论文标题
关于Zakharov和Klein-Gordon Zakharov Systems的衰减问题
On the decay problem for the Zakharov and Klein-Gordon Zakharov systems in one dimension
论文作者
论文摘要
我们对解决标量Zakharov系统的长期渐近行为感兴趣\ [I U_ {t} +ΔU= NU，\] \ [N_ {TT} - ΔN=δ| n_ {tt} - Δn=δ| u |^2 \] 在空间的一个维度中。对于这两个系统，我们给出了两个结果，证明了能量空间中初始数据的解决方案的衰减。第一个结果涉及衰减，超过紧凑的间隔，使小小和平等条件（$ u $奇数）。第二个结果证明了在远场区域沿曲线衰减的解决方案，这些解决方案可以由$ c^1 $函数增加的解决方案主导。不需要小小的条件来证明Zakharov系统的最后结果。我们认为依靠适当的方程式使用适当的病毒身份，并遵循Kowalczyk-Martel-Muñoz和Muñoz-Ponce-Saut的技术。
We are interested in the long time asymptotic behavoir of solutions to the scalar Zakharov system \[ i u_{t} + Δu = nu,\] \[n_{tt} - Δn= Δ|u|^2\] and the Klein-Gordon Zakharov system \[ u_{tt} - Δu + u = - nu,\] \[ n_{tt} - Δn= Δ|u|^2\] in one dimension of space. For these two systems, we give two results proving decay of solutions for initial data in the energy space. The first result deals with decay over compact intervals asuming smallness and parity conditions ($u$ odd). The second result proves decay in far field regions along curves for solutions whose growth can be dominated by an increasing $C^1$ function. No smallness condition is needed to prove this last result for the Zakharov system. We argue relying on the use of suitable virial identities appropiate for the equations and follow the technics of Kowalczyk-Martel-Muñoz and Muñoz-Ponce-Saut.
