Existence and Blowup of Solutions for Neutral Partial Integro-Differential Equations with State-Dependent Delay (Journal of Nonlinear Modeling and Analysis, 2024)
In this paper, we study the existence and blowup of solutions fora neutral partial functional integro-differential equation with state-dependentdelay in Banach space. The mild solutions are obtained by Sadovskii fixedpoint theorem under compactness condition for the resolvent operator, thetheory of fractional power and $α$-norm are also used in the discussion sincethe nonlinear terms of the system involve spacial derivatives. The strongsolutions are obtained under the lipschitz condition. In addition, based on thelocal existence result and a piecewise extended method, we achieve a blowupalternative result as well for the considered equation. Finally, an example isprovided to illustrate the application of the obtained results.