| Abstract: |
| In this talk, I will develop a unified theory of well-posedness for delay differential equations (DDEs) by ``history spaces`` and ``prolongations.`` One of the main results is the following: under the assumption that the history space $H$ is $C^0$-prolongable and $C^0$-regulated, the trivial equation $\dot{x} = 0$ generates a continuous semiflow on $H$ is necessary and sufficient for the property that the initial value problem of $\dot{x}(t) = F(t, x_t)$ is well-posed for any history functional F which is continuous and uniformly locally Lipschitzian about $C^0$-prolongations. I will also generate this result to the setting that $H$ is $C^1$-prolongable and $C^1$-regulated in order to cover DDEs with general state-dependent delays. |
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