On a class of differential equations of parabolic type with impulsive action
For the class of semi-linear abstract differential equations with impulsive action of the form \[ \begin{gathered} \frac{d}{dt}\,[ Au(t)]+Bu(t)= f(t,u(t)),\quad t>t_0, \,\, t\notin\{t_k\},\\ \Delta u|_{t=t_{k}}=u(t_k+0)-u(t_k-0)=I_k(u(t_k-0)), \,\,\, k=1,2,\ldots, \end{gathered} \] the existence and uniqueness of solutions are studied. Here \(A,B\) are closed linear operators mapping a complex Banach space \(X\) into another complex Banach space \(Y,\) \(f: [t_0,\infty)\times X\to Y\) is a continuous function, \(I_k:\Omega_k\subset X\to X\) and \(t_0<t_1<t_2<\ldots.\) The abstract results are applied to partial differential equations with impulsive action.
Impulsive partial differential equations
scientific article; zbMATH DE number 5659345