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

Karen Yagdjian

University of Texas Rio Grande Valley

Microlocal and Global Analysis, Interactions with Geometry Colloquium 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-times Microlocal and Global Analysis, Interactions with

The Integral Transform: purpose and structure

The purpose: target problem (partial differential equations) The structure:

The function subject to transformation The kernel function

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

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

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

The Target Equation:

∂2

t u − a2(t)A(x, ∂x)u − M2u = f ,

t ∈ (0, T), x ∈ Ω ⊆ Rn . Here M ∈ C and A(x, ∂x) =

• |α|≤m

aα(x)∂α

x ,

∂α

x = ∂α1 x1 · · · ∂αn xn ,

|α| = α1 + . . . + αn The Goal : Explicit representation for the solutions of that equation The Tool : The new integral transform

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Why this equation?

∂2

t u − a2(t)A(x, ∂x)u − M2u = f ,

t ∈ (0, T), x ∈ Ω ⊆ Rn . Here M ∈ C and A(x, ∂x) =

• |α|≤m

aα(x)∂α

x

Equations of Gas Dynamics Equations of Physics in Expanding Universe

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Gas Dynamics

Tricomi equation (Chaplygin’1909, Tricomi’1923): ∂2

t u − t∆u = f ,

t ∈ R, x ∈ Ω ⊆ Rn . The equation representing in hodograph variables a steady transonic flow (flight) of ideal gas. The small disturbance equations for the perturbation velocity potential of a near sonic uniform flow of dense gases (Kluwick, Tarkenton, Cramer’93) ∂2

t u − t3∆u = f ,

t ∈ R, x ∈ Ω ⊆ Rn . Here ∆u = ∂2u ∂x2

1

+ ∂2u ∂x2

2

+ · · · + ∂2u ∂x2

n

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

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

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

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 e2Ht e2Ht e2Ht     c is the speed of light, H is the Hubble constant. We set c = 1 and H = 1. ds2 = −dt2 + a2

sc(t)dσ2, where asc(t) = eHt is the scale factor.

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

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

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

The Covariant Wave and Klein-Gordon Equations

The covariant wave equation 1

• |g(x)|

∂ ∂xi

• |g(x)|gik(x) ∂ψ

∂xk

• =

f . The covariant Klein-Gordon Equation 1

• |g(x)|

∂ ∂xi

• |g(x)|gik(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.

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Waves in the Universe (Cosmological Models)

The (non-covariant) wave equation in the radiation dominated universe: utt − t−1A(x, ∂x)u = f . The wave equation in the Einstein-de Sitter space-time (matter dominated universe). The covariant d’Alambert’s operator, after the change 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 spatial geometry, respectively.

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

The Klein-Gordon Equation in Expanding Universe

The 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

• | det σ(x)|

n

• i,j=1

∂ ∂xi

• | det σ(x)|σij(x) ∂

∂xj ψ

• If u = ent/2ψ, then

utt − e−2tA(x, ∂x)u − M2u = f , where M2 = n2

4 − m2 is curved (or effective) mass. This example

includes equations in the metric with hyperbolic, flat or spherical spatial geometry.

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

The Klein-Gordon Equation of Self-Interacting Field in Expanding Universe

In the spatially flat de Sitter universe the equation for the scalar field with mass m and potential function V is 1 c2 φtt + 1 c2 nHφt − e−2Ht∆φ + m2c2 h2 φ = 1 c2 V ′(φ) . Here x ∈ Rn, t ∈ R, and ∆ is the Laplace operator, ∆ := n

j=1 ∂2 ∂x2

j ,

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.

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

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)∂α x .

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 − → 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 .

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

The New Integral Transform

The integral operator K : w − → u is u(x, t) = K[w](x, t) := t db |φ(t)−φ(b)| K(t; r, b; M)w(x, r; b)dr, x ∈ Ω, t ∈ (0, T). Here φ(t) = t a(τ) dτ is a distance function produced by a = a(t), M ∈ C is a constant. Integral transform is applicable to the distributions and fundamental solutions as well. In fact, u = u(x, t) takes initial values u(x, 0) = 0, ut(x, 0) = 0, x ∈ Ω .

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

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 db t−b wf (x, r; b) dr , (1) is the solution of the Cauchy problem

• utt − ∆u = f (x, t),

in Rn+1 u(x, 0) = 0, ut(x, 0) = 0 in Rn , if wf = wf (x; t; b) solves

• wtt − ∆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 the inner integral is generated by the propagation phenomena with the speed =1. In fact, t − b is a distance function.

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Our third observation is that the solution operator G : f − → u can be regarded as a composition of two operators G = K ◦ WE. The first one WE : f − → w is a Fourier Integral Operator, which is a solution operator of the Cauchy problem for wave equation. The second operator K : w − → u is the integral operator (1).

Figure: Case of A(x, ∂x) = ∆

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Figure: Case of A(x, ∂x) = ∆

We introduce the distance function φ(t) and provide the integral

• perator with the kernel

u(x, t) = t db |φ(t)−φ(b)| K(t; r, b; M)w(x, r; b)dr, x ∈ Ω, t ∈ (0, T). This operator generates solutions of different well-known equations with x-independent coefficients. We have generated a class of operators for which we have obtained explicit representation formulas for the solutions of the equations with a(t) = tℓ, ℓ ∈ R, a(t) = e±t

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Figure: (b) Case of general A(x, ∂x)

By varying the first mapping, we extend the class of the equations for which we can generate the solutions. More precisely, consider the diagram (b), where w = wA,ϕ(x, t; b) is a solution 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 applying integral transform, we can generate solutions of new equations. Thus, GA = K ◦ EEA. The new class of equations contains operators with x-dependent coefficients.

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

The Kernel. The Klein-Gordon Equation in de Sitter space-time

For given x0 ∈ Rn, t0 ∈ R define a chronological future D+(x0, t0) and a 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

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

The Kernel. The Klein-Gordon Equation in de Sitter space-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)2M− 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 a chronological 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.

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

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 > ln M M − 1 .

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

The Kernel K0(r, t; M)

K0(r, t; M) := − ∂ ∂b E(r, t; 0, b; M)

• b=0

= 4−MetM (1 + e−t)2 − r 2− 1

2 +M

1 (1 − e−t)2 − r 2 ×

• e−t − 1 + M(e−2t − 1 − r 2)
• F

1 2 − M, 1 2 − M; 1; (1 − e−t)2 − r 2 (1 + e−t)2 − r 2

• +
• 1 − e−2t + r 21

2 + M

• F
• − 1

2 − M, 1 2 − M; 1; (1 − e−t)2 − r 2 (1 + e−t)2 − r 2

• Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times

Microlocal and Global Analysis, Interactions with

The Kernel K0(r, t; M)

The graph of the K0(r, t; 3

4) 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; 1

6) shows that the K0 does not change a

sign.

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

The Kernel K0(r, t; M)

For M = 1/2 the kernels are E

• r, t; 0, b; 1

2

• = 1

2e

1 2 (b+t), K0

• r, t; 1

2

• = −1

4e

1 2 t, K1

• r, t; 1

2

• = 1

2e

1 2 t

The graph of the K0(r, t; 1

6) 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 .

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

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 the

function 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 .

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Theorem (continuation) [K.Y.’15]

Then the function u = u(x, t) defined by u(x, t) = t db φ(t)−φ(b) wf (x, r; b)E(r, t; 0, b; M) dr +e

t 2 wϕ0(x, φ(t)) +

φ(t) wϕ0(x, s)K0(s, t; M) ds + φ(t) wϕ1(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 that de Sitter model possesses the horizon.

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

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 real C ∞-coefficients such that A(x, D) = A(∞, D) for |x| large enough. Let u(t) = G0(t)g0 + G1(t)g1 be the solution of ∂2

t 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)gBs′,q

p′

≤ C(T)tν+s−s′−2nδgBs,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.

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

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

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)ϕ0Bs′,q

p′

≤ CM(1 + t)1−sgnM(1 − e−t)s−s′−2nδe

t 2 ϕ0Bs,q p ,

G1,dS(t)ϕ1Bs′,q

p′

≤ CM(1 + t)1−sgnM(1 − e−t)1+s−s′−2nδϕ1Bs,q

p ,

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

p + 1 p′ = 1,

s − s′ − 2nδ > −1, and δ = 1/p − 1/2.

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

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

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′,q

p′

≤ CM t db f (x, b)Bs,q

p eb

e−b − e−t1+s−s′−2nδ (1 + t − b)1−sgnM db for all t > 0, provided that 1 < p ≤ 2,

1 p + 1 p′ = 1, s − s′ − 2nδ > −1,

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

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Knot Points

For 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 −

1

2 + 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.

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Application: Huygens’ Principle

The knot points are linked to the Huygens’ principle. Recall that a hyperbolic equation is said to satisfy Huygens principle if the solution vanishes at all points which cannot be reached from the 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,

• beys the Huygens’ principle, whenever the wave equation in the

Minkowski space-time does, that is n ≥ 3 is an odd number. If n = 3, then m = √ 2 What fundamental particle has m = √ 2?

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Definition [K.Y.’13]

We say that the equation obeys the incomplete Huygens’ principle with respect to the first initial datum, if it obeys the Huygens’ principle provided that the second datum vanishes, ϕ1 = 0. If equation obeys the Huygens’ principle, then it obeys also the incomplete 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 incomplete Huygens’ principle with respect to the first initial datum, if and only if the equation is massless, m = 0 (the left knot point), and either n = 1 or n = 3.

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Corollary

Assume that the equations Φtt + nΦt − e−2t∆Φ + m2

1Φ = 0,

Φtt + nΦt − e−2t∆Φ + m2

2Φ = 0

describe two fields with m1 = m2. Then they obey the incomplete Huygens’ principle if and only if the dimension 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. In

quantum 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 field theory in de Sitter space-time because negative probability appears if

• ne introduce interactions. (Atsushi Higuchi: Nuclear Phys. B (1987))

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Paul Ehrenfest’s Question’ 1917:

In that way does it become manifest in the fundamental laws of physics that 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 our space just three dimensions?” or in other words: “By which singular characteristics do geometries and physics in R3 distinguish themselves from those in the other Rn’s?”. He discussed physical laws that critically depend on the number of space 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.

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Application: Self-interacting scalar field in the de Sitter space-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. Seng¨ ul ’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

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Application: The linear and semi-linear generalized Tricomi

• equation. 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

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Semilinear equation in the de Sitter space-time

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 u with exponent α in the space H(s)(Rn) if there is C ≥ 0 such that F(u) − F(v)H(s)(Rn) ≤ Cu − vH(s)(Rn)

• uα

H(s)(Rn) + vα H(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:= sup

t∈[0,∞)

eγt Φ(t) H(s)(Rn) ≤ R

• with the metric

d(Φ1, Φ2) := sup

t∈[0,∞)

eγt Φ1(t) − Φ2(t) H(s)(Rn) .

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Semilinear equation in the de Sitter space-time

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

1 2 .

Theorem [K.Y., ArXiv’2017]

Assume that the nonlinear term F(x, Φ) is a Lipschitz continuous in H(s)(Rn), s > n/2 ≥ 1, F(x, 0) ≡ 0, and α > 0. (i) Assume that 0 < ℜM < 1/2. Then, there exists ε0 > 0 such that, for every ψ0, ψ1 ∈ H(s)(Rn), such that ψ0H(s)(Rn) + ψ1H(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−1

2 ).

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Semilinear equation in the de Sitter space-time

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

(ii) Assume 1/2 ≤ ℜM < n/2 and γ ∈ (0,

1 α+1( n 2 − ℜM)). Then there

exists ε0 > 0 such that for every ψ0, ψ1 ∈ H(s)(Rn), such that ψ0H(s)(Rn) + ψ1H(s)(Rn) ≤ ε < ε0, there exists a solution ψ ∈ X(2ε, s, γ). (iii) If ℜM > n/2, then the lifespan Tls can be estimated Tls ≥ − 1 ℜM − n

2

ln

• ψ0H(s)(Rn) + ψ1H(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

• r

F(Φ) = ±|Φ|α+1

• r

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-times Microlocal and Global Analysis, Interactions with

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)

1 2 .

If µ > 0 and F(Φ) is a Lipschitz continuous, then the lifespan Tls of the solution can be estimated from below as follows Tls ≥ − 1 ℜM − n

2

ln

• ϕ0H(s)(Rn) + ϕ1H(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 up solution for arbitrary small initial data. (K.Y.’09).

Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

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Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with