Existence and stability of solutions to partial functional-differential equations with delay.

*(English)*Zbl 0949.35141Summary: Results on (a) the existence and (b) asymptotic stability of mild and of strong solutions to the nonlinear partial functional differential equation with delay
\[
\dot u(t)+ Bu(t)\ni F(u_t),\quad t\geq 0,\quad u_0= \varphi\in E,\tag{FDE}
\]
are presented. The ‘partial differential expression’ \(B\) will be a generally multivalued, accretive operator, and the history-responsive operator \(F\) will be allowed to be (defined and) Lipschitz continuous on ‘thin’ subsets of the initial-history space \(E\) of functions from an interval \(I\subset(-\infty,0]\) to the state Banach space \(X\). As one of the main results, it is shown that the well-established solution theory on strong, mild and integral solutions to the underlayed counterpart to (FDE) of the nonlinear initial-value problem
\[
\dot u(t)+ Bu(t)\ni f(t),\quad t\geq 0,\quad u(0)= u_0\in X,\tag{CP}
\]
can fully be extended to the more general initial-history problem (FDE). The results are based on the relation of the solutions to (FDE) to those of an associated nonlinear Cauchy problem in the initial-history space \(E\). Applications to models from population dynamics and biology are presented.

##### MSC:

35R10 | Partial functional-differential equations |

35B35 | Stability in context of PDEs |

47H06 | Nonlinear accretive operators, dissipative operators, etc. |

47H20 | Semigroups of nonlinear operators |