A novel integral transform approach to solving partial ...personal.psu.edu/~tuk14/Potsdam2019/Slides/Yagdjian.pdf · Semilinear equation in the de Sitter space-time Karen Yagdjian

A novel integral transform approach to solving partialdifferential equations in the curved space-times

Karen Yagdjian

University of Texas Rio Grande Valley

Microlocal and Global Analysis, Interactions with GeometryColloquium in honor of Professor Schulze’s 75th birthday

University of Potsdam, March 4-8, 2019

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-timesMicrolocal and Global Analysis, Interactions with Geometry Colloquium in honor of Professor Schulze’s 75th birthday University of Potsdam, March 4-8, 2019 1 / 45

The Integral Transform: purpose and structure

The purpose: target problem (partial differential equations)

The structure:

The function subject to transformationThe kernel function


Outline

The Target Equations

Motivation. Gas Dynamics. The Expanding Universe

From Duhamel’s Principle to Integral Transform

The Kernel of Integral Transform

Applications

The Klein-Gordon Equation in the de Sitter Space-time

Maximum principle for hyperbolic equations

Estimates for solution

Huygens’ Principle.

Semilinear equation in the de Sitter space-time


The Target Equation:

∂2t u − a2(t)A(x , ∂x)u −M2u = f , t ∈ (0,T ), x ∈ Ω ⊆ Rn .

Here M ∈ C andA(x , ∂x) =

∑|α|≤m

aα(x)∂αx ,

∂αx = ∂α1x1· · · ∂αn

xn , |α| = α1 + . . .+ αn

The Goal : Explicit representation for the solutions of that equation

The Tool : The new integral transform


Why this equation?

∂2t u − a2(t)A(x , ∂x)u −M2u = f , t ∈ (0,T ), x ∈ Ω ⊆ Rn .

Here M ∈ C andA(x , ∂x) =

∑|α|≤m

aα(x)∂αx

Equations of Gas Dynamics

Equations of Physics in Expanding Universe


Gas Dynamics

Tricomi equation (Chaplygin’1909, Tricomi’1923):

∂2t u − t∆u = f , t ∈ R, x ∈ Ω ⊆ Rn .

The equation representing in hodograph variables a steady transonicflow (flight) of ideal gas.

The small disturbance equations for the perturbation velocitypotential of a near sonic uniform flow of dense gases (Kluwick,Tarkenton, Cramer’93)

∂2t u − t3∆u = f , t ∈ R, x ∈ Ω ⊆ Rn .

Here

∆u =∂2u

∂x21

+∂2u

∂x22

+ · · ·+ ∂2u

∂x2n


Einstein’s Equations with Cosmological Term, 1917

The metric (tensor) gµν = gµν(x0, x1, x2, x3) , where µ, ν = 0, 1, 2, 3

Rµν −1

2gµνR = 8πGTµν − Λgµν

Rµν is the Ricci tensor

Scalar curvature R = gµνRµν

Energy-momentum tensor Tµν

Λ is the cosmological constant


The de Sitter space-time

The line element in the spatially flat de Sitter space-time has the form

ds2 = − c2dt2 + e2Ht(dx2 + dy2 + dz2) ,

gik =

−c2 0 0 0

0 e2Ht 0 00 0 e2Ht 00 0 0 e2Ht

c is the speed of light, H is the Hubble constant. We set c = 1 andH = 1.

ds2 = −dt2 + a2sc(t)dσ2, where asc(t) = eHt is the scale factor.


Big Bang and evolution of the Universe

de Sitter model

radiation dominated universe

Einstein-de Sitter spacetime (matter dominated universe)

Big Bang

Time

scale factor t

scale factor t2/3

scale factor et

1 2 3 4 5

50

100

150


The Covariant Wave and Klein-Gordon Equations

The covariant wave equation

1√|g(x)|

∂

∂x i

(√|g(x)|g ik(x)

∂ψ

∂xk

)= f .

The covariant Klein-Gordon Equation

1√|g(x)|

∂

∂x i

(√|g(x)|g ik(x)

∂ψ

∂xk

)−m2ψ = f ,

where |g(x)| := | det(gik(x))| and x = (x0, x1, x2, x3) ∈ R4, x0 = t.The Einstein’s summation notation convention is used.


Waves in the Universe (Cosmological Models)

The (non-covariant) wave equation in the radiation dominateduniverse:

utt − t−1A(x , ∂x)u = f .

The wave equation in the Einstein-de Sitter space-time (matterdominated universe). The covariant d’Alambert’s operator, after thechange ψ = t−1u of the unknown function, leads to

utt − t−4/3A(x , ∂x)u = f .

Here

A(x , ∂x)u =

√1− Kr2

r2

∂

∂r

(r2√

1− Kr2∂u

∂r

)+

1

r2 sin θ

∂

∂θ

(sin θ

∂u

∂θ

)+

1

r2 sin2 θ

(∂

∂φ

)2

u ,

where K = −1, 0, or +1, for a hyperbolic, flat or spherical spatialgeometry, respectively.


The Klein-Gordon Equation in Expanding UniverseThe metric g00 = −1, g0j = 0, gij = e2tσij(x), i , j = 1, 2, . . . , n,

Scale factor asc(t) = et (accelerating expansion).

The covariant Klein-Gordon equation in the de Sitter space-time:

ψtt − e−2tA(x , ∂x)ψ + nψt + m2ψ = f .

Here m is a physical mass of the field (particle) while

A(x , ∂x)ψ =1√

| detσ(x)|

n∑i ,j=1

∂

∂x i

(√| detσ(x)|σij(x)

∂

∂x jψ

)

If u = ent/2ψ, then

utt − e−2tA(x , ∂x)u −M2u = f ,

where M2 = n2

4 −m2 is curved (or effective) mass. This exampleincludes equations in the metric with hyperbolic, flat or sphericalspatial geometry.


The Klein-Gordon Equation of Self-Interacting Field inExpanding Universe

In the spatially flat de Sitter universe the equation for the scalar field withmass m and potential function V is

1

c2φtt +

1

c2nHφt − e−2Ht∆φ+

m2c2

h2φ =

1

c2V ′(φ) .

Here x ∈ Rn, t ∈ R, and ∆ is the Laplace operator, ∆ :=∑n

j=1∂2

∂x2j

,

H =√

Λ/3 is the Hubble constant,

Λ is the cosmological constant.

In the case of Higgs potential (Higgs boson)

φtt + 3Hφt − e−2Htc2∆φ = µ2φ− λφ3

with λ > 0 and µ > 0 while n = 3.


The New Integral Transform

Let f = f (x , t) be a given function of t ∈ (0,T ), x ∈ Ω.

Ω is a domain in Rn, A(x , ∂x) =∑|α|≤m aα(x)∂αx .

The function w = w(x , t; b) is a solution of the problem

wtt − A(x , ∂x)w = 0, t ∈ (0,T1), x ∈ Ω,

w(x , 0; b) = f (x , b), wt(x , 0; b) = 0, x ∈ Ω,

with the parameter b ∈ (0,T ) and 0 < T1 ≤ ∞.

We introduce the integral operator

K : w 7−→ u,

which maps function w = w(x , t; b) into solution of the equation

utt − a2(t)A(x , ∂x)u −M2u = f , t ∈ (0,T ), x ∈ Ω .


The New Integral Transform

The integral operator K : w 7−→ u is

u(x , t) = K[w ](x , t)

:=

∫ t

0db

∫ |φ(t)−φ(b)|

0K (t; r , b;M)w(x , r ; b)dr , x ∈ Ω, t ∈ (0,T ).

Here φ(t) =

∫ t

0a(τ) dτ is a distance function produced by a = a(t),

M ∈ C is a constant.

Integral transform is applicable to the distributions and fundamentalsolutions as well.

In fact, u = u(x , t) takes initial values

u(x , 0) = 0, ut(x , 0) = 0, x ∈ Ω .


From Duhamel’s principle to the New Integral Transform

The revised Duhamel’s principle:

Our first observation is that the function

u(x , t) =

∫ t

0db

∫ t−b

0wf (x , r ; b) dr , (1)

is the solution of the Cauchy problemutt −∆u = f (x , t), in Rn+1

u(x , 0) = 0, ut(x , 0) = 0 in Rn ,

if wf = wf (x ; t; b) solveswtt −∆w = 0, (x , t) ∈ Rn+1,

w(x , 0; b) = f (x , b), wt(x , 0) = 0, x ∈ Rn.

The second observation is that in (1) the upper limit t − b of theinner integral is generated by the propagation phenomena with thespeed =1. In fact, t − b is a distance function.


Our third observation is that the solution operator

G : f 7−→ u

can be regarded as a composition of two operators G = K WE .The first one

WE : f 7−→ w

is a Fourier Integral Operator, which is a solution operator of theCauchy problem for wave equation. The second operator

K : w 7−→ u

is the integral operator (1).

Figure: Case of A(x , ∂x) = ∆


Figure: Case of A(x , ∂x) = ∆

We introduce the distance function φ(t) and provide the integraloperator with the kernel

u(x , t) =

∫ t

0db

∫ |φ(t)−φ(b)|

0K (t; r , b;M)w(x , r ; b)dr ,

x ∈ Ω, t ∈ (0,T ).

This operator generates solutions of different well-known equationswith x-independent coefficients.

We have generated a class of operators for which we have obtainedexplicit representation formulas for the solutions of the equations with

a(t) = t`, ` ∈ R, a(t) = e±t


Figure: (b) Case of general A(x , ∂x)

By varying the first mapping, we extend the class of the equations forwhich we can generate the solutions.

More precisely, consider the diagram (b), where w = wA,ϕ(x , t; b) is asolution to

wtt − A(x , ∂x)w = 0, t ∈ (0,T1), x ∈ Ω,

w(x , 0; b) = f (x , b), x ∈ Ω,

with the parameter b ∈ (0,T ).If we have a resolving operator of this problem, then, by applyingintegral transform, we can generate solutions of new equations.

Thus, GA = K EEA.

The new class of equations contains operators with x-dependentcoefficients.


The Kernel. The Klein-Gordon Equation in de Sitterspace-time

For given x0 ∈ Rn, t0 ∈ R define a chronological future D+(x0, t0) anda chronological past D−(x0, t0) of (x0, t0) :

D±(x0, t0) := (x , t) ∈ Rn+1 ; |x − x0| ≤ ±(e−t0 − e−t) .

For (x0, t0) ∈ Rn × R the dependence and influence domains


The Kernel. The Klein-Gordon Equation in de Sitterspace-time

For (x0, t0) ∈ Rn × R, M ∈ C, we define

E (x , t; x0, t0;M) := 4−MeM(t0+t)(

(e−t0 + e−t)2 − (x − x0)2)M− 1

2

×F(1

2−M,

1

2−M; 1;

(e−t0 − e−t)2 − (x − x0)2

(e−t0 + e−t)2 − (x − x0)2

),

where (x , t) ∈ D+(x0, t0) ∪ D−(x0, t0)

Here D−(x0, t0) is a chronological future and D−(x0, t0) is achronological past of (x0, t0):

D±(x0, t0) := (x , t) ∈ Rn+1 ; |x − x0| ≤ ±(e−t0 − e−t) .

F(a, b; c ; ζ

)is the Gauss’ hypergeometric function.

We use x2 := |x |2 for x ∈ Rn.


The Kernels K0(r , t;M) and K1(r , t;M)

are defined by

K0(r , t;M) = −[∂

∂bE (r , t; 0, b;M)

]b=0

, (2)

K1(r , t;M) = E (r , t; 0, 0;M) (3)

The positivity of the kernel functions E , K0 and K1.

Proposition [A.Balogh-K.Y.’18]

Assume that M ≥ 0. Then

E (r , t; 0, b;M) > 0, for all 0 ≤ b ≤ t, r ≤ e−b − e−t , t ∈ [0,∞),

K1(r , t;M) > 0 for all r ≤ 1− e−t , t ∈ [0,∞) .

If we assume that M > 1, then

K0(r , t;M) > 0 for all r ≤ 1− e−t and for all t > lnM

M − 1.


The Kernel K0(r , t;M)

K0(r , t;M) := −[∂

∂bE (r , t; 0, b;M)

]b=0

= 4−MetM((1 + e−t)2 − r2

)− 12 +M 1

(1− e−t)2 − r2

×

[(e−t − 1 + M(e−2t − 1− r2)

)F(1

2−M,

1

2−M; 1;

(1− e−t)2 − r2

(1 + e−t)2 − r2

)+(1− e−2t + r2

)(1

2+ M

)F(− 1

2−M,

1

2−M; 1;

(1− e−t)2 − r2

(1 + e−t)2 − r2

)]



The graph of the K0(r , t; 34 ) shows that the K0 changes a sign.

Figure: The graph of K0

(z , t, 3

4

), t ∈ (0, 3) and t ∈ (0, 15)

The graph of the K0(r , t; 16 ) shows that the K0 does not change a

sign.



For M = 1/2 the kernels are

E

(r , t; 0, b;

1

2

)=

1

2e

12 (b+t), K0

(r , t;

1

2

)= −1

4e

12 t , K1

(r , t;

1

2

)=

1

2e

12 t

The graph of the K0(r , t; 16 ) shows that the K0 does not change a sign.

Conjecture

Assume that M ∈ [0, 1/2]. Then

K0(r , t;M) ≤ 0 for all r ≤ 1− e−t and for all t > 0 .


Application: Representation Theorem

The Klein-Gordon equation with complex mass, M ∈ C

utt − a2(t)A(x , ∂x)u −M2u = f , t ∈ (0,T ), x ∈ Ω .

Theorem [K.Y.’15]

For f ∈ C∞(Ω× [0,T ]), 0 < T ≤ ∞, and ϕ0, ϕ1 ∈ C∞0 (Ω), let thefunction wf (x , t; b) be a solution to the problem

wtt − A(x , ∂x)w = 0 , t ∈ [0, 1− e−T ], x ∈ Ω , (4)

w(x , 0; b) = f (x , b) , wt(x , 0; b) = 0 , b ∈ [0,T ], x ∈ Ω ,

and vϕ = vϕ(x , s) be a solution of the problem

wtt − A(x , ∂x)w = 0, t ∈ [0, 1− e−T ], x ∈ Ω ,

w(x , 0) = ϕ(x), wt(x , 0) = 0 , x ∈ Ω .


Theorem (continuation) [K.Y.’15]

Then the function u = u(x , t) defined by

u(x , t) =

∫ t

0db

∫ φ(t)−φ(b)

0wf (x , r ; b)E (r , t; 0, b;M) dr

+et2wϕ0(x , φ(t)) +

∫ φ(t)

0wϕ0(x , s)K0(s, t;M) ds

+

∫ φ(t)

0wϕ1(x , s)K1(s, t;M) ds, x ∈ Ω ⊆ Rn, t ∈ [0,T ] ,

where φ(t) := 1− e−t , solves the problem

utt − e−2tA(x , ∂x)u −M2u = f , t ∈ [0,T ], x ∈ Ω ,

u(x , 0) = ϕ0(x) , ut(x , 0) = ϕ1(x), x ∈ Ω .

E , K0 and K1 have been defined in (2), (2) and (3), respectively.

[0, 1− e−T ] ⊆ [0, 1], which appears in (4), reflects the fact thatde Sitter model possesses the horizon.


Application: estimates for eq. in de Sitter space-time

Theorem [Brenner’79]

Let A = A(x ,D) be a second order negative elliptic differential operator with realC∞-coefficients such that A(x ,D) = A(∞,D) for |x | large enough. Letu(t) = G0(t)g0 + G1(t)g1 be the solution of

∂2t u − A(x ,D)u = 0, x ∈ Rn, t ≥ 0,

u(x , 0) = g0(x), ut(x , 0) = g1(x), x ∈ Rn .

Then for each T <∞ there is a constant C = C (T ) such that if(n + 1)δ ≤ ν + s − s ′,

‖Gν(t)g‖Bs′,qp′≤ C (T )tν+s−s′−2nδ‖g‖Bs,q

p, 0 < t ≤ T .

Here s, s ′ ≥ 0, q ≥ 1, 1 ≤ p ≤ 2, 1/p + 1/p′ = 1, and δ = 1/p − 1/2.



Theorem [A.Galstian-K.Y.’17]

Let u(t) = G0,dS(t)ϕ0 + G1,dS(t)ϕ1 be the solution of the Cauchy problem

utt − e−2tA(x , ∂x)u −M2u = 0, t ∈ [0,T ], x ∈ Ω ,

u(x , 0) = ϕ0(x) , ut(x , 0) = ϕ1(x), x ∈ Ω .

Then the operators G0,dS(t) and G1,dS(t) satisfy the following estimates

‖G0,dS(t)ϕ0‖Bs′,qp′

≤ CM(1 + t)1−sgnM(1− e−t)s−s′−2nδe

t2 ‖ϕ0‖Bs,q

p,

‖G1,dS(t)ϕ1‖Bs′,qp′

≤ CM(1 + t)1−sgnM(1− e−t)1+s−s′−2nδ‖ϕ1‖Bs,qp,

for all t ∈ (0,∞), provided that (n + 1)δ ≤ s − s ′, 1 < p ≤ 2, 1p + 1

p′ = 1,

s − s ′ − 2nδ > −1, and δ = 1/p − 1/2.



Theorem [A.Galstian-K.Y.’17]

Let u = u(x , t) be solution of the Cauchy problem

utt − e−2tA(x , ∂x)u −M2u = f , x ∈ Rn , t > 0,

u(x , 0) = 0 , ut(x , 0) = 0, x ∈ Rn .

Then for n ≥ 2 one has the following estimate

‖u(x , t)‖Bs′,qp′≤

CM

∫ t

0db ‖f (x , b)‖Bs,q

peb(e−b − e−t

)1+s−s′−2nδ(1 + t − b)1−sgnM db

for all t > 0, provided that 1 < p ≤ 2, 1p + 1

p′ = 1, s − s ′ − 2nδ > −1,

s, s ′ ≥ 0, (n + 1)δ ≤ s − s ′, and δ = 1/p − 1/2.


Knot PointsFor m ∈ [0, n/2] we have M =

√n2

4 −m2 and

E (z , t; 0, b;M) = (4e−b−t)−M(

(e−t + e−b)2 − z2)− 1

2+M

×F(1

2−M,

1

2−M; 1;

(e−b − e−t)2 − z2

(e−b + e−t)2 − z2

).

Let 1

2−M = −k, k = 0, 1, . . . ,

[n − 1

2

],

then

F (−k ,−k ; 1; z) =k∑

j=0

(k(k − 1) · · · (k + 1− j)

k!

)2

z j .

Definition. We call m2 = n2

4 −(

12 + k

)2, k = 0, 1, . . . ,

[n−1

2

]the

knot points for the physical mass m.

For n = 1, 2 there is only one knot point.For n = 3 there are two knot points: m = 0,

√2.


Application: Huygens’ Principle

The knot points are linked to the Huygens’ principle.

Recall that a hyperbolic equation is said to satisfy Huygens principleif the solution vanishes at all points which cannot be reached fromthe support of initial data by a null geodesic.

Theorem [K.Y.’13]

The right knot point m =√n2 − 1/2 is the only value of the physical

mass m, such that the equation

Φtt + nΦt − e−2t∆Φ + m2Φ = 0,

obeys the Huygens’ principle, whenever the wave equation in theMinkowski space-time does, that is n ≥ 3 is an odd number.

If n = 3, then m =√

2

What fundamental particle has m =√

2?


Definition [K.Y.’13]

We say that the equation obeys the incomplete Huygens’ principle withrespect to the first initial datum, if it obeys the Huygens’ principleprovided that the second datum vanishes, ϕ1 = 0.

If equation obeys the Huygens’ principle, then it obeys also theincomplete Huygens’ principle with respect to the first initial datum.

The string equation (n = 1) obeys the incomplete Huygens’ principle.

Theorem [K.Y.’13]

Suppose that equation

Φtt + nΦt − e−2t∆Φ + m2Φ = 0

does not obey the Huygens’ principle. Then, it obeys the incompleteHuygens’ principle with respect to the first initial datum, if and only if theequation is massless, m = 0 (the left knot point), and either n = 1 orn = 3.


Corollary

Assume that the equations

Φtt + nΦt − e−2t∆Φ + m21Φ = 0,

Φtt + nΦt − e−2t∆Φ + m22Φ = 0

describe two fields withm1 6= m2.

Then they obey the incomplete Huygens’ principle if and only if thedimension n of the spatial variable x is 3 and m1 = 0, m2 =

√2.

The case of n = 3: there are only two knot points m = 0,√

2. Inquantum field theory they are the endpoints of the interval (0,

√2)

known as the so-called Higuchi bound.

Higuchi bound (0,√

2) is the forbidden mass range for spin-2 fieldtheory in de Sitter space-time because negative probability appears ifone introduce interactions. (Atsushi Higuchi: Nuclear Phys. B (1987))


Paul Ehrenfest’s Question’ 1917:

In that way does it become manifest in the fundamental laws of physicsthat space has three dimensions?, KNAW, Proceedings Royal Acad.Amsterdam, Vol. XX, I, 1918, Amsterdam, 1918, pp. 200-209

Paul Ehrenfest in that article addressed the question: “Why has ourspace just three dimensions?” or in other words: “By which singularcharacteristics do geometries and physics in R3 distinguish themselvesfrom those in the other Rn’s?”.

He discussed physical laws that critically depend on the number ofspace dimensions:? Newton’s Law of gravitation and planetary motion;? Electro-magnetic field;? The wave equation and Huygens’ principle.

This question was raised on May, 1917.


Application: Self-interacting scalar field in the de Sitterspace-time. Semilinear equation

Small data blow up: K. Y. ’09

Small data global solution: K. Y. ’12

Energy Spaces: D. Baskin ’13

Energy Spaces: M. Nakamura ’14 ,

Energy Spaces: A. Galstian & K.Y. ’15

Energy Spaces: P. Hintz, A. Vasy ’15

Life span for massless equation: A. Galstian ’15

Numerical results: M. Yazici and S. Sengul ’16

More applications A. Galstian, T. Kinoshita ’16

Energy Spaces: M. Ebert, W. N. Do Nascimento ’17

Maximum principle: A. Balogh, K. Y. ’17

Energy Spaces: M. Ebert & M. Reissig ’18

Higgs Boson Equation: A. Balogh, J. Banda, K. Y. ’18

L∞ decay estimates M. Yazici ’18


Application: The linear and semi-linear generalized Tricomiequation. Einstein-de Sitter model.

Fundamental solution K. Y. ’04

Small data global solution: K. Y. ’06

Self-similar solutions K.Y.’07

Einstein de Sitter model. A. Galstian, T. Kinoshita, K.Y. ’10

A.Palmieri & M. Reissig ’17

Small data global solution: Daoyin He, I.Witt, Huicheng Yin ’17

Z.Ruan, I.Witt, Huicheng Yin’18



Let H(s)(Rn) be a Sobolev space with the norm ‖ · ‖H(s)(Rn).

To estimate the nonlinear term F (u) we use

Condition (L). The function F is said to be Lipschitz continuous in uwith exponent α in the space H(s)(Rn) if there is C ≥ 0 such that

‖F (u)− F (v)‖H(s)(Rn) ≤ C‖u − v‖H(s)(Rn)

(‖u‖αH(s)(Rn) + ‖v‖αH(s)(Rn)

)for all u, v ∈ H(s)(Rn).

Define the complete metric space

X (R, s, γ) :=

Φ ∈ C ([0,∞);H(s)(Rn)) |

‖ Φ ‖X := supt∈[0,∞)

eγt ‖ Φ(t) ‖H(s)(Rn)≤ R

with the metric

d(Φ1,Φ2) := supt∈[0,∞)

eγt ‖ Φ1(t)− Φ2(t) ‖H(s)(Rn) .



Denote the principal square root M := (n2/4−m2)12 .

Theorem [K.Y., ArXiv’2017]

Assume that the nonlinear term F (x ,Φ) is a Lipschitz continuous inH(s)(Rn), s > n/2 ≥ 1, F (x , 0) ≡ 0, and α > 0.(i) Assume that 0 < <M < 1/2. Then, there exists ε0 > 0 such that, forevery ψ0, ψ1 ∈ H(s)(Rn), such that

‖ψ0‖H(s)(Rn) + ‖ψ1‖H(s)(Rn) ≤ ε, ε < ε0 , (5)

there exists Φ ∈ C ([0,∞);H(s)(Rn)) of the Cauchy problem

ψtt + nψt − e−2tA(x , ∂x)ψ + m2ψ = F (x , ψ) ,

ψ(x , 0) = ψ0(x) , ψt(x , 0) = ψ1(x) .

The solution ψ(x , t) belongs to the space X (2ε, s, n−12 ).



Theorem (continuation) [K.Y., ArXiv’2017]

(ii) Assume 1/2 ≤ <M < n/2 and γ ∈ (0, 1α+1 (n2 −<M)). Then there

exists ε0 > 0 such that for every ψ0, ψ1 ∈ H(s)(Rn), such that‖ψ0‖H(s)(Rn) + ‖ψ1‖H(s)(Rn) ≤ ε < ε0, there exists a solution

ψ ∈ X (2ε, s, γ).(iii) If <M > n/2, then the lifespan Tls can be estimated

Tls ≥ − 1

<M − n2

ln(‖ψ0‖H(s)(Rn) + ‖ψ1‖H(s)(Rn)

)− C (m, n, α)

with some constant C (m, n, α).

The theorem covers the case of m ∈ (√n2 − 1/2, n/2).

For

F (Φ) = ±|Φ|αΦ or F (Φ) = ±|Φ|α+1 or F (Φ) = λΦ3,

the small data Cauchy problem is globally solvable for every α > 0.Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-timesMicrolocal and Global Analysis, Interactions with Geometry Colloquium in honor of Professor Schulze’s 75th birthday University of Potsdam, March 4-8, 2019 40 / 45

Higgs boson equation in the de Sitter space-time

Open problem: Existence of global in time solution for small data for

ψtt + nψt − e−2t∆ψ = µ2ψ − F (ψ) , t ∈ [0,∞), x ∈ Rn

ψ(x , 0) = ψ0(x) , ψt(x , 0) = ψ1(x) , x ∈ Rn .

Case of µ > 0 and F (ψ) = λ|ψ|2ψ with λ > 0 is the Higgs boson equation.

What is known: Denote M := (n2/4 + µ2)12 .

If µ > 0 and F (Φ) is a Lipschitz continuous, then the lifespan Tls ofthe solution can be estimated from below as follows

Tls ≥ − 1

<M − n2

ln(‖ϕ0‖H(s)(Rn) + ‖ϕ1‖H(s)(Rn)

)− C (m, n, α)

with some constant C (m, n, α). (K.Y.’17)

If µ > 0 and F (ψ) = −|ψ|p, and p > 1, then there is a blowing upsolution for arbitrary small initial data. (K.Y.’09).


Thank you for your time!


Documents

A novel integral transform approach to solving partial ...personal.psu.edu/~tuk14/Potsdam2019/Slides/Yagdjian.pdf · Semilinear equation in the de Sitter space-time Karen Yagdjian