数理解析研究所講究録 1211 巻 2001 年 34-43

34

Edge Sobolev Spaces,
Weakly Hyperbolic Equations, andBranching of Singularities
MICHAEL DREHER and INGO WITT *Edge Sobolev spaces are proposed as amain new tool for the in-posedness of thevestigation of weakly hyperbolic quations. The
lnear and the semilnear Cauchy problem in the class of such edge
Sobolev spaces is proved. Applications to the propagation of singubitiae for solutions to semilnear problems are considered.
$\mathrm{w}\mathrm{e}\mathrm{U}$1Introduction
We consider the two semilnear Cauchy problems,$(\dot{F}_{t}u)(0,x)=u_{j}(x)$ ,$Lu=f(u)$ ,$(\dot{\theta}_{t}u)(0,x)=u_{j}(x)$$Lu=f(u, \partial_{t}u, \#\cdot\nabla_{x}u)$

where

$L$

,

$j=0,1$ ,(1.1)$j=0,1$ ,(1.2)

is the weakly hyperbolic operator$L= \partial_{t}^{2}+2\sum_{\dot{g}=1}^{n}\lambda(t)\mathrm{c}_{j}(t)\ \partial_{x_{j}}-\sum_{\dot{|}\dot{s}=1}^{||}\lambda(t)^{2}a_{j}(t)\partial_{x_{t}}\partial_{x_{j}}$(1.3)$+ \sum_{j=1}^{n}\lambda’(t)b_{j}(t)\partial_{x_{j}}+\alpha(t)\mathrm{d}$with coefficientssorne$a_{t\mathrm{j}}$,$b_{\dot{f}}$,$\mathrm{c}_{\mathrm{j}}$belonging to$l_{*}\in \mathrm{N}_{+}=\{1,2,3, \ldots\}$.

$C^{\infty}([-T_{0},T_{0}],\mathrm{R})$and$\lambda(t)=t^{l}$. with, Ibaraki*M. Dreher: Institute of Mathematics, University of Tsukuba,
Mathematics,
305-8571, Japan, email: doeh\mbox{\boldmath $\varpi$}@math.t\epsilon ukuba.ac.jp; I. Witt: Institute of, -14415Potsdam, Germany, email: ingo@mth.uniUniversity of Potsdam,$\mathrm{T}\mathrm{s}\mathrm{u}\mathrm{k}\mathrm{u}\mathrm{b}\mathrm{a}-\mathrm{s}\mathrm{h}\mathrm{i}$$\mathrm{P}\mathrm{F}601553$

potsdam.de.

$\mathrm{D}$35
1INTRODUCTION
The variables and satisfy $(t,x)\in[0,T_{0}]\mathrm{x}\Psi$ ;in the end of this paper wewill also consider the case $(t,x)\in[-T_{0},T_{0}]\mathrm{x}\Psi$ . The operator is supposed
to be weakly hyperbolic with degeneracy for $t=0$ only, i.e.,
$t$

$x$

$L$

$( \sum_{j=1}^{n}c_{\mathrm{j}}(t)\xi_{j})^{2}+\sum_{\dot{o}=1}^{||}a_{i\mathrm{j}}(t)\xi_{}\xi_{j}\geq\alpha_{0}|\xi|^{2}$,$\alpha_{0}>0$,$\forall(t,\xi)$.

The choice of the exponents of in (1.3) reflects s0-called Levi conditions
well-posedness ofwhich are necessary and sufficient conditions for thethe linear Cauchy problem, see [8], [10]. If, for instance, the -exponent of
the coefficient of
were less than $l_{*}-1$ , the linear Cauchy problem for thatwould be well-posed only in certain Gevrey spaces, see [14].
We list some known results. The Cauchy problems (1.1), (1.2) are 10large enough ([9], [10]) andcally well-posed in $C^{k}([0,T], H^{s}(\mathrm{P}))$ for$t$$C^{\infty}$$t$$\partial_{x_{\mathrm{j}}}$

$L$

$s$

$C^{k}([0,T],C^{\infty}(\mathrm{f}\mathrm{f}))([1], [2])$.

Furthermore, singularities of the initial data may propagate in an astonishingway: in [11], it has been shown that the solution $v=v(t, x)$ of,$Lv=v_{tt}-t^{2}v_{xx}-(4m+1)v_{x}=0$ ,(1.4)$m\in \mathrm{N}$with initial data $v(0,x)=u_{0}(x)$ , $v_{t}(0,x)=0$ is given by$v(t,x)= \sum_{j=0}^{m}C_{jm}t^{2j}(\partial_{x}^{j}u_{0})(x+t^{2}/2)$,$C_{\mathrm{j}m}\neq 0$

.

(1.5)

This shows that singularities of propagate only to the left.
Taniguchi and Tozaki discovered branching phenomena for similar operatorsin [15]. They have studied the Cauchy problem$u_{0}$$v_{\mathrm{f}1}-t^{2l}.v_{xx}-bl_{*}t^{l_{\mathrm{r}}-1}v_{x}=0$,$(\partial_{t}^{j}v)(-1, x)=u_{j}(x)$,$j=0,1$ ,and assumed that the initial data have asingularity at some point . Since
the equation is strictly hyperbolic for $t<0$ , this singularity propagates,.
in general, along each of the two characteristic curves starting at -1,
When these characteristic curves cross the line $t=0$ , they split, and thesingularities then propagate along four characteristics for $t>0$ . However, incertain cases, determined by adiscrete set of values for , one or two of thesefour characteristic curves do not carry any singularities.
, for which
The function spaces $C^{k}([0,T], H^{s}(\mathbb{P}))$ andlocal well-posedness could be proved, have the disadvantage that their ele
ments have different smoothness with respect to and . We do not know
$x_{0}$$($$b$$C^{k}([0, T], C^{\infty}(\mathrm{R}^{\iota}))$

$t$

$x$

$x_{0})$

36

any previous result concerning the weakly hyperbolic Cauchy problem stating that solutions belong to afunction space that embeds into the Sobolev
spaces, for someunder the assumption that theinitial data and the right-hand side belong to appropriate function spaces ofthe same kind.
In this paper, solutions to (1.1) and (1.2) are sought in edge Sobolev spaces,
aconcept which has been initially invented in the analysis of elliptic pseud0differential equations near edges, see [7], [13].
The operator can be written ae, whereand $P(t,\tau,\xi)$ is apolynomial in , of degree $\mu=2$ with coefficients depending on smoothly up to $t=0$ . Operators with such astructurearise in the investigation of edge pseudodifferential problems on manifolds
with cuspidal edges, where cusps are described by means of the function. The singularity of the manifold requires the use of adapted classes ofSobolev spaces s0-called edge Sobolev spaces.
We shall define edge Sobolev spaces, where $s\geq 0$ denotes
the Sobolev smoothness with respect to $(t,x)$ for $t>0$ andis anadditional parameter. More precisely, we have continuous embeddings$H_{1\mathrm{o}\mathrm{c}}^{s}((0, T)\mathrm{x}$$s\in \mathrm{R}$$\mathrm{f}\Psi)$$L$$L=t^{-\mu}P(t,t\partial_{t},\Lambda(t)\partial_{x})$$\int_{0}^{t}\lambda(t)dt$$\tau$$\Lambda(t)=$

$\xi$

$t$

$\lambda(t)$$H.,\delta_{j}\lambda((0,T)\mathrm{x}\Psi)$$\delta\in \mathrm{R}$$H_{\infty \mathrm{m}\mathrm{p}}^{s}(\mathrm{R}_{+}\mathrm{x}\mathrm{R}^{n})|_{(0,T)\mathrm{x}\mathrm{B}^{n}}\subset H^{s,\delta;\lambda}((0, T)\mathrm{x}\mathrm{R}^{||})$$\subset H_{1\mathrm{o}\mathrm{c}}^{s}(\mathrm{R}_{+}\mathrm{x} \mathrm{R}^{||})|_{(0,T)\mathrm{x}\mathrm{B}^{*}}.$.

The elements of the spaces
have different Sobolev smoothnear at $t=0$ in the following sense: There are traces ,,with continuous mappings$H^{s,\delta_{j}\lambda}((0,T)\mathrm{x}\mathrm{R}^{b})$$\tau_{j}$$\tau_{j}$:$H^{s,\delta;\lambda}((0,T)\mathrm{x}\mathrm{R}^{11})arrow H^{-\beta j+\beta\delta l.-\beta/2}.(\mathrm{R}^{b})$,$\tau_{j}u(x)=(\partial_{t}^{j}u)(0, x)$$\beta=\frac{1}{l_{*}+1}$, $j<s-1/2$ . This reflects the loss of Sobolev regularity observed
for allwhen passing from the Cauchy data at $t=0$ to the solution. Namely, (1.5)
implies $v(t, .)\in H.(\mathrm{R})$ only, since $C_{mm}\neq 0$ .
shows that
This phenomenon has consequences for the investigation of the nonlinear
problems (1.1), (1.2). The usual iteration procedure giving the existenceof solutions for small times cannot be applied in the case of the standard, since we have no longer amapping whichfunction spacemaps this Banach space into itself.
However, it turns out, that the iteration approach is applicable if we employ. Roughly speakthe specially chosen edge Sobolev spaces
ing, the iteration algorithm does not feel the loss of regularity, because it has$j\in \mathrm{N}$$u_{0}\in H^{s+m}(\mathrm{R})$$C([0,T], H^{s}(\mathrm{R}^{\iota}))$$H^{s,\delta;\lambda}((0,T)\mathrm{x}\Psi)$

37
2EDGE SOBOLEV SPACES
been absorbed in the function spaces. The idea to choose aspecial function
space adapted to the weakly hyperbolic operator has also been used in [3],[4], and [12].
Our results are the following. We claim thewell-posednessof (1.1) and (1.2). In Section 4, we consider the hyperbolic equation from
(1.1), but with data prescribed at $t=-T_{0}$ , and show that the strongest
singularities of the solution propagate in the same way as the singularitiesof the solution solving $Lv=0$ and having the same initial data as for$t=-T$ . The propagation of the singularities of was discussed in [15]. The
proofs of the results mentioned here can be found in [5] and [6].
$H^{s,\delta;\lambda}((0,T)\mathrm{x}\mathrm{R}^{\iota})$

$u$

$v$

$u$

$v$

2Edge Sobolev Spaces
Details on the abstract approach to edge Sobolev spaces can be found, e.g.,in [7], [13]. Proofs of the results listed here are given in [5].

2.1

Weighted Sobolev Spaces on

We say that$u=u(t)\in \mathcal{H}^{\epsilon,\delta}(\mathbb{R}_{+})$,$s\in \mathrm{N}$,$\delta\in \mathrm{R}$$\mathrm{R}_{+}$

if$||u||_{\mathcal{H}^{s,\delta}(\mathrm{n}_{+})}^{2}= \sum_{k=0}’\int_{0}^{\infty}|t^{-\delta}(t\partial_{t})^{k}u(t)|^{2}dt<\infty$.

For arbitrary ,this Mellin Sobolev spacecan be defined byrequirement
interpolation
duality,bythatmeans ofandor the$s$$\mathcal{H}^{s,\delta}(\mathbb{R}_{+})$$\delta\in \mathrm{R}$$||u||_{H^{s,\delta}(\mathrm{R}_{\dagger})}^{2}= \frac{1}{2\pi i}\int_{{\rm Re} z=1/2-\delta}\langle z\rangle^{2s}|Mu(z)|^{2}dz<\infty$,

where $Mu(z)= \int_{0}^{\infty}t^{z-1}u(t)dt$ denotes the Mellin transform. (Both norms.
) Furthermore, the spaceis dense incoincide if
We introduce the notations, $n\geq 0$ ,$s\in \mathrm{N}.$$\mathcal{H}^{s,\delta}(\mathrm{R}_{+})$$C_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}^{\infty}(\mathbb{R}_{+})$$H^{s}(\mathbb{R}_{\vdash}\mathrm{x}\mathrm{R}^{n})=\{v|_{\mathrm{R}\mathrm{x}1\mathrm{R}^{n}}+ :v\in H^{s}(\mathrm{R}^{1+n})\}$$H_{0}^{s}(\overline{\mathbb{R}}_{+}\mathrm{x}\mathrm{R}^{n})=\{v\in H^{s}(\mathrm{R}^{1+n}):\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{w}\subseteq\overline{\mathrm{R}}_{+}\mathrm{x}\mathrm{f}\mathrm{f}\}$$S(\overline{\mathrm{R}}_{+}\mathrm{x}\mathrm{R}^{n})=\{v|_{\mathrm{R}\mathrm{x}\mathrm{R}^{n}}+ :Example 2.1. For $s\geq 0$ ,v\in S(\mathrm{R}^{1+n}\},$$n\geq 0$,$n\geq 0$,.

$H_{0}^{s}(\overline{\mathrm{R}}_{+})=\mathcal{H}^{0,0}(\mathbb{R})\cap \mathcal{H}^{s,s}(\mathbb{R}_{+})$.

functionbe aandDefinition 2.2. Let $s\geq 0$ ,is bounded and $\omega(t)=1$ for close to 0. Then theclose to $t=0$ , i.e.,,
are defined bycone Sobolev spaces,$\delta\in \mathrm{R}$$\omega$$\in C^{\infty}(\overline{\mathrm{R}}_{+})$$\mathrm{c}\mathrm{u}\mathrm{t}-\mathrm{o}\mathrm{f}\mathrm{f}$$t$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\omega$$H^{s,\delta_{j}\lambda}(\mathrm{R}_{+})$$H_{0}^{s,\delta;\lambda}(\overline{\mathrm{R}}_{+})$$H^{s,\delta;\lambda}(\mathbb{R}_{+})=\{\omega u_{0}+(1-\omega)u_{1} :u_{0}\in H^{s}(\mathbb{R}_{+}), u_{1}\in \mathcal{H}_{\#}^{s,\delta;\lambda}(\mathbb{R}_{+})\}$

38
2.2 The Spaces$H^{s,\delta;\lambda}(\mathrm{R}_{+}\mathrm{x}\Psi)$$H_{0}^{s,\delta;\lambda}(\overline{\mathrm{R}}_{+})=\{\omega u_{0}+(1-\omega)u_{1} :u_{0}\in H_{0}^{s}(\overline{\mathrm{R}}_{+}), u_{1}\in \mathcal{H}_{\#}^{s,\delta_{\dot{1}}\lambda}(\mathrm{R}_{+})\}$

.

where
equipped with the norm$\mathcal{H}_{\#}^{s,\delta;\lambda}(\mathrm{R}_{+})=\mathcal{H}^{0,\delta l}\cdot(\mathrm{R}_{+})\cap \mathcal{H}^{s,s(l.+1)+\delta l}$$(\mathrm{R}_{+})$. The space,$H^{s,\delta;\lambda}(\mathrm{R}_{+})$$||u||_{H}^{2}.,\iota_{i}\mathrm{x}_{(\mathrm{r}_{+})(\mathrm{n}_{+})(\mathrm{n}_{+})\cap \mathcal{H}\cdot(\mathrm{n}_{+})}=||\omega u_{0}||_{H}^{2}.+||(1-\omega)u_{1}||_{\mathcal{H}^{0,\iota\iota_{*}(l.+1)+\iota\iota}}^{2}.,.$

2.2

The Spacesis.

$H^{\epsilon,\delta;\lambda}(\mathbb{R}_{+}\mathrm{x}\mathbb{R}^{n})$be astrongly continDefinition 2.3. Let be aHilbert space andfor , $\nu’>0$ anduous group of isomorphisms acting on with. For
the abstract edge Sobolev space$u\inS’(\Psi;E)$andconsists of all
such that$E$$\{\kappa_{\nu}\}_{\nu>0}$$E$$\kappa_{\nu}\kappa_{\nu’}=\kappa_{\nu\sqrt}$$\nu$$\mathcal{W}^{s}(\mathrm{R}^{1}; (E, \{\kappa_{\nu}\}_{\nu>0}))$$s\in \mathrm{R}$$\kappa_{1}=\mathrm{i}\mathrm{d}_{B}$$\hat{u}\in L_{1\mathrm{o}\mathrm{c}}^{2}(\mathrm{R}^{b}; E)$.

$||u||_{\mathcal{W}(\mathrm{B}^{*};(E,\{n_{\nu}\}_{\nu>}0))}^{2}..= \int_{\mathrm{B}^{n}}\langle\xi\rangle^{\mathfrak{U}}||\kappa_{(\xi)}^{-1}\hat{u}(\xi)||_{B}^{2}d\xi<\infty$

Ddnition 2.4. Let $s\geq 0$ ,$\delta\in \mathrm{R}$Then we define the group$\{\kappa_{\nu}^{(\delta)}\}_{\nu>0}$by, $\nu>0$ ,$\kappa_{\nu}^{(\delta)}w(t)=\nu^{\beta/2-\beta\delta l}.w(\nu^{\beta}t)$where $\beta=1/(l_{*}+1)$ , and set$H^{\cdot}’\delta;\lambda(\mathrm{R}_{+}\mathrm{x}\Psi)=\mathcal{W}^{\cdot}(\Psi;(H^{s,\delta;\lambda}(\mathrm{R}_{+}), \{\kappa_{\nu}^{(\delta)}\}_{\nu>0}))$$H_{0}^{s,\delta;\lambda}(\overline{\mathrm{R}}_{+}\mathrm{x}\mathrm{R}^{||})=\mathcal{W}^{s}(\mathrm{R}_{\mathrm{i}}^{||}(H_{0}^{\delta_{j}\lambda}.,(\overline{\mathrm{R}}_{+}),$

Proposition 2.5. (a)$S(\overline{\mathrm{R}}_{+}\mathrm{x}W)$$\{\kappa_{\nu}^{(\delta)}\}_{\nu>0}))$is dense in,.

$H.,\delta;\lambda(\mathrm{R}_{+}\mathrm{x}\mathrm{R}^{l})$.

(b) For every fixed
forms an interpolation
scale with respect to the complex interpolation method.
$\delta\in \mathrm{R}$(c) If $l_{*}=0$ , then$\{H^{s,\delta_{j}\lambda}(\mathrm{R}_{+}\mathrm{x}\mathrm{R}^{b}):s\geq 0\}$$H.,\delta;\lambda(\mathrm{R}_{\vdash}\mathrm{x}\mathrm{N}^{\iota})=H^{s}(\mathrm{R}_{+}\mathrm{x}\mathrm{R}^{l})$.

(d) We have the continuous embeddings$H_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}^{s}(\mathrm{R}_{+}\mathrm{x}\Psi)\subset H^{\cdot}’\delta;\lambda(\mathrm{R}_{+}\mathrm{x}\Psi)\subset H_{1\mathrm{o}\mathrm{c}}^{s}(\mathrm{R}_{+}\mathrm{x}\Psi)$.

admit traces at $t=0$ in the following sense.
, $j<s-1/2$ , theThen, for each
Proposition 2.6. Let $s\geq 0$ ,continuity
to a map, edends by,map

The spaces$H^{s,\delta;\lambda}(\mathrm{R}_{+}\mathrm{x}\Psi)$$j\in \mathrm{N}$$\delta\in \mathrm{R}$$S(\overline{\mathrm{R}}_{+}\mathrm{x}\mathrm{R}^{1})arrow S(\Psi)$$\tau_{j}$:$u\vdash\rangle(\dot{\theta}_{t}u)(0,x)$.

$H^{s,\delta;\lambda}(\mathrm{R}_{+}\mathrm{x}\mathrm{R}^{1l})arrow H^{\cdot}-\beta j+\beta\delta l.-\beta/2(\mathrm{R}^{n})$$R\iota\hslash hemore$, we have a surjective map$H^{s,\delta_{j}\lambda}(\mathrm{R}_{\vdash}\mathrm{x}\Psi)$$arrow\prod_{j<\cdot-1/2}H^{s-\beta j+\beta\delta l.-\beta/2}(\Psi)$,$u\ovalbox{\tt\small REJECT}\mapsto\{\tau_{j}u\}_{j<s-1/2}$.

39
2EDGE SOBOLEV SPACES
Proposition 2.7. For $s\geq 0$ ,, the following maps are continuous:(a) :;(b) 1:
for $l=0,1$ , , ;(c):for $1\leq j\leq n$ ;(d).
for each
Here means the operator of multiplication by . Similarly for .
$\delta\in \mathrm{R}$$H^{s+1,\delta;\lambda}(\mathrm{R}_{+}\mathrm{x}\Psi)arrow H^{s,\delta+1;\lambda}(\mathrm{R}_{\star}\mathrm{x}\mathrm{R}^{\iota})$$\partial_{t}$$H^{s,\delta_{j}\lambda}(\mathrm{R}_{+}\mathrm{x}\Psi)arrow H^{s,\delta+l/l_{*;}\lambda}(\mathrm{R}_{+}\mathrm{x}\Psi)$$l_{*}$$\ldots$$H^{s+1,\delta;\lambda}(\mathrm{R}_{\vdash}\mathrm{x}\Psi)arrow H^{s,\delta;\lambda}(\mathrm{R}_{+}\mathrm{x}\Psi)$$\partial_{x_{\mathrm{j}}}$$\varphi=\varphi(t)\in S(\overline{\mathrm{R}}_{+})$$\varphi:H^{s,\delta;\lambda}(\mathrm{R}_{\vdash}\mathrm{x}\Psi)arrow H^{s,\delta;\lambda}(\mathrm{R}_{\vdash}\mathrm{x}\Psi)$$t^{l}$$t^{l}$$\varphi$

2.3

The Spaces$H^{s,\delta;\lambda}((0,$T)$\mathrm{x}\mathbb{R}^{n})$For $T>0$ , we set
and equip
this space with its infimum norm. There is an dtemative description of thisspace provided that.
$H^{s,\delta_{j}\lambda}((0,T)\mathrm{x}\mathbb{P})=H^{s,\delta;\lambda}(\mathrm{R}_{\vdash}\mathrm{x}\Psi)|_{(0,T)\mathrm{x}\mathrm{R}^{n}}$$s\in \mathrm{N}$

Lemma 2.8. Let
space$s\in \mathrm{N}$,$H^{s,\delta;\lambda}((0, T)\mathrm{x}\Psi)$

and $T>0$ . Then the infimum norm of the, whereis equivalent to the norm$\delta\in \mathrm{R}$$||.||_{s,\delta_{j}T}$$||u||_{s,\delta T}^{2}= \sum_{l=0}^{s}T^{21-1}J_{0}^{T}\int_{\mathrm{R}_{\xi}^{\hslash}}\theta_{l}(t,\xi)^{2}|d_{t}\hat{u}(t,\xi)|^{2}d\xi dt$,$\theta_{l}(t,\xi)=\{\begin{array}{l}\langle\xi\rangle^{s-\mathrm{t}}\lambda(t_{\xi})^{-\delta-l}..0\leq t\leq t_{\xi}\langle\xi\rangle^{s-l}\lambda(t)^{-\delta-l}..t_{\xi}\leq t\leq T\end{array}$Here we have introduced the notation
Lemma 2.9. For , $s’\geq 0$ , ,$\delta$$s$$\delta’\in \mathrm{R}$$t_{\xi}=\langle\xi\rangle^{-\beta}$, $\beta=1/(l_{*}+1)$ .

, and $T>0$ ,$H^{s,\delta;\lambda}((0, T)\mathrm{x}\mathrm{R}^{n})\subseteq H^{s’,\theta\cdot\lambda}|((0,T)\mathrm{x}\mathrm{R}^{n})$if and only if$s\geq s’$,$s+\beta\delta l_{*}\geq s’+\beta\delta’l_{*}$.

The two conditions on are related to the fact that the elements of the edge
Sobolev spaces have different smoothness for $t>0$ and $t=0$ , respectively.
The following result provides acriterion when the superposition operators
defined by the right-hand sides of the hyperbolic equations in (1.1) and (1.2)
map an edge Sobolev space into itself. This result is related to the fact thatthe usual Sobolev spaces are Banach algebras for sufficiently high smoothness.
$s$

Proposition 2.10. Let $f=f(u)$ be an entire function with $f(0)=0$ , $i.e.$ ,,andAssume thatfor alleryiththe$\beta\delta l_{*}\}>(n+2)/2$ . Then there is,for each $R>0$ , a constant $C_{1}(R)$property that$f(u)= \sum_{j=1}^{\infty}f_{j}u^{j}$$u\in \mathbb{R}$$||f(u)||_{s,\delta T}\leq C_{1}(R)||u||_{s,\delta;T}$

provided that ,$u$$\min\{\lfloor s\rfloor$$\lfloor s\rfloor+\delta\geq 0$,$\lfloor s\rfloor+$$||f(u)-f(v)||_{s,\delta T}\leq C_{1}(R)||u-v||_{s,\delta_{j}T}$$v\in H^{s,\delta;\lambda}((0, T)\mathrm{x}\mathbb{R}^{n})$and$||u||_{s,\delta T}\leq R$,$||v||_{s,\delta T}\leq R$

.

40

3

Linear and Semilinear Cauchy Problems

Our considerations start with the linear Cauchy problem$Lw(t,x)=f(t,x)$ ,$(\dot{\theta}_{t}w)(0,x)=w_{\dot{f}}(x)$,$j=0,1$ .

(3.1)

’

(3.2)

We introduce the number$Q_{0}=- \frac{1}{2}+\sup_{\xi}\frac{|\sum_{j}(-b_{j}(0)\xi_{j}+c_{\mathrm{j}}(0)\xi_{\mathrm{j}})|}{2\sqrt{(\sum_{j}\mathrm{C}_{j(0)\xi_{j})^{2}+\sum_{i}\mathrm{h}j(0)\xi_{}\xi_{\dot{f}}}}}.$

and fix $A_{0}=Q_{0}l_{*}/(l_{*}+1)=\beta Q_{0}l_{l}$ .

Theorem 3.1. Let ,, andthere is a solution
is unique in the space$s$$Q\in \mathrm{R}$$s>1$ ,.

,
let
, where $A=\beta Ql_{*}$ . Thento (3.1). Moreover, the solution$Q\geq Q\mathrm{o}$$R\iota\hslash her$$w_{0}\in H^{s+A}(\mathrm{P})$$f\in H.-1\overline{Q},+1;\lambda((0,T)\mathrm{x}\mathrm{R}^{b})$$w_{1}\in H^{s+A-\beta}(\Psi)$$w\in H^{s,Q;\lambda}((0,T)\mathrm{x}\mathrm{P}^{\iota})$$w$$H^{s,Q_{0;}\lambda}((0,T)\mathrm{x}\Psi)$.

Remark S. . The parameter
describes the loss of regularity. The explicit
representations of the solutions for special model operators in [11] and [15]show that the statement of the Theorem becomes false if $A<A_{0}$ .
$B$$A_{0}$

Theorem 3.3. Let
and assume that $\min\{s, s+\beta Q_{0}l_{*}\}>(n+2)/2$ ,wherebe the number from (3.2). Suppose that $f=f(u)$ is an entire
function with $f(0)=0$ . Let $Q\geq Q_{0}$ and $A=\beta Ql_{*}$ . Then, for$H.+A(\Psi)$ ,, there is a number $T>0$ with the property thata solution
to the Cauchy problem (1.1) exists. This.
solution is unique in the space$s\in \mathrm{N}$$Q_{0}$$u_{0}\in$$u_{1}\in H^{s+A-\beta}(\mathrm{R}’)$$u\in H^{s,Q;\lambda}((0, T)\mathrm{x}\mathrm{R}^{b})$$H.,Q_{0;}\lambda((0, T)\mathrm{x}\mathrm{R}^{b})$$u$Theorem 3.4. Let
and assume that $s-1>(n+2)/2$ . Suppose$f=f(u,v,v_{1},\ldots,v_{1},)$thatis entire with $f(0, \ldots, 0)=0$ . Let $Q\geq Q_{0}$, $u_{1}\in H^{s+A-\beta}(\Psi)$ , there is. Then, forandnumber $T>0$ with the property that a solution
to the Cauchy problem (1.2) exists. This solution is unique in the space$s\in \mathrm{N}$$u_{0}\in H.+A(\mathrm{R}^{1})$$A=\beta Ql_{\mathrm{s}}$$a$$u\in H.,Q;\lambda((0,T)\mathrm{x}\mathrm{P})$$u$$H^{s,Q\mathrm{o};\lambda}((0, T)\mathrm{x}\Psi)$.

Eventually, we state result concerning the propagation of mild singularities.

Theorem 3.5. Let satisfy the assumptions of Theorem 3.3. Assume, whereis given by (3.2). Let,
the solution to$s$$H^{s+\beta Q_{0}l}\cdot(\mathrm{R}^{b})$$Lv=0$ ,$u_{1}\in H^{s+\beta Q_{0}l.-\beta}(\mathrm{N})$$(\dot{\theta}_{t}v)(0,x)=u_{j}(x)$Then the solutions ,$u$$j=0,1$ .

,$v\in H.,Q\mathrm{o};\lambda((0,T)\mathrm{x}\mathrm{R}^{b})$$u-v\in H^{s+\beta,Q\mathrm{o};\lambda}((0, T)\mathrm{x}\mathrm{R}^{||})$$u_{0}\in$$Q_{0}$

.

$v$

be

(3.3)

to (1.1) and (3.3) satisfy

41
4BRANCHING PHENOMENA-You’soperator from (1.4). Then $l_{*}=1$ ,Example 3.6. Consider Qi$\beta=1/2$ , and $Q_{0}=2m$ . Theorems 3.1, 3.3, and 3.5 state that the solutions, to (1.1), (3.3) satisfy$L$${\rm Min}$

$u$

$v$

$u$

if

,

$v\in H^{s,2m;\lambda}((0,T)\mathrm{x}\mathrm{R})$$u_{0}\in H^{s+m}(\mathrm{R})$

$u$

,

,

,

$u-v\in H^{s+1/2,2m;\lambda}((0,T)\mathrm{x}\mathrm{R})$$u_{1}\in H^{s+m-1/2}(\mathrm{R})$$v\in H_{1\mathrm{o}\mathrm{c}}^{s}((0,T)\mathrm{x}\mathrm{R})$,. Proposition 2.5 then implies$u-v\in H_{1\mathrm{o}\mathrm{c}}^{s+1/2}((0,T)\mathrm{x}\mathrm{R})$.

We find that the strongest singularities of coincide with the singularities.
of . The latter can be looked up in (1.5) in case$u$$u_{1}\equiv 0$$v$

4Branching Phenomena for Solutions to
Semilinear Equations
In this section, we consider the Cauchy problems$Lu=f(u)$ ,$(\partial_{t}^{j}u)(-T_{0}, x)=\epsilon w_{j}(x)$$Lv=0$ ,$(\partial_{t}^{j}v)(-T_{0}, x)=\epsilon w_{j}(x)$,,$j=0,1$ ,(4.1)$j=0,1$ ,(4.2)

with from (1.3), and we are interested in branching phenomena for singularities of the solution . Our main result is Theorem 4.2.
-You that we have to expect aloss
We know, e.g., from the example of Qi
of regularity when we pass from the Cauchy data at $\{t=0\}$ to the solution
at $\{t\neq 0\}$ . However, we also will observe aloss of smoothness if we prescribe
Cauchy data at, say, $t=$ -To and look at the solution for $t=0$ .
$L$

$u$

${\rm Min}$

Definition 4.1. Let $s\geq 0$ ,,if

We say that$\delta\in \mathbb{R}$$u(t, x)\in H^{s,\delta;\lambda}((0,T)\mathrm{x}\mathbb{P}^{b})$$u(-t, x)\in H_{0}^{s,\delta;\lambda}((0,T)\mathrm{x}\mathrm{R}^{n})$

,
Let ,
say
that
We
and
$s_{-}$$s_{+}\geq 0$$\delta_{-}$,$\delta_{+}\in \mathrm{R}$$u\in H^{s,\delta;\lambda}((-T, T)\mathrm{x}\mathrm{F}^{\mathrm{r}})$$u(-t,x)\in H^{s,\delta;\lambda}((0, T)\mathrm{x}\mathrm{F})$.

and suppose that$s_{-}+\beta\delta_{-}l_{*}=s_{+}+\beta\delta_{+}l_{*}$,$s_{+}\leq s_{-}$.

if. We define the norm by$u\in H^{S_{-},i}+^{\delta_{-},\delta}’+;\lambda((-T,T)\mathrm{x}\mathrm{f}\mathrm{f})$$u(-t,x)\in H^{s_{-},\delta_{-;}\lambda}((0,T)\mathrm{x}\mathbb{P})$, and $u(t, x)-$$u\in H^{s}+^{\delta_{+;}\lambda}’((-T, T)\mathrm{x}\mathbb{R}^{1})$$||u(t,x)||_{H-\cdot+^{\delta\delta;\lambda}}..’-\cdot+((-\tau,\tau)\mathrm{x}\mathrm{R}^{n})$$=||u(t,x)||_{H^{*}+^{\delta;\lambda}}’+((0,\tau)\mathrm{x}\mathrm{B}^{n})+||u(-t,x)||_{H^{*}-\cdot s_{-i}\lambda}((0,T)\mathrm{x}\mathrm{B}^{n})$This choice of the norm is possible, since, compare Lemma 2.9$H^{s\delta\lambda}+,+j((0,T)\mathrm{x}\mathrm{R}^{\mathrm{r}})$.

$H^{s_{-},\delta_{-j}\lambda}((0,T)\mathrm{x}\mathrm{R}^{l})$$\subset$

42
REFERENCES
The next theorem relates branching phenomena for the semilinear problem(4.1) with branching phenomena for the linear reference problem (4.2). This
relation between asemilinear Cauchy problem and an associated linear reference problem has already been discussed in Example 3.6. The explicit representations of solutions in [15] show that the statements about the smoothnessof solutions in the following theorem are optimal.

Theorem 4.2. Let be the operator ffom (1.3),
be the number from
(3.2), and suppose that,,
, when $Q_{+}=Q_{0}$ , $Q_{-}=-1-Q_{0}$ , and $s_{+}=s_{-}+\beta Q_{-}l_{*}-\beta Q_{+}l_{*}$ .
,, and that $f=f(u)$ is an entire
Assume that$f(0)=f’(0)=0$.
function with
Then there is ansuch that for every
there are unique s0lutions
to (4.1) and (4.2), respectively,
which, in addition, satisfy$L$$Q_{0}$$\min\{\lfloor s_{\pm}\rfloor, \lfloor s_{\pm}\rfloor+\beta Q_{\pm}l_{*}\}>(n+2)/2$$\lfloor s_{\pm}\rfloor+Q_{\pm}\geq 0$$s_{\pm}\geq 1$$w_{0}\in H.-(\mathrm{R}’)$$w_{1}\in H^{s_{-}-1}(\mathrm{R}^{b})$$0<\epsilon$$\epsilon_{0}>0$$\leq\epsilon_{0}$$u,v\in H.-,.+,q_{-},q_{+;\lambda}((-T_{0},T_{0})\mathrm{x}\mathrm{N}^{\iota})$$u-v\in H^{\cdot}-+\beta,.++\beta,Q_{-},g_{+j}\lambda((-T_{0},T_{0})\mathrm{x}\mathrm{R}^{l})$Remark$\mathit{4}\cdot S$.

. Due to (3.2), $Q_{0}\geq-1/2$ , which is equivalent to(4.3)$s_{+}\leq s_{-}$. If, no loss of regularity occurs when we cross the line of degeneracy.
The case of alinear hyperbolic operator with this property and countably
many points of degeneracy (or singularity) accumulating at $t=0$ has been
discussed in [16].
$s_{+}=s_{-}$

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