Nonlinear Volterra Equations for Heat Flow in Materials with Memory.
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WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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Consider the nonlinear Volterra equation ut bAu not an element of ft. This paper discusses existing and recent results for the following problems concerning this equation the global existence and uniqueness of solutions and their continuous dependence on the data the boundedness and asymptotic behavior as t approaches infinity in th special cases when X H is a real Hilbert space and A is either a maximal monotone operator on H or A is a subdifferential of a proper, convex, lower semicontinuous function and the existence, boundedness, and asymptotic behavior of positive solutions in the general settting. The theory is used to study one possible model problem for heat flow in a material with memory which can be transformed to the equivalent from of the equation under physically reasonable assumptions the latter provide a motivation for the natural setting of much of the theory developed here.
- Theoretical Mathematics