A first observation





∂_t²u−(t+|x|²)∆_xu=f(t, x), in(0, T)×R^d, u(0, x) =u₀(x), u_t(0, x) =u₁(x) onR^d,

for a suitable source functionf and initial datau0 andu1. Now, this second-order scalar equation can be generalized to the higher-dimensional case, that is







P u =f(t, x), in(0, T)×R^d,

(∂_t^ju)(0, x) =u₀(x) onR^d,j= 0, . . . m−1, for

P =D^m_t +

m

X

j=1

aj(t, x, Dx)D^m−j_t .

Here, thea_j are pseudodifferential operators of orderjthat degenerate like a power oft+|x|² as(t, x)−→(0,0). Such higher-dimensional equation then can be reduced to a first order system of sizem×mas







D_tU =A(t, x, D_x)U +F(t, x), in(0, T)×R^d, U(0, x) =U₀(x), onR^d,

whereA(t, x, Dx)is again a pseudodifferential operator of order1with same dege-neracy.

In Chapter 3 we will develop a pseudodifferential calculus, where we analyze these kinds of operators.

2.2 A first observation

When dealing with the equation itself, the following result arises:

Lemma 2.3. Letu₀, u₁ ∈Cc^∞(R^d), ^u_|x|¹ ∈L²(R^d)andf ∈L²((0, T)×R^d). A solution uto the Cauchy problem







∂²_tu−(t+|x|²)∆_xu=f(t, x), in(0, T)×R^d, u(0, x) =u₀(x), u_t(0, x) =u₁(x) onR^d,

2.2 A first observation 11

satisfies on(0, T)the energy inequality

Proof. Writer =^pt+|x|². We can show this inequality by direct formal computati-ons. Since

Combining (2.7) and (2.8) yields d

Integrating (2.10) with respect totyields

This is the desired estimate.

Remark2.4. Using the sequel notation of function spaces, this lemma would imply u∈CH^1,1∩C¹H^0,1. By settings=δ = 0in Theorem 1.2, our theory will give

u∈CH^1,0∩C¹H^0,0 and thus provide a better result.

This inequality is just a first heuristical observation and will not play any role in the latter analysis. However, since we have to assume that the function _|x|^u1 is square-integrable, this shows a connection to the 2-microlocal spaces, see the Appendix for an introduction.

2.2 A first observation 13

3

Pseudodifferential Calculus

Developing a pseudodifferential calculus to study a certain kind of partial differential equations means to introduce a special class of symbols, which are closely related to the differential operator of the given problem. In our setting we are not interested in long-time behavior, but in existence and uniqueness of solutions near the origin.

Therefore, we define the functionσ: [0, T]×R^d−→Rvia

σ(t, x) =







pt+|x|², t+|x|²≤ ¹₂, 1, t+|x|²≥1.

The function σ is positive for (t, x) 6= (0,0) and belongs to the function space C¹²([0, T]×R^d)∩C^∞(([0, T]×R^d)\ {(0,0)})and describes the kind of singularity near the origin.

3.1 The symbol class Σ

^m,p

We useσas a weight function in the following symbol estimate.

Definition 3.1. For (m, p) ∈ R², the symbol class Σ^m,p consists of all functions a∈C^∞([0, T]×R^2d, MN×N(C))such that for each multiindex(j, α, β)∈N^1+2d₀ the estimate

|∂_t^j∂_x^α∂_ξ^βa(t, x, ξ)|.hξip+2j−|β|+|α|hσξi^{m−p−|α|−2j} holds for all(t, x, ξ)∈[0, T]×R^2d.

For`∈N0let us also define a system of semi-norms via

|a|_m,p;` = sup

(t,x,ξ)∈[0,T]×R^2d j+|α|+|β|≤`

hξi−p−2j+|β|−|α|hσξi−m+p+|α|+2j|∂^j_t∂_x^α∂_ξ^βa|.

It is not difficult to check, thatΣ^m,p together with these semi-norms forms a Fréchet space.

Away form (t, x) = (0,0)the symbol classΣ^m,p coincides withS_1,0^m. Moreover, by restricting tot = 0, we obtain the 2-microlocal estimates in variables(x, ξ) with

15

respect to the LagrangianT₀^∗R^d. More precisely, we get symbolsa(0, x, ξ) ∈Σ^m,p₀ , that is

|∂_x^α∂_ξ^βa(0, x, ξ)|.h|x|ξi^m−p−|α|hξi^{p−|β|+|α|}.

For more details on this clas, see the Appendix. Another difficulty arises directly in the origin(t, x) = (0,0), since we then get the estimate

|∂_x^α∂^β_ξa(0,0, ξ)|.hξi^{p−|β|+|α|}.

Thus, we havea(0,·,·)∈S_1,1^p , or, more generally,(∂_t^ja)(0,·,·)∈S_1,1^p+2j over(x, ξ) = (0,0). The analysis of operators of type (1,1) is technically difficult, because in general they are notL²-continuous. The problems stem from the behavior of the twisted diagonal of the Fourier transform. We will recall some results on operators of type(1,1)in the Appendix. Furthermore, by definition, the weight functionshξiand hσξiare symbols inΣ^1,1andΣ^1,0, respectively. Moreover, we have the embedding

S_1,0^m([0, T]×R^2d)⊆Σ^m,m, which follows directly from the estimate

|∂_t^j∂_x^α∂_ξ^βa|.hξi^m−β .hσξi^−2j−|α|hξim+2j+|α|−|β

for ana∈S_1,0^m([0, T]×R^2d), sincehσξi.hξi.

Example 3.2. We also want to discuss the example ofa(t, x, ξ) =σ² near the origin.

It isσ² ∈Σ^0,−2. Indeed, we have σ²= 1 +|ξ|²

1 +|ξ|²σ²= σ²+σ²|ξ|²

1 +|ξ|² ≤ 1 +σ²|ξ|²

1 +|ξ|² =hσξi²hξi⁻². Moreover, we easily compute

∂tσ²= 1 and ∂xiσ²= 2xi .σ . hσξi hξi . All higher derivatives vanish.

For later use, we also set

Σ^−∞,p:= ^\

m∈R

Σ^m,p and Σ^{−∞,−∞}:= ^\

(m,p)∈R²

Σ^m,p=Cb^∞([0, T]×R^dx,S(R^dξ)).

Let us prove our first result, an approximation lemma:

Lemma 3.3. Leta∈Σ^0,0,0≤ε≤1and setaε=a(t, x, εξ). Thenaεis bounded in Σ^0,0and

aε Σ^m,p

−→ a0

for allp≥m >0asε−→0.

Proof. We prove this with the additional assumption thatm, p∈(0,1], the general case follows immediately. Sincea0 =a(t, x,0) ∈ Σ^0,0, we are finished, if we can prove that for all (t, x, ξ) ∈ [0, T]×R^2d and all (j, α, β) ∈ N^1+2d₀ there exists a constantCsatisfying

hξi−p−2j−|α|+|β|hσξi−m+p+|α|+2j|∂_t^j∂^α_x∂_ξ^β(a(t, x, εξ)−a(t, x,0))| ≤Cε^m. Consider first|β| ≥1. Then the left-hand side can be estimated by

hξi−p−2j−|α|+|β|hσξi−m+p+|α|+2j|∂_t^j∂_x^α∂^β_ξ(a(t, x, εξ)−a(t, x,0))| and further, by algebraic manipulations,

hξi−p−2j−|α|+|β|hσξi−m+p+|α|+2j|∂_t^j∂_x^α∂_ξ^β(a(t, x, εξ)−a(t, x,0))|

Consider nowβ = 0. By using Taylor’s formula, we obtain (∂_t^j∂_x^αa)(t, x, εξ)−(∂_t^j∂^α_xa)(t, x,0) = ^X

|β|=1

(εξ)^β∂_t^j∂_x^α∂^β_ξa(t, x, ϑεξ) β!

3.1 The symbol classΣ^m,p 17

for a certainϑ∈[0,1]. Estimating the right side brings us to

|(∂_t^j∂_x^αa)(t, x, εξ)−(∂_t^j∂_x^αa)(t, x,0)| ≤

d

X

i=1

|εξ| · |∂_t^j∂_x^α∂_ξia(t, x, ϑεξ)|

≤εhξihϑεξi^2j−1+|α|hϑεσξi^−|α|−2j. By similar arguments as in the previous case, we get

hξi−p−2j−|α|+|β|hσξi−m+p+|α|+2j|∂_t^j∂_x^α∂_ξ^β(a(t, x, εξ)−a(t, x,0))|

≤Cεhξihξi^−phϑεξi⁻¹hσξi^−m+p

≤Cε^m εhξi

hϑεξi 1−m

≤Cε^m. This completes the proof.

We are now going to introduce the corresponding pseudodifferential operators, namely the classOp(Σ^m,p).

Theorem 3.4. Ifa∈Σ^m,p andu∈C^∞([0, T],S(R^d)), then Op(a)u:=a(t, x, Dx)u(t, x) = 1

(2π)^d Z

R^d

e^ixξa(t, x, ξ)ˆu(t, ξ)dξ

defines a functiona(t, x, Dx)u∈C^∞([0, T],S(R^d))and the bilinear map

Σ^m,p×C^∞([0, T],S(R^d))−→C^∞([0, T],S(R^d)), (a, u)7−→a(t, x, Dx)u is continuous. Moreover[Dt,Op(a)]u= Op(Dta).

Proof. For every fixedt∈[0, T]we haveu(t, ξˆ )∈S(R^d)and the function (Op(a)u)(t, x) = 1

(2π)^d Z

R^d

e^ixξa(t, x, ξ)ˆu(t, ξ)dξ

is continuous. Moreover,

|(Op(a)u)(t, x)|. Z

R^d

hξi^phσξi^m−p|ˆu(t, ξ)|dξ· sup

(t,x,ξ)∈[0,T]×R^2d

|a(t, x, ξ)|hξi^−phσξi^p−m,

which shows, that(Op(a)u)(t,·)belongs toS(R^d). Sinceaanduˆdepend smoothly

Definition 3.5. We call a(t, x, D_x) a pseudodifferential operator of order m and bi-order(m, p)and set

Op(Σ^m,p) :={Op(a)|a∈Σ^m,p}.

For later use, we also define Σ^m,p,† := {a ∈ Σ^m,p|Op(a)^† ∈ Op(Σ^m,p)}, where † denotes the operation of taking adjoint with respect toL².

We now want to prove the following lemma:

Lemma 3.6. Letχ∈C^∞(R)with χ(z) = 0if|z| ≤ 1/2andχ(z) = 1for|z| ≥1be

are bounded and compactly supported with same support. So now fix an multiindex (j, α, β)∈N^1+2d₀ withj+|α|+|β| ≥1. By higher multi-dimensional chain rule for composed function the function(∂_t^j∂_x^α∂_ξ^βχ⁺)(t, x, ξ)can be written as a sum, where each summand is basically a product of derivatives ofχ(z)and derivatives ofσhξi up to combinatorial constants. Since all derivativesχ⁽ⁿ⁾(z)are zero outside the set {1/2 ≤ |z| ≤1}, also(∂_t^j∂_x^α∂_ξ^βχ)(σhξi)vanishes outside{1/2≤σhξi ≤ 1}. Hence, we can find a constantC_jαβ such that

|∂_t^j∂_x^α∂^β_ξχ⁺| ≤Cjαβhξi^{2j−|β|+|α|}hσξi^−|α|−2j. This means thatχ⁺∈Σ^0,0.

3.1 The symbol classΣ^m,p 19

Sinceχ⁻ is compactly supported, we immediately get

|χ⁻(t, x, ξ)|.hσξi^m for allm∈R. Similar arguments as above give usχ⁻∈Σ^−∞,0.

In the next proposition we list properties of the symbol classesΣ^m,p. Proposition 3.7.

(i) Leta∈Σ^m,pand(j, α, β)∈N^1+2d0 , then∂_t^j∂_x^α∂_ξ^βa∈Σm−|β|,p+2j+|α|−|β|. (ii) For(m, p),(m⁰, p⁰)∈R²the compositionΣ^m,p·Σ^m0,p0 ⊆Σ^m+m0,p+p0 holds.

(iii) We have the embeddingΣ^m,p ⊆Σ^m0,p0 ⇐⇒ m≤m⁰ andp≤p⁰.

Proof. Statement (i) is proven by definition of the symbol class, (ii) by a calculation using to product rule for derivatives.

Let us now prove (iii)and firstΣ^m,p⊆Σ^m0,p0. Then we get that hσξi^m−phξi^p.hσξi^m0−p0hξi^p0

holds for all(t, x, ξ)∈[0, T]×R^2d. For the particular point(0,0, ξ)we havehξi^p. hξi^p0 and sincehξi ≥1for allξ ∈R^d, we finally get p≤ p⁰. By setting|ξ|= 1 we arrive athσi^m−p .hσi^m0−p0. Since this is true for all(t, x)∈[0, T]×R^d, we conclude m−p≤m⁰−p⁰. Putting these two conditions together, we getm≤m⁰andp≤p⁰. Conversely letm≤m⁰andp≤p⁰. Observe that

hξi

hσξi ≥1 =⇒

hξi hσξi

p

≥ hξi

hσξi p⁰

=⇒ hξi^phσξi^−p≤ hξi^p0hσξi^−p0. Moreover,hσξi^m≤ hσξi^m0 and so

hξi^phσξi^m−p ≤ hξi^p0hσξi^m0−p0. This provesΣ^m,p⊆Σ^m0,p0.

Next, for the sake of completeness, we want to prove asymptotic completeness.

Proposition 3.8. Let ak ∈ Σ^mk,p, k ∈ N, be an arbitrary sequence, where mk is monotone decreasing withm_k −→ −∞ask−→ ∞. Then there is a symbola∈Σ^m,p withm:=m1 such that for everyM, there is anN(M)so that for allN ≥N(M):

a(t, x, ξ)−

N

X

k=0

ak(t, x, ξ)∈Σ^m−M,p.

The elementa∈Σ^m,pis uniquely determined by this property moduloΣ^−∞,p. Definition 3.9. We call any suchaasymptotic sum of thea_k,k∈N, and write

a(t, x, ξ)∼ ^X

k∈N

a_k(t, x, ξ).

Proof. In order to apply Theorem A.20, let us setE^k := Σ^mk,pfork∈N, with the natural system of semi-norms

|a|_mk,p;` := sup

(t,x,ξ)∈[0,T]×R^2d j+|α|+|β|≤l

hξi−p−2j+|β|−|α|hσξi^{−mk+p+|α|+2j}|∂_t^j∂_x^α∂_ξ^βa|.

Let again

χ(z) =







0, |z| ≤1/2 1, |z| ≥1

be an excision function and defineχ^k(c) : Σ^mk,p−→Σ^mk,p,c∈R+, as the operator of multiplication byχ(c⁻¹σhξi). Then it is clear, that the diagram

Σ^{mk+1,p  //}

χ(c⁻¹σhξi)

Σ^mk,p

χ(c⁻¹σhξi)

Σ^{mk+1,p  //}Σ^mk,p

commutes. By previous results, we haveχ⁻∈Σ^−∞,0and so we get (1−χ(c⁻¹σhξi))a∈Σ^−∞,p

for alla ∈ Σ^mk,p, k ∈ N. It suffices now to check, that for everyk ∈ Nand all (j, α, β)∈N^1+2d₀ there is an indexi, such thatb(t, x, ξ)∈Σ^µ,pforµ≤miimplies

sup

(t,x,ξ)∈[0,T]×R^2d

hξi−p−2j+|β|−|α|hσξi^{−mk+p+|α|+2j}|∂_t^j∂_x^α∂_ξ^β{χ(c⁻¹σhξi)b(t, x, ξ)}| −→0

3.1 The symbol classΣ^m,p 21

asc−→ ∞. Let us show this for arbitrayµ < mk. We have

suphξi−p−2j+|β|−|α|hσξi^{−mk+p+|α|+2j}|∂^j_t∂_x^α∂_ξ^β{χ(c⁻¹σhξi)b(t, x, ξ)}|

= suphξi−p−2j+|β|−|α|hσξi^µ−mkhσξi−µ+p+|α|+2j|∂_t^j∂_x^α∂_ξ^β{χ(c⁻¹σhξi)b(t, x, ξ)}|

with supremum taken over(t, x, ξ)∈[0, T]×R^2d. Sinceχ(z)is an excision function, we haveχ(z) = 0 for |z| < R with some R > 0. Then χ(c⁻¹σhξi) vanishes for

|c⁻¹σhξi|< R. Thus, it is permitted to take the supremum of|σhξi| ≥cR, and so suphξi−p−2j+|β|−|α|hσξi^{−mk+p+|α|+2j}|∂_t^j∂_x^α∂_ξ^β{χ(c⁻¹σhξi)b(t, x, ξ)}| ≤K(c)hcRi^µ−mk with

K(c) = suphξi−p−2j+|β|−|α|hσξi−µ+p+|α|+2j|∂_t^j∂_x^α∂_ξ^β{χ(c⁻¹σhξi)b(t, x, ξ)}|.

Now, K(c)is uniformly bounded forc ≥εfor allε > 0. So there exists anLwith K(c)≤Lforcsufficently large, for which

|χ(c⁻¹σhξi)b|_mk,p,`≤ L hcRi^mk−µ. Sincem_k−µ >0, we get

|(χ(c⁻¹σhξi)b|_mk,p,`−→0 asc−→ ∞.

In view of Theorem A.20, there now exists a sequence of constantsc_k ∈R+, such that

a(t, x, ξ) =

∞

X

k=1

χ(c⁻¹_k σhξi)a_k(t, x, ξ) converges inΣ^m,p, and has the property

a(t, x, ξ)−

N

X

k=1

a_k(t, x, ξ)∈Σ^mN+1,p

for allN ∈N. Furthermore,ais unique moduloΣ^−∞,pand so a(t, x, ξ)∼^X

k∈N

ak(t, x, ξ).

This completes the proof.

Im Dokument On the Cauchy problem for a class of degenerate hyperbolic equations (Seite 21-33)