Abstract
The initial value problem for a multivalued differential equation is studied, which is governed by the sum of a monotone, hemicontinuous, coercive operator fulfilling a certain growth condition and a Volterra integral operator in time of convolution type with exponential decay. The two operators act on different Banach spaces where one is not embedded in the other. The set-valued right-hand side is measurable and satisfies certain continuity and growth conditions. Existence of a solution is shown via a generalisation of the Kakutani fixed-point theorem.
Highlights
1.1 Problem Statement and Main ResultWe consider the multivalued differential equation1 v (t) + Av(t) + (BKv)(t) ∈ F (t, v(t)), t ∈ (0, T ), v(0) = v0, (1)A
The initial value problem for a multivalued differential equation is studied, which is governed by the sum of a monotone, hemicontinuous, coercive operator fulfilling a certain growth condition and a Volterra integral operator in time of convolution type with exponential decay
The two operators act on different Banach spaces where one is not embedded in the other
Summary
The operator F : [0, T ] × H → Pf c(H ) is measurable, fulfils a certain growth condition in the second argument and the graph of v → F (t, v) is sequentially closed in H × Hw for almost all t ∈ (0, T ), where Hw denotes the Hilbert space H equipped with the weak topology. Physical applications of the system we are considering in this work are, e.g., heat flow in materials with memory (see, e.g., MacCamy [25], Miller [30]) or viscoelastic fluid flow (see, e.g., Desch, Grimmer, and Schappacher [11], MacCamy [26]) Another application related to that are non-Fickian diffusion models which describe diffusion processes of a penetrant through a viscoelastic material (see, e.g., Edwards [13], Edwards and Cohen [14], Shaw and Whiteman [39]). The limit yields a second-order in time equation for u (see, e.g., Emmrich and Thalhammer [16])
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
