Barbu V., Colli P., Gilardi G., Grasselli M. Existence, uniqueness, and longtime behavior for a nonlinear Volterra
integrodifferential equation. Technical report ercim.cnr.ian//1998-1083, 1998. |

Abstract (English) |
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 $theta$, which are mutually related by a maximal monotone graph $gamma$ in $erretimeserre.$ 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 $(u,theta)$ 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 $erre$. 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. | |

Subject | Nonlinear Volterrra integrodifferential equation, Stefan problem |

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