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The $C$-regularized semigroup method for partial differential equations with delays

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  • This paper is devoted to study the abstract functional differential equation (FDE) of the following form $$\dot{u}(t)=Au(t)+\Phi u_t,$$ where $A$ generates a $C$-regularized semigroup, which is the generalization of $C_0$-semigroup and can be applied to deal with many important differential operators that the $C_0$-semigroup can not be used to. We first show that the $C$-well-posedness of a FDE is equivalent to the $\mathscr{C}$-well-posedness of an abstract Cauchy problem in a product Banach space, where the operator $\mathscr{C}$ is related with the operator $C$ and will be defined in the following text. Then, by making use of a perturbation result of $C$-regularized semigroup, a sufficient condition is provided for the $C$-well-posedness of FDEs. Moreover, an illustrative application to partial differential equation (PDE) with delay is given in the last section.
    Mathematics Subject Classification: Primary: 47D60, 47D06; Secondary: 35R10.


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