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Existence and stability of solutions to partial functional-differential equations with delay. (English) Zbl 0949.35141
Summary: 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