evaluation of the Nemytskii operator $L^{p}_{\rm loc}(\mathbb{R}^{n})\to
L^{1}_{\rm loc}(\mathbb{R}^{n})$.

We note that for $\lambda\in\mathbb{R}\setminus\\{-n-2k:k\in\mathbb{N}\\}$,
$R_{\lambda}$ is homogeneous of degree $\lambda$, and $R_{\lambda}\in
H^{1}_{\rm loc}(\mathbb{R}^{n})$, if $\lambda>(2-n)/2$.

As examples to be discussed further, we only consider two cases in which $p$
is a positive integer. In the context of propagation of singularities,
fractional powers are not interesting for our purpose, because they represent
non-smooth nonlinearities. We use the solutions $R_{\lambda}$ as examples of
peculiar rotationally symmetric stationary solutions to nonlinear wave
equations.

######  Example 4.1.

Let $n=3$ and $\lambda=-1/2$ (then $\lambda(\lambda+n-2)=-1/4$). Let
$u_{0}(x)=|x|^{-1/2}$. Then $u_{0}\in L^{5}_{\rm loc}(\mathbb{R}^{3})$, and
$u(x,t)\equiv u_{0}(x)$ satisfies the nonlinear wave equation

$\frac{1}{c^{2}}\partial_{t}^{2}u-\Delta u-\frac{1}{4}u^{5}=0,\quad
u(x,0)=u_{0}(x),\ \partial_{t}u(x,0)=0$ (13)

for whatever $c>0$.

######  Example 4.2.

Let $n=4$ and $\lambda=-1$ (then $\lambda(\lambda+n-2)=-1$). Let
$u_{0}(x)=|x|^{-1}$. Then $u_{0}\in L^{3}_{\rm loc}(\mathbb{R}^{4})$, and
$u(x,t)\equiv u_{0}(x)$ satisfies the nonlinear wave equation

$\frac{1}{c^{2}}\partial_{t}^{2}u-\Delta u-u^{3}=0,\quad u(x,0)=u_{0}(x),\
\partial_{t}u(x,0)=0$ (14)

for whatever $c>0$.

In all these cases, derivatives are understood in the weak sense and the
powers of $u$ exist as locally integrable functions, actually as evaluations
of the continuous map $u\to u^{p}$ from $L^{p}_{\rm loc}\to L^{1}_{\rm loc}$.
Note that the nonlinear operation is taken outside the space of distributions,
and the result is embedded afterwards.

######  Remark 4.3.

(a) As $u_{0}$ is nonnegative, we might replace $u^{5}$ by $|u|^{5}$ or
$|u|^{4}u$. In any case, we are dealing with so-called focusing
nonlinearities.

(b) Recall that $u(x,t)$ is a self-similar solution to the nonlinear wave
equation

$\frac{1}{c^{2}}\partial_{t}^{2}u-\Delta u\pm|u|^{p}=0,$ (15)

if $u(x,t)=\mu^{\alpha}u(\mu t,\mu x)$ for all $\mu>0$, where necessarily
$\alpha=2/(p-1)$. On the other hand, $u_{0}=R_{\lambda}$ is homogeneous of
degree $\lambda$, that is, $u_{0}(sx)=s^{\lambda}u_{0}(x)$. It also satisfies
equation (15) when $\lambda-2=\lambda p$, i.e., $\lambda=-2/(p-1)$. Thus the
special solutions exhibited here are self-similar solutions to the nonlinear
wave equation. However, they do not fall into the classes of functions
considered e.g. in [4, 19, 25, 26, 31]. It should be noted that solutions to
nonlinear elliptic equations have also been used in the literature. They can
serve for constructing solutions of finite life span, but also for proving the
existence of (time-dependent) self-similar solutions [7, 12, 20, 21].

##  5 Type III – weak asymptotic solutions

A net of smooth functions $(u_{\varepsilon})_{\varepsilon>0}$ is a called a
_weak asymptotic solution_ [8] to a nonlinear partial differential equation,
such as equation (15), if it has a limit in the space of distributions and if
it satisfies the equation up to an error term which tends to zero weakly as
$\varepsilon\to 0$.

The basic example derives again from a nonlinear elliptic equation. Indeed, in
$\mathbb{R}^{n}$, we start from the relation

$\Delta(|x|^{2}+\varepsilon^{2})^{q}=\big{(}\big{(}2qn+4q(q-1)\big{)}|x|^{2}+2qn\varepsilon^{2}\big{)}(|x|^{2}+\varepsilon^{2})^{q-2}.$

We will simply work out two special cases that correspond to the ones in
Examples 4.1 and 4.2.

######  Example 5.1.

Let $n=3$ and $q=-1/4$. By simple arithmetic,

$\big{(}2qn+4q(q-1)\big{)}|x|^{2}+2qn\varepsilon^{2}=-\frac{1}{4}(|x|^{2}-\varepsilon^{2})-\frac{5}{4}\varepsilon^{2}$

and so

$\Delta(|x|^{2}+\varepsilon^{2})^{-1/4}=-\frac{1}{4}(|x|^{2}+\varepsilon^{2})^{-5/4}-\frac{5}{4}\varepsilon^{2}(|x|^{2}+\varepsilon^{2})^{-9/4}.$

Thus

$u_{\varepsilon}(x,t)=(|x|^{2}+\varepsilon^{2})^{-1/4}$

satisfies the nonlinear wave equation

$\frac{1}{c^{2}}\partial^{2}_{t}u_{\varepsilon}-\Delta
u_{\varepsilon}-\frac{1}{4}u_{\varepsilon}^{5}-\frac{5}{4}\varepsilon^{2}u_{\varepsilon}^{9}=0$
(16)

for whatever $c>0$. An easy calculation shows that
$\varepsilon^{2}u_{\varepsilon}^{9}$ converges to zero in
${\mathcal{D}}^{\prime}(\mathbb{R}^{3})$ as $\varepsilon\to 0$. Thus
$u_{\varepsilon}$ is a weak asymptotic solution to the nonlinear wave equation
(13) with initial data converging to $u_{0}(x)=|x|^{-1/2}$. As in Example 4.1
we set $u(x,t)=u_{0}(x)$. By the continuity assertions for Type II solutions,

$u_{\varepsilon}\to u,\quad u_{\varepsilon}^{5}\to u^{5}\quad{\rm in}\quad
L_{\rm loc}^{1}(\mathbb{R}^{3})\quad{\rm as\ }\varepsilon\to 0,$

thus each term in equation (16) converges to the corresponding term in
equation (13). Further, $u_{\varepsilon}$ is a smooth approximation to $u$; as
$\varepsilon\to 0$, a singularity emerges at $x=0$.

It is of interest to note that the solution to the regularized equation (16)
is unique. This emphasizes again the anomaly in the propagation of
singularities in the initial value problem (13).

######  Lemma 5.2.

Let $n=1$, $n=2$ or $n=3$. Assume that
$u_{0}\in{\mathcal{C}}^{1}_{b}(\mathbb{R}^{n})$,
$u_{1}\in{\mathcal{C}}^{0}_{b}(\mathbb{R}^{n})$ and let $f$ be smooth. Given
any $T>0$, the initial value problem

$\frac{1}{c^{2}}\partial_{t}^{2}u-\Delta u=f(u),\quad u(x,0)=u_{0}(x),\
\partial_{t}u(x,0)=u_{1}(x)$ (17)

has at most one weak solution in
${\mathcal{C}}^{0}_{b}(\mathbb{R}^{n}\times[0,T])$.

###### Proof.

Let $S(t)$ be the fundamental solution of the Cauchy problem, that is, $S(t)$
is the inverse Fourier transform of $\sin(c|\xi|t)/|\xi|$. In space dimensions
$n=1,2,3$, $S(t)$ is a finite measure of total mass $ct$. The solution is
given by

$u(.,t)=\frac{d}{dt}S(t)\ast u_{0}+S(t)\ast u_{1}+\int_{0}^{t}S(t-s)\ast
f(u(.,s))\hskip 1.0pt{\rm d}\hskip 0.5pts.$

By Young’s inequality, the $L^{\infty}$-estimate

$\|u(.,t)\|_{L^{\infty}(\mathbb{R}^{n})}\leq C(t)\|u_{0},\nabla
u_{0},u_{1}\|_{L^{\infty}(\mathbb{R}^{n})}+\int_{0}^{t}(ct-
cs)\|u(.,s)\|_{L^{\infty}(\mathbb{R}^{n})}\hskip 1.0pt{\rm d}\hskip 0.5pts$

holds, where C(t) is a constant depending linearly on $t$. Applying this
estimate to the difference $u-v$ of two solutions with the same initial data,
writing $f(u)-f(v)=(u-v)g(u,v)$ with $g$ smooth and applying Gronwall’s
inequality shows that $u=v$. ∎

######  Example 5.3.

Let $n=4$ and $q=-1/2$ and let

$u_{\varepsilon}(x,t)=(|x|^{2}+\varepsilon^{2})^{-1/2}.$

By the same arguments as in Example 5.1 one sees that $u_{\varepsilon}$
satisfies the nonlinear wave equation

$\frac{1}{c^{2}}\partial^{2}_{t}u_{\varepsilon}-\Delta
u_{\varepsilon}-u_{\varepsilon}^{3}-3\varepsilon^{2}u_{\varepsilon}^{5}=0$
(18)

for whatever $c>0$. Again, one shows that $\varepsilon^{2}u_{\varepsilon}^{5}$
converges to zero in ${\mathcal{D}}^{\prime}(\mathbb{R}^{4})$ as
$\varepsilon\to 0$, and $u_{\varepsilon}$ is a weak asymptotic solution to the
nonlinear wave equation (14) with initial data converging to
$u_{0}(x)=|x|^{-1}$. With $u(x,t)\equiv u_{0}(x)$, one has again

$u_{\varepsilon}\to u,\quad u_{\varepsilon}^{3}\to u^{3}\quad{\rm in}\quad
L_{\rm loc}^{1}(\mathbb{R}^{4})\quad{\rm as\ }\varepsilon\to 0,$

thus each term in equation (18) converges to the corresponding term in
equation (14). The same behavior as in Example 5.1 is observed.

Due to the continuity of the Nemytskii operators, the weak asymptotic
solutions constructed here are consistent with the solutions presented in
Section 4.

##  6 Type IV – sequential solutions

In this section, we address solutions defined by nets of smooth functions
which do not necessarily converge. To introduce the concept, let $\Omega$ be
an open subset of $\mathbb{R}^{n}$ and let $P$ be a possibly nonlinear partial
differential operator which is a smooth function of its arguments,
$Pu=P(x,u,\partial u,\ldots)$. Let $(u_{\varepsilon})_{\varepsilon>0}$ a net
of functions belonging to ${\mathcal{C}}^{\infty}(\Omega)$. If
$Pu_{\varepsilon}=0$ for all sufficiently small $\varepsilon>0$, then the net
$(u_{\varepsilon})_{\varepsilon>0}$ is called a _sequential solution_ of the
equation $Pu=0$, following e.g. [32]. The net
$(u_{\varepsilon})_{\varepsilon>0}$ may or may not converge. Even if
$(u_{\varepsilon})_{\varepsilon>0}$ converges, individual terms in
$P(x,u,\partial u,\ldots)$ may or may not converge. However, if
$(u_{\varepsilon})_{\varepsilon>0}$ converges to a distribution $u$, together
with all individual terms in $P(x,u,\partial u,\ldots)$, then $u$ can be
called a _proper weak solution_ to $Pu=0$ [22].

Restricting the class of sequential solutions to _moderate nets_ allows one to
establish a regularity theory for sequential solutions, even if they diverge.
A net of smooth functions $(u_{\varepsilon})_{\varepsilon>0}$ on $\Omega$ is
called _moderate_ , if for all compact subsets $K$ of $\Omega$ and all multi-
idices $\alpha\in\mathbb{N}_{0}^{n}$ there exists $b\geq 0$ such that

$\textstyle\sup_{x\in
K}|\partial^{\alpha}u_{\varepsilon}(x)|=O(\varepsilon^{-b})\quad{\rm as}\
\varepsilon\to 0.$

The net of smooth functions $(u_{\varepsilon})_{\varepsilon>0}$ on $\Omega$ is
called _negligible_ , if for all compact subsets $K$ of $\Omega$, all multi-
idices $\alpha\in\mathbb{N}_{0}^{n}$ and all $a\geq 0$,

$\textstyle\sup_{x\in
K}|\partial^{\alpha}u_{\varepsilon}(x)|=O(\varepsilon^{a})\quad{\rm as}\
\varepsilon\to 0.$

Following [15, 33], a moderate net satisfying $Pu_{\varepsilon}=0$ for all
sufficiently small $\varepsilon>0$ is called a _very weak solution_ to the
equation $Pu=0$. If $(u_{\varepsilon})_{\varepsilon>0}$ is moderate and
$Pu_{\varepsilon}=n_{\varepsilon}$ where $(n_{\varepsilon})_{\varepsilon>0}$
is a negligible net, then $(u_{\varepsilon})_{\varepsilon>0}$ is a _Colombeau
solution_ to the equation $Pu=0$. (As a matter of fact, its equivalence class
in the Colombeau algebra ${\mathcal{G}}(\Omega)$ is a solution in the
differential-algebraic sense [6, 17, 24].)

Finally, a net $(u_{\varepsilon})_{\varepsilon>0}$ is said to possess the
${\mathcal{G}}^{\infty}$-property, if for all compact subsets $K$ of $\Omega$
there is $b\geq 0$ such that for all multi-idices
$\alpha\in\mathbb{N}_{0}^{n}$,

$\textstyle\sup_{x\in
K}|\partial^{\alpha}u_{\varepsilon}(x)|=O(\varepsilon^{-b})\quad{\rm as}\
\varepsilon\to 0.$

(Note the change in quantifiers: the local order of growth is the same for all
derivatives.) The significance of this notion is that it generalizes
${\mathcal{C}}^{\infty}$-smoothness from distributions to moderate nets. In
fact, if $w\in{\mathcal{E}}^{\prime}(\Omega)$ is a compactly supported
distribution and $\varphi_{\varepsilon}$ is a mollifier
($\varphi_{\varepsilon}(x)=\varepsilon^{-n}\varphi(x/\varepsilon)$ with
$\varphi$ smooth, rapidly decaying and $\int\varphi(x)\hskip 1.0pt{\rm
d}\hskip 0.5ptx=1$), then

  * •

$w_{\varepsilon}=w\ast\varphi_{\varepsilon}|\Omega$ defines a moderate net;

  * •

$(w_{\varepsilon})_{\varepsilon>0}$ has the ${\mathcal{G}}^{\infty}$-property
if and only if $w\in{\mathcal{C}}^{\infty}(\Omega)$.

The ${\mathcal{G}}^{\infty}$-singular support of a moderate net
$(u_{\varepsilon})_{\varepsilon>0}$ is defined as the complement of the
largest open subset $\omega\subset\Omega$ such that
$(u_{\varepsilon}|\omega)_{\varepsilon>0}$ has the
${\mathcal{G}}^{\infty}$-property on $\omega$. With this notion,
${\mathcal{C}}^{\infty}$-regularity theory and propagation of singularities
for linear partial differential equations can be literally transferred to the
setting of moderate nets in the case of linear equations (with possibly non-
smooth coefficients) see e.g. [10, 14, 24].

The same notions can be introduced for nets of smooth functions defined on the
closure of an open subset of $\mathbb{R}^{n}$, thereby enabling the study of
initial value problems or boundary value problems.

###  6.1 Moderate sequential solutions to an advection-reaction equation

We are going to construct moderate sequential solutions to the advection-
reaction equation in one space dimension

$\frac{1}{c}\partial_{t}u+\partial_{x}u+\frac{2}{p}\,x\,u^{p+1}=0,\quad
u(x,0)=u_{0}(x)$ (19)

where – for simplicity – $p$ is a positive integer. We first note that for
continuous initial data, there is at most one solution.

######  Lemma 6.1.

Assume that $u_{0}\in{\mathcal{C}}^{0}_{b}(\mathbb{R})$, $c\neq 0$ and let $f$
be smooth. Given any $T>0$, the initial value problem

$\frac{1}{c}\partial_{t}u+\partial_{x}u=f(x,t,u),\quad u(x,0)=u_{0}(x)$ (20)

has at most one weak solution in
${\mathcal{C}}^{0}_{b}(\mathbb{R}^{n}\times[0,T])$.

###### Proof.

Indeed, if $u$ is a solution, it solves the integral equation

$u(x,t)=u_{0}(x-ct)+\int_{0}^{t}f(x-ct+cs,s,u(x-ct+cs,s))\hskip 1.0pt{\rm
d}\hskip 0.5pts.$

Uniqueness follows by the same argument as in the proof of Lemma 5.2. ∎

It is immediately checked that, for each $\varepsilon>0$, the smooth function

$u_{\varepsilon}(x,t)\equiv
u_{0\varepsilon}(x)=(x^{2}+\varepsilon^{2})^{-1/p}$ (21)

is a solution to the initial value problem

$\frac{1}{c}\partial_{t}u_{\varepsilon}+\partial_{x}u_{\varepsilon}+\frac{2}{p}\,x\,u_{\varepsilon}^{p+1}=0,\quad
u_{\varepsilon}(x,0)=(x^{2}+\varepsilon^{2})^{-1/p}.$ (22)

According to Lemma 6.1, the solution is unique. It is clear that the net
$(u_{\varepsilon})_{\varepsilon>0}$ is moderate, hence it defines a moderate
sequential solution to (19).

######  Lemma 6.2.

The net $(u_{0\varepsilon})_{\varepsilon>0}$ converges for $p\geq 3$ and
diverges for $p=1,2$. In particular,
$(u_{0\varepsilon}^{p+1})_{\varepsilon>0}$ diverges for every $p$.

###### Proof.

For $p\geq 3$, $u_{0}(x)=|x|^{-2/p}$ belongs to the space of locally
integrable functions, and $u_{0\varepsilon}(x)=(x^{2}+\varepsilon^{2})^{-1/p}$
converges to it in that space.

Let $p=2$ and take a test function $\varphi\geq 0$ such that $\varphi(x)=1$ on
$[-1,1]$. Then

$\displaystyle\langle
u_{0\varepsilon},\varphi\rangle=\int_{-\infty}^{\infty}\frac{\varphi(x)}{\sqrt{x^{2}+\varepsilon^{2}}}\hskip
1.0pt{\rm d}\hskip
0.5ptx\geq\int_{-1}^{1}\frac{1}{\sqrt{x^{2}+\varepsilon^{2}}}\hskip 1.0pt{\rm
d}\hskip
0.5ptx=\int_{-1/\varepsilon}^{1/\varepsilon}\frac{1}{\sqrt{y^{2}+1}}\hskip
1.0pt{\rm d}\hskip 0.5pty\to\infty$

as $\varepsilon\to 0$. A similar argument shows that
$(x^{2}+\varepsilon^{2})^{-q}$ diverges for $q>1/2$. Thus
$u_{0\varepsilon}(x)=(x^{2}+\varepsilon^{2})^{-1/p}$ diverges when $p<2$ as
well, in particular, for $p=1$. Further,
$u_{0\varepsilon}^{p+1}(x)=(x^{2}+\varepsilon^{2})^{-1-1/p}$ diverges for
every nonnegative $p$. ∎

This shows that even in the convergent case $p\geq 2$, the limit
$u=\lim_{\varepsilon\to 0}u_{\varepsilon}$ is not a proper solution of
equation (19).

####  6.1.1 The special case $p=2$

Let us have a more detailed look at the (divergent) case $p=2$. Then the
function

$u_{\varepsilon}(x,t)=(x^{2}+\varepsilon^{2})^{-1/2},$ (23)

at fixed $\varepsilon>0$, is a solution to the advection-reaction equation

$\frac{1}{c}\partial_{t}u_{\varepsilon}+\partial_{x}u_{\varepsilon}+x\,u_{\varepsilon}^{3}=0,\quad
u_{\varepsilon}(x,0)=(x^{2}+\varepsilon^{2})^{-1/2}.$ (24)

According to Lemma 6.1, this solution is unique. We may study its
${\mathcal{G}}^{\infty}$-regularity properties.

######  Proposition 6.3.

The ${\mathcal{G}}^{\infty}$-singular support of
$(u_{\varepsilon})_{\varepsilon>0}$ is $\\{(0,t):t\geq 0\\}$.

###### Proof.

Let $\chi(x)=(x^{2}+1)^{-1/2}$. Then
$(u_{\varepsilon}(x,t)=(x^{2}+\varepsilon^{2})^{-1/2}=\chi_{\varepsilon}(x)=\varepsilon^{-1}\chi(x/\varepsilon)$.
It is straightforward to show that the $k$th derivative of $\chi$ is of the
form

$\chi^{(k)}(x)=P_{k}(x)(x^{2}+1)^{-k-1/2}$

where $P_{k}$ is a polynomial of degree $k$. Therefore,

$\chi_{\varepsilon}^{(k)}(x)=\varepsilon^{-k-1}P_{k}\big{(}\frac{x}{\varepsilon}\big{)}\big{(}\frac{x^{2}}{\varepsilon^{2}}+1\big{)}^{-k-1/2}=\varepsilon^{k}P_{k}\big{(}\frac{x}{\varepsilon}\big{)}(x^{2}+\varepsilon^{2})^{-k-1/2}.$

When $|x|\geq x_{0}>0$, the latter expression is bounded independently of
$\varepsilon>0$. Thus $(u_{\varepsilon})_{\varepsilon>0}$ has the
${\mathcal{G}}^{\infty}$-property in the region $\\{(x,t):|x|>0,t\geq 0\\}$.

On the other hand, $\chi(x)$ is the derivative of $\mathop{\mathrm{arsinh}}x$,
whose Taylor expansion shows that $\chi^{(k)}(x)\neq 0$ when $k$ is an even
integer. Thus

$\chi_{\varepsilon}^{(k)}(0)=\varepsilon^{-k-1}\chi^{(k)}(0)$

does not have the ${\mathcal{G}}^{\infty}$-property: the line $x=0$ is
contained in the ${\mathcal{G}}^{\infty}$-singular support. ∎

This shows that the moderate sequential solution to (24) exhibits anomalous
propagation of singularities. The initial ${\mathcal{G}}^{\infty}$-singularity
at $x=0$ is not propagated along the line $x=ct$ as in the linear case, but
rather remains at $x=0$ for all times.

######  Remark 6.4.

Actually, the classical initial value problem
$\frac{1}{c}\partial_{t}v+\partial_{x}v+x\,v^{3}=0$, $v(x,0)=v_{0}(x)$ can be
solved explicitly. Transformation to characteristic coordinates $s=t$,
$y=x-ct$ leads to an ordinary differential equation and to the solution

$v(x,t)=\frac{v_{0}(x-ct)}{\sqrt{\big{(}x^{2}-(x-ct)^{2}\big{)}v_{0}^{2}(x-ct)+1}}.$

Inserting $v_{0}(x)=(x^{2}+\varepsilon^{2})^{-1/2}$ it turns out that by
simple arithmetic, $v(x,t)=(x^{2}+\varepsilon^{2})^{-1/2}$, supporting the
fact that $u_{\varepsilon}(x,t)$ as given above by (23) is indeed the
solution. The same phenomenon also happens for $p\neq 2$ in (21) and (22).

###  6.2 Moderate sequential solutions to a nonlinear wave equation

Taking a further $x$-derivative, it is seen that $u_{\varepsilon}(x,t)$ given
by (23) also solves the one-dimensional nonlinear wave equation

$\frac{1}{c^{2}}\partial_{t}^{2}u_{\varepsilon}-\partial_{x}^{2}u_{\varepsilon}+u_{\varepsilon}^{3}+3x^{2}u_{\varepsilon}^{5}=0,\quad
u_{\varepsilon}(x,0)=(x^{2}+\varepsilon^{2})^{-1/2},\
\partial_{t}u_{\varepsilon}(x,0)=0$

for every $c>0$. In this case, standard energy estimates can be used to show
that the solution is unique.

######  Lemma 6.5.

Given $v_{0}\in H^{1}(\mathbb{R})$, $v_{1}\in L^{2}(\mathbb{R})$ of finite
energy (defined by (26) below), the equation

$\frac{1}{c^{2}}\partial_{t}^{2}v-\partial_{x}^{2}v+v^{3}+3x^{2}v^{5}=0,\quad
v(x,0)=v_{0}(x),\ \partial_{t}v(x,0)=v_{1}(x)$ (25)

has a unique solution
$v\in{\mathcal{C}}([0,\infty):H^{1}(\mathbb{R}))\cap{\mathcal{C}}^{1}([0,\infty):L^{2}(\mathbb{R}))$
of finite energy, where $c>0$.

###### Proof.

It is quite obvious that the energy

$E(t)=\frac{1}{2}\int_{-\infty}^{\infty}\Big{(}|\partial_{t}v|^{2}+c^{2}|\partial_{x}v|^{2}+|v|^{4}+3x^{2}|v|^{6}\Big{)}\hskip
1.0pt{\rm d}\hskip 0.5ptx$ (26)

is conserved. The proof follows standard arguments (see e.g. [35]). ∎

At fixed $\varepsilon>0$,
$u_{\varepsilon}(x,0)=(x^{2}+\varepsilon^{2})^{-1/2}$ belongs to
$H^{1}(\mathbb{R})$ and, together with $\partial_{t}u_{\varepsilon}(x,0)=0$,
forms initial data of finite energy. Thus the stationary solution
$u_{\varepsilon}(x,t)=u_{\varepsilon}(x,0)$ is the unique solution in this
sense. The net $(u_{\varepsilon})_{\varepsilon>0}$ provides a moderate
sequential solution to the nonlinear wave equation (25). Its
${\mathcal{G}}^{\infty}$-singular support $\\{(x,t),x=0,t\geq 0\\}$ has been
computed in Proposition 6.3. Again, this differs from the linear case [14] and
the nonlinear, classical case (Propositions 2.1, 2.3), according to which the
singular support should be $\\{(x,t),|x|=ct,t\geq 0\\}$.

Anomalous propagation of singularities persists for sequential solutions.

Acknowledgements: I wish to thank the organizers of the INdAM Workshop
_Anomalies in Partial Differential Equations_ for providing an attractive
environment for presenting the results of the paper. Discussions with various
participants led to further insight. In particular, I would like to thank Lavi
Karp, Sandra Lucente, Alberto Parmeggiani, Michael Reissig, Luigi Rodino and
Michael Ruzhansky for helpful remarks. Thanks go to Hideo Deguchi for joint
work on the topic since 2016.

## Appendix: On multiplication of distributions

Let $S,T\in{\mathcal{S}}^{\prime}(\mathbb{R}^{n})$. The
${\mathcal{S}}^{\prime}$-convolution of $S$ and $T$ is said to exist, if

$(\varphi\ast\check{S})T\in{\mathcal{D}}_{L^{1}}^{\prime}(\mathbb{R}^{n}),\quad{\rm
for\ all}\quad\varphi\in{\mathcal{S}}(\mathbb{R}^{n}).$

where $\check{S}(x)=S(-x)$. In this case, the convolution is defined by
$\langle S\ast T,\varphi\rangle=\langle(\varphi\ast\check{S})T,1\rangle$, and
$S\ast T$ belongs to ${\mathcal{S}}^{\prime}(\mathbb{R}^{n})$.

Let $u,v\in{\mathcal{S}}^{\prime}(\mathbb{R}^{n})$. If the
${\mathcal{S}}^{\prime}$-convolution of ${\mathcal{F}}u$ and ${\mathcal{F}}v$
exists, one may define the _Fourier product_

$u\cdot v={\mathcal{F}}^{-1}({\mathcal{F}}u\ast{\mathcal{F}}v).$

The definition can be localized [1] as follows. Assume that for every
$x\in\mathbb{R}^{n}$ there is a neighborhood $\Omega_{x}$ and
$\chi_{x}\in{\mathcal{D}}(\mathbb{R}^{n})$, $\chi_{x}\equiv 1$ on
$\Omega_{x}$, such that the ${\mathcal{S}}^{\prime}$-convolution of
${\mathcal{F}}(\chi_{x}u)$ and ${\mathcal{F}}(\chi_{x}v)$ exists. Locally near
$x$, the product $u\cdot v$ is defined to be
${\mathcal{F}}^{-1}({\mathcal{F}}(\chi_{x}u)\ast{\mathcal{F}}(\chi_{x}v))$.
Globally, it is defined by a partition of unity argument.

A special case arises when the distributions satisfy Hörmander’s wave front
set criterion [18], requiring that for every
$(x,\xi)\in\mathbb{R}^{n}\times\mathbb{R}^{n}\setminus\\{0\\}$,
$(x,\xi)\in\mathrm{WF}(u)$ implies $(x,-\xi)\not\in\mathrm{WF}(v)$.

In space dimension $n=1$, a very convenient case arises when
$\mathop{\mathrm{supp}}{\mathcal{F}}u$ and
$\mathop{\mathrm{supp}}{\mathcal{F}}v$ are contained in $[0,\infty)$. (In
particular, Hörmander’s criterion is fulfilled.) The basic example used in
Section 3 is

$u_{0}(x)=\frac{1}{x+{\rm i}\hskip 1.0pt0}=\lim_{\varepsilon\to
0}\frac{1}{x+{\rm i}\hskip 1.0pt\varepsilon}={\rm vp}\frac{1}{x}-{\rm i}\hskip
1.0pt\pi\delta(x)$

whose Fourier transform is $({\mathcal{F}}u_{0})(\xi)=-2\pi{\rm i}\hskip
1.0ptH(\xi)$. The auto-convolution results in
$({\mathcal{F}}u_{0}\ast{\mathcal{F}}u_{0})(\xi)=-4\pi^{2}\xi H(\xi)$. Thus
$u_{0}^{2}={\mathcal{F}}^{-1}({\mathcal{F}}u_{0}\ast{\mathcal{F}}u_{0})$
exists as Fourier product, and the formula shows that
$u_{0}^{2}(x)=-u_{0}^{\prime}(x)$. The remaining formulas used in Section 3
follow in the same way.

A more general definition of the product of distributions on $\mathbb{R}^{n}$
can be obtained by regularization and passage to the limit. The _model
product_ of $u$ and $v$ is defined as

$[u\cdot v]=\lim_{\varepsilon\to
0}(u\ast\varphi_{\varepsilon})(v\ast\varphi_{\varepsilon})$

provided the limit exists for all mollifiers $\varphi_{\varepsilon}$ of the
form $\varphi_{\varepsilon}(x)=\varepsilon^{-n}\varphi(x/\varepsilon)$ with
$\varphi\in{\mathcal{D}}(\mathbb{R}^{n})$, $\int\varphi(x)\hskip 1.0pt{\rm
d}\hskip 0.5ptx=1$, and is independent of the chosen mollifier. If the Fourier
product exists, so does the model product.

In the one-dimensional case ($n=1$), a yet more general definition is obtained
by using the representation by boundary values of analytic functions, which
was discussed in Section 3. Let $u\in{\mathcal{D}}^{\prime}(\mathbb{R})$. Let

$\widetilde{u}_{\varepsilon}(x)=\widehat{u}(x+{\rm i}\hskip
1.0pt\varepsilon)-\widehat{u}(x-{\rm i}\hskip 1.0pt\varepsilon),$

with the right-hand side as in (9). It was seen in Section 3 that
$u(x)=\lim_{\varepsilon\to 0}\widetilde{u}_{\varepsilon}(x)$. If
$u\in{\mathcal{D}}^{\prime}_{L^{1}}(\mathbb{R})$,
$\widetilde{u}_{\varepsilon}$ is obtained by convolving $u$ with the special
mollifier $\psi_{\varepsilon}(x)=\varepsilon/(\pi(x^{2}+\varepsilon^{2}))$.
The _Tillmann product_ [37] of two distributions $u,v$ is defined by

$u\circ v=\lim_{\varepsilon\to
0}\widetilde{u}_{\varepsilon}\cdot\widetilde{v}_{\varepsilon}$

provided the limit exists. The definition does not work in higher space
dimensions; there, harmonic regularization should be used [5]. In any case,
the powers in (6) and (7) can also be understood in the sense of the Tillmman
product.

Hörmander’s criterion implies the existence of the Fourier product, which
implies the existence of the model product and in turn also the existence of
the Tillmann product. None of the implications can be reversed.

The other products used in this paper enter at different levels. For example,
the most basic product of a smooth function with a distributions enters below
Hörmander’s criterion. The product in $H^{s}_{\rm loc}(\mathbb{R}^{n})$ when
this space is an algebra ($s>n/2$) enters as a subcase of the Fourier product,
but is independent of Hörmander’s criterion. The Nemytskii operators in the
form of a continuous map $L^{p}_{\rm loc}\times L^{q}_{\rm loc}\to L^{1}_{\rm
loc}$, $1/p+1/q=1$, enter at the level of the model product, but are
independent of the Fourier product criterion. For more details on these circle
of ideas, see [24].

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