Consider the initial value problem for cubic derivative nonlinear Schrödinger equations in one space dimension. We provide a detailed lower bound estimate for the lifespan of the solution, which can be computed explicitly from the initial data and the nonlinear term. This is an extension and a refinement of the previous work by one of the authors [H. Sunagawa: Osaka J. Math. 43 (2006), 771-789] where the gauge-invariant nonlinearity was treated.

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<p style='text-indent:20px;'>We consider the initial value problem for cubic derivative nonlinear Schrödinger equations possessing weakly dissipative structure in one space dimension. We show that the small data solution decays like <inline-formula><tex-math id="M2">\begin{document}$ O((\log t)^{-1/4}) $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M3">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M4">\begin{document}$ t\to +\infty $\end{document}</tex-math></inline-formula>. Furthermore, we find that this <inline-formula><tex-math id="M5">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-decay rate is optimal by giving a lower estimate of the same order.</p>