Advances in Differential Equations

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

Wolfgang M. Ruess

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Abstract

Results on (a) the existence and (b) asymptotic stability of mild and of strong solutions to the nonlinear partial functional differential equation with delay $ (FDE) \; \, \dot{u} (t) + B u(t) \ni F(u_t), \; t \geq 0 , \; u_0 = \varphi \in E, $ 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 undelayed counterpart to (FDE) of the nonlinear initial-value problem $ (CP) \; \, \dot{u} (t) + B u(t) \ni f(t), \; t \geq 0 , \; u(0) = u_0 \in X, $ 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.

Article information

Source
Adv. Differential Equations Volume 4, Number 6 (1999), 843-876.

Dates
First available in Project Euclid: 15 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1366030749

Mathematical Reviews number (MathSciNet)
MR1729393

Zentralblatt MATH identifier
0949.35141

Subjects
Primary: 34K30: Equations in abstract spaces [See also 34Gxx, 35R09, 35R10, 47Jxx]
Secondary: 35R10: Partial functional-differential equations 47J05: Equations involving nonlinear operators (general) [See also 47H10, 47J25] 47N20: Applications to differential and integral equations

Citation

Ruess, Wolfgang M. Existence and stability of solutions to partial functional-differential equations with delay. Adv. Differential Equations 4 (1999), no. 6, 843--876. https://projecteuclid.org/euclid.ade/1366030749.


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