Search

We consider the defocusing, energy subcritical wave equation ∂ t 2 u − Δ u = − | u | p − 1 u in 4 to 6 dimensional spaces with radial initial data. We define w = r (d − 1) / 2 u , reduce the equation above to one-dimensional equation of w and apply method of characteristic lines. This gives scattering of solutions outside any given light cone as long as the energy is finite. The scattering in the whole space can also be proved if we assume the energy decays at a certain rate as x → + ∞. This generalizes the 3-dimensional results in Shen [32] to higher dimensions. [ABSTRACT FROM AUTHOR]

We consider the scattering theory for the defocusing energy subcritical wave equations with an inverse square potential. By employing the energy flux method we establish energy flux estimates on the light cone. Then by the characteristic line method and radiation theorem, we show that the radial finite-energy solutions scatter to free waves outside of light cones. Using Morawetz estimates we then obtain the scattering theory for radial solutions with finite weighted energy initial data. [ABSTRACT FROM AUTHOR]

The topic of this paper is a semi-linear, energy subcritical, defocusing wave equation ∂t2 u − Δ u = − |u|p−1 u in the 3-dimensional space (3 ≤ p < 5) whose initial data are radial and come with a finite energy. We split the energy into inward and outward energies, then apply the energy flux formula to obtain the following asymptotic distribution of energy: Unless the solution scatters, its energy can be divided into two parts: ''scattering energy'', which concentrates around the light cone |x| = |t| and moves to infinity at the light speed, and ''retarded energy'', which is at a distance of at least |t|β behind when |t| is large. Here β is an arbitrary constant smaller than β0(p) = 2(p-2)/p+1. A combination of this property with a more detailed version of the classic Morawetz estimate gives a scattering result under a weaker assumption on initial data (u0,u1) than previously known results. More precisely, we assume ∫R3 (|x|κ + 1) (½|∇u0|2 + ½|u1|2 + 1/p+1 |u|p+1) dx < + ∞. Here β > β0(p) = 1−β0(p) = 5−p/p+1 is a constant. This condition is so weak that the initial data may be outside the critical Sobolev space of this equation. This phenomenon is not covered by previously known scattering theory, as far as the author knows. [ABSTRACT FROM AUTHOR]

In this work we consider an energy subcritical semi-linear wave equation (3 < p < 5) { ∂ t 2 u − Δ u = ϕ (x) | u | p − 1 u , (x , t) ∈ R 3 × R ; u | t = 0 = u 0 ∈ H ˙ s p (R 3) ; ∂ t u | t = 0 = u 1 ∈ H ˙ s p − 1 (R 3) ; where s p = 3 / 2 − 2 / (p − 1) and the function ϕ : R 3 → − 1 , 1 is a radial continuous function with a limit at infinity. We prove that unless the elliptic equation − Δ W = ϕ (x) | W | p − 1 W has a nonzero radial solution W ∈ C 2 (R 3) ∩ H ˙ s p (R 3) , any radial solution u with a finite uniform upper bound on the critical Sobolev norm ‖ (u (⋅ , t) , ∂ t u (⋅ , t)) ‖ H ˙ s p × H ˙ s p (R 3) for all t in the maximal lifespan must be a global solution in time and scatter. [ABSTRACT FROM AUTHOR]

In this paper, we consider a semi-linear, energy sub-critical, defocusing wave equationin the three-dimensional space withp∈[3,5). We prove that if initial dataare radial so that, wheredμ = (|x|+1)1+2𝜀with𝜀>0, then the corresponding solutionumust exist for all timet∈ℝ and scatter. The key ingredients of the proof include a transformationTso thatv = Tusolves the equationwith a finite energy, and a Morawetz-type estimate regarding a solutionvas above. [ABSTRACT FROM AUTHOR]

In this paper we consider an energy critical wave equation ( 3 ≤ d ≤ 5 , ζ = ± 1 ) ∂ t 2 u − Δ u = ζ ϕ ( x ) | u | 4 / ( d − 2 ) u , ( x , t ) ∈ R d × R with initial data ( u , ∂ t u ) | t = 0 = ( u 0 , u 1 ) ∈ H ˙ 1 × L 2 ( R d ) . Here ϕ ∈ C ( R d ; ( 0 , 1 ] ) converges as | x | → ∞ and satisfies certain technical conditions. We generalize Kenig and Merle's results on the Cauchy problem of the equation ∂ t 2 u − Δ u = | u | 4 / ( d − 2 ) u . Following a similar compactness-rigidity argument we prove that any solution with a finite energy must scatter in the defocusing case ζ = − 1 . While in the focusing case ζ = 1 we give a criterion for global behaviour of the solutions, either scattering or finite-time blow-up when the energy is smaller than a certain threshold. As an application we give a similar criterion on the global behaviour of radial solutions to the focusing, energy critical shifted wave equation ∂ t 2 v − ( Δ H 3 + 1 ) v = | v | 4 v on the hyperbolic space H 3 . [ABSTRACT FROM AUTHOR]

The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation ∂ t 2 u − Δ u = − | u | p − 1 u in the 3-dimensional space with 3 ≤ p < 5. We generalize inward/outward energy theory and weighted Morawetz estimates for radial solutions to the non-radial case. As an application we show that if 3 < p < 5 and κ > 5 − p 2 , then the solution scatters as long as the initial data (u 0 , u 1) satisfy ∫ R 3 (| x | κ + 1) (1 2 | ∇ u 0 | 2 + 1 2 | u 1 | 2 + 1 p + 1 | u 0 | p + 1) d x < + ∞. If p = 3 , we can also prove the scattering result if initial data (u 0 , u 1) are contained in the critical Sobolev space and satisfy the inequality ∫ R 3 | x | (1 2 | ∇ u 0 | 2 + 1 2 | u 1 | 2 + 1 4 | u 0 | p + 1) d x < + ∞. These assumptions on the decay rate of initial data as | x | → ∞ are weaker than previously known scattering results in the non-radial case. [ABSTRACT FROM AUTHOR]

Ketamine has been reported to impair human cognitive function as a recreational drug of abuse. However, chronic effects of ketamine on central nervous system need to be further explored. We set out to establish chronic ketamine addiction models by giving mice a three or six month course of daily intraperitoneal injections of ketamine, then examined whether long-term ketamine administration induced cognition deficits and changed hippocampal post-synaptic protein expression in adult mice. Behavior tests results showed that mice exhibited dose- and time-dependent learning and memory deficits after long-term ketamine administration. Western blot results showed levels of GluA1, p-S845 and p-S831 proteins demonstrated significant decline with ketamine 60 mg/kg until six months administration paradigm. But levels of p-S845 and p-S831 proteins exhibited obvious increase with ketamine 60 mg/kg three months administration paradigm. NR1 protein levels significantly decrease with ketamine 60 mg/kg three and six months administration paradigm. Our results indicate that reduced expression levels and decreased phosphorylation levels of hippocampal post-synaptic membrane GluA1- containing AMPA receptors maybe involved in cognition impairment after long-term ketamine administration. These findings provide further evidence for the cognitive damage of chronic ketamine addiction as a recreational drug. [ABSTRACT FROM AUTHOR]