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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Weak solutions of hyperbolic-parabolic Volterra equations


Author: Gustaf Gripenberg
Journal: Trans. Amer. Math. Soc. 343 (1994), 675-694
MSC: Primary 45K05; Secondary 35D05, 35K60, 45D05, 73F15
DOI: https://doi.org/10.1090/S0002-9947-1994-1216335-0
MathSciNet review: 1216335
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Abstract: The existence of a global weak solution, satisfying certain a priori $ {L^\infty }$-bounds, of the equation $ {u_t}(t,x) = \int _0^tk(t - s){(\sigma ({u_x}))_x}(s,x)ds + f(t,x)$ is established. The kernel k is locally integrable and log-convex, and $ \sigma \prime $ has only one local minimum which is positive.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1216335-0
Keywords: Volterra equation, weak solution, $ {L^\infty }$-bound, viscoelasticity, parabolic, hyperbolic
Article copyright: © Copyright 1994 American Mathematical Society