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Existence and Blowup of Solutions for Neutral Partial Integro-differential Equations with State-dependent Delay
  
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KeyWord:Neutral partial integro-differential equation  Analytic semigroup  Resolvent operator  Fractional power operator  State-dependent delay
Author NameAffiliation
Jianbo Zhu School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China 
Xingxing Wang School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China 
Xianlong Fu School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China 
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Abstract:
      In this paper, we study the existence and blowup of solutions for a neutral partial functional integro-differential equation with state-dependent delay in Banach space. The mild solutions are obtained by Sadovskii fixed point theorem under compactness condition for the resolvent operator, the theory of fractional power and α-norm are also used in the discussion since the nonlinear terms of the system involve spacial derivatives. The strong solutions are obtained under the lipschitz condition. In addition, based on the local existence result and a piecewise extended method, we achieve a blowup alternative result as well for the considered equation. Finally, an example is provided to illustrate the application of the obtained results.