Differential and Integral Equations

Existence, uniqueness, and longtime behavior for a nonlinear Volterra integrodifferential equation

Viorel Barbu, Pierluigi Colli, Gianni Gilardi, and Maurizio Grasselli

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We consider an initial and boundary value problem for a nonlinear Volterra integrodifferential equation. This equation governs the evolution of a pair of state variables, $u$ and $\vartheta$, which are mutually related by a maximal monotone graph $\gamma$ in ${{\Bbb R}}\times{{\Bbb R}}.$ The model can be viewed, for instance, as a generalized Stefan problem within the theory of heat conduction in materials with memory. Besides, it can be used for describing some diffusion processes in fractured media. The relation defined by $\gamma$ is properly interpreted and generalized in terms of a subdifferential operator associated with $\gamma$ and acting from $H^1(\Omega)$ to its dual space. Then, the generalized problem is formulated as an abstract Cauchy problem for a perturbation of a nonlinear semigroup, and existence and uniqueness of a solution $(u,\vartheta)$ can be proved via a fixed-point argument whatever the maximal monotone graph $\gamma$ is. Moreover, the meaning of $\gamma$ as a pointwise relationship is recovered almost everywhere, in the case when $\gamma$ is bounded on bounded subsets of ${{\Bbb R}}$. Finally, under some other restrictions on $\gamma$, the longtime behavior of the solution is investigated, in a more specific context related to the generalized Stefan problem.

Article information

Differential Integral Equations, Volume 13, Number 10-12 (2000), 1233-1262.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20]
Secondary: 45N05: Abstract integral equations, integral equations in abstract spaces


Barbu, Viorel; Colli, Pierluigi; Gilardi, Gianni; Grasselli, Maurizio. Existence, uniqueness, and longtime behavior for a nonlinear Volterra integrodifferential equation. Differential Integral Equations 13 (2000), no. 10-12, 1233--1262. https://projecteuclid.org/euclid.die/1356061125

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