A scale-invariant Klein–Gordon model with time-dependent potential

Böhme, Christiane; Reissig, Michael

doi:10.1007/s11565-012-0153-9

Cited by 22 publications

(32 citation statements)

References 10 publications

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“…In this paper we prove results in the following cases: (1)The case

δ \leq 0

. In this case the mass term is predominant and not non‐effective in the sense of . (2)The case

δ \geq {(n + 1)}^{2}

. In this case the dissipation term is predominant and not non‐effective in the sense of . (3)The case

δ = 1

.…”

Section: Resultsmentioning

confidence: 63%

“…To reach this goal we will proceed the following strategy: (1)If

δ \leq 0,

in other words, if the mass is predominant and not non‐effective , then after the dissipative change of variables

v false(t, x false) = {(1 + t)}^{- \frac{μ_{1}}{2}} w false(t, x false)

the Cauchy problem becomes

\begin{matrix} true \\ left w_{t t} - normalΔ w + \frac{μ}{{(1 + t)}^{2}} w = 0, \\ left w (s, x) = {false(1 + s false)}^{\frac{μ_{1}}{2}} v_{0} (x), w_{t} (s, x) = {false(1 + s false)}^{\frac{μ_{1}}{2}} v_{1} (x) + 0false \frac{μ_{1}}{2} {false(1 + s false)}^{\frac{μ_{1}}{2} - 1} v_{0} (x), \end{matrix}

where

μ = \frac{μ_{1}}{2} - \frac{μ_{1}^{2}}{4} + μ_{2}^{2}

. Note that

δ = 1 - 4 μ \leq 0

if, and only if,

μ \geq \frac{1}{4}

, i.e., we are dealing with not non‐effective masses and we can use special function theory to derive estimates for solutions to this Cauchy problem, as it was done in a special case in . (2)If

δ \geq {(n + 1)}^{2}

, i.e., if the dissipation is predominant and not non‐effective , then after the change of variables …”

Section: Philosophy Of Our Approachmentioning

confidence: 99%

“…Performing the partial Fourier transform with respect to x in the Cauchy problem and denoting by

\hat{w} = \hat{w} false(t, ξ false)

the partial Fourier transform

F_{x \to ξ} false(w false) false(t, ξ false)

we obtain

{\overset{w}{true}}_{t t} + {| ξ |}^{2} \hat{w} + \frac{μ}{{false(1 + t false)}^{2}} \hat{w} = 0 .

The scale‐invariant property allows us to derive explicit representations of solutions in terms of known special functions. In this case we will perform, like in , a change of variables to reduce the ordinary differential equation to a confluent hypergeometric equation. If

τ = false| ξ false| (1 + t), \hat{w} false(t, ξ false) = τ^{ρ} \tilde{v} false(τ false),

then choosing

2 ρ = 1 + \sqrt{1 - 4 μ}

and putting

z = 2 i τ, v false(z false) = e^{i τ} \tilde{v} false(τ false),

we can reduce Equation to confluent hypergeometric equation

z v_{z z} + false(2 ρ - z false) v_{z} - ρ v = 0 .

Its fundamental solutions are called confluent hypergeometric functions and depend on the parameter ρ.…”

Section: Linear Estimatesmentioning

confidence: 99%

“…From the paper the solution of can be represented by

\hat{w} false(t, ξ false) = H_{1, 0} false(t, s, ξ false) \hat{w} false(s, ξ false) + H_{2, 0} false(t, s, ξ false) {\overset{w}{true}}_{t} false(s, ξ false),

and the derivative of

\hat{w} = \hat{w} false(t, ξ false)

with respect to t can be represented by

{\overset{w}{true}}_{t} false(t, ξ false) = H_{1, 1} false(t, s, ξ false) \hat{w} false(s, ξ false) + H_{2, 1} false(t, s, ξ false) {\overset{w}{true}}_{t} false(s, ξ false),

where

H_{k, 1} = \partial_{t} H_{k, 0} false(t, s, ξ false),

k = 1, 2

. These fundamental solutions satisfy

H_{k, ℓ} false(s, s, ξ false) = false(δ_{k, ℓ + 1} false), k = 1, 2

ℓ = 0, 1

, and are given by the next lemma.…”

Section: Linear Estimatesmentioning

confidence: 99%

See 3 more Smart Citations

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation

Nascimento

Palmieri

Reissig

2016

Mathematische Nachrichten

Self Cite

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“…In this paper we prove results in the following cases: (1)The case

δ \leq 0

. In this case the mass term is predominant and not non‐effective in the sense of . (2)The case

δ \geq {(n + 1)}^{2}

. In this case the dissipation term is predominant and not non‐effective in the sense of . (3)The case

δ = 1

.…”

Section: Resultsmentioning

confidence: 63%

“…To reach this goal we will proceed the following strategy: (1)If

δ \leq 0,

in other words, if the mass is predominant and not non‐effective , then after the dissipative change of variables

v false(t, x false) = {(1 + t)}^{- \frac{μ_{1}}{2}} w false(t, x false)

the Cauchy problem becomes

\begin{matrix} true \\ left w_{t t} - normalΔ w + \frac{μ}{{(1 + t)}^{2}} w = 0, \\ left w (s, x) = {false(1 + s false)}^{\frac{μ_{1}}{2}} v_{0} (x), w_{t} (s, x) = {false(1 + s false)}^{\frac{μ_{1}}{2}} v_{1} (x) + 0false \frac{μ_{1}}{2} {false(1 + s false)}^{\frac{μ_{1}}{2} - 1} v_{0} (x), \end{matrix}

where

μ = \frac{μ_{1}}{2} - \frac{μ_{1}^{2}}{4} + μ_{2}^{2}

. Note that

δ = 1 - 4 μ \leq 0

if, and only if,

μ \geq \frac{1}{4}

, i.e., we are dealing with not non‐effective masses and we can use special function theory to derive estimates for solutions to this Cauchy problem, as it was done in a special case in . (2)If

δ \geq {(n + 1)}^{2}

, i.e., if the dissipation is predominant and not non‐effective , then after the change of variables …”

Section: Philosophy Of Our Approachmentioning

confidence: 99%

“…Performing the partial Fourier transform with respect to x in the Cauchy problem and denoting by

\hat{w} = \hat{w} false(t, ξ false)

the partial Fourier transform

F_{x \to ξ} false(w false) false(t, ξ false)

we obtain

{\overset{w}{true}}_{t t} + {| ξ |}^{2} \hat{w} + \frac{μ}{{false(1 + t false)}^{2}} \hat{w} = 0 .

τ = false| ξ false| (1 + t), \hat{w} false(t, ξ false) = τ^{ρ} \tilde{v} false(τ false),

then choosing

2 ρ = 1 + \sqrt{1 - 4 μ}

and putting

z = 2 i τ, v false(z false) = e^{i τ} \tilde{v} false(τ false),

we can reduce Equation to confluent hypergeometric equation

z v_{z z} + false(2 ρ - z false) v_{z} - ρ v = 0 .

Its fundamental solutions are called confluent hypergeometric functions and depend on the parameter ρ.…”

Section: Linear Estimatesmentioning

confidence: 99%

“…From the paper the solution of can be represented by

\hat{w} false(t, ξ false) = H_{1, 0} false(t, s, ξ false) \hat{w} false(s, ξ false) + H_{2, 0} false(t, s, ξ false) {\overset{w}{true}}_{t} false(s, ξ false),

and the derivative of

\hat{w} = \hat{w} false(t, ξ false)

with respect to t can be represented by

{\overset{w}{true}}_{t} false(t, ξ false) = H_{1, 1} false(t, s, ξ false) \hat{w} false(s, ξ false) + H_{2, 1} false(t, s, ξ false) {\overset{w}{true}}_{t} false(s, ξ false),

where

H_{k, 1} = \partial_{t} H_{k, 0} false(t, s, ξ false),

k = 1, 2

. These fundamental solutions satisfy

H_{k, ℓ} false(s, s, ξ false) = false(δ_{k, ℓ + 1} false), k = 1, 2

ℓ = 0, 1

, and are given by the next lemma.…”

Section: Linear Estimatesmentioning

confidence: 99%

See 2 more Smart Citations

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation

Nascimento

Palmieri

Reissig

2016

Mathematische Nachrichten

Self Cite

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show abstract

“…for some C ϵ > 0, where u(t, x) is the solution to the Cauchy problem (14) with initial data (u 0 , u 1 ) ∈ F ϵ , and v(t, x) is the solution to the Cauchy problem (23) with initial data (v 0 , v 1 ) = W ϵ + (u 0 , u 1 ).…”

Section: Corollarymentioning

confidence: 99%

A class of dissipative wave equations with time-dependent speed and damping

D’Abbicco

Ebert

2013

Journal of Mathematical Analysis and Applications

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We study the long time behavior of the energy for wave-type equations with time-dependent speed and damping: utt-λ(t)2δu+b(t)ut=0. We investigate the interaction between the speed of propagationλ (t) and the damping coefficient. b(t), showing how to describe the dissipative effect on the energy. We study a class of dissipations for which the equation keeps its hyperbolic structure and properties. © 2012 Elsevier Ltd

show abstract

The threshold of effective damping for semilinear wave equations

D’Abbicco

2014

Math. Meth. Appl. Sci.

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A scale-invariant Klein–Gordon model with time-dependent potential

Cited by 22 publications

References 10 publications

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation

A class of dissipative wave equations with time-dependent speed and damping

The threshold of effective damping for semilinear wave equations

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