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Upper and lower $L^2$-decay bounds for a class of derivative nonlinear Schr\"odinger equations
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math.AP
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decayderivativeequationslowernonlinearodingerschrbounds
abstract
We consider the initial value problem for cubic derivative nonlinear Schr\"odinger equations possessing weakly dissipative structure in one space dimension. We show that the small data solution decays like $O((\log t)^{-1/4})$ in $L^2$ as $t\to +\infty$. Furthermore, we find that this $L^2$-decay rate is optimal by giving a lower estimate of the same order.
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