Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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An integro-differential equation with application in heat flow


Author: R. C. MacCamy
Journal: Quart. Appl. Math. 35 (1977), 1-19
MSC: Primary 80.45; Secondary 35L65
DOI: https://doi.org/10.1090/qam/452184
MathSciNet review: 452184
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Abstract | References | Similar Articles | Additional Information

Abstract: The problem

$\displaystyle {u_t}\left( {x, t} \right) = \int_0^t {} a\left( {t - \tau } \rig... ...t( {1,t} \right) \equiv 0 \qquad u\left( {x, 0} \right) = {u_0}\left( x \right)$

is considered. Asymptotic stability theorems for the solution are established under appropriate conditions on $ a$, $ \sigma $ and $ f$. The conditions on $ a$ are of frequency domain type and are related to ones used previously in the study of Volterra integral equations,

$\displaystyle \dot u = - \int_0^t a \left( {t - \tau } \right)g\left( {u\left( \tau \right)} \right)d\tau + f\left( t \right)$

on a Hilbert space. An existence theorem for the problem is established under smallness assumptions on $ f$ and $ {u_0}$ This theorem is related to one by Nishida for the damped non-linear wave equation,

$\displaystyle {u_{tt}} + \alpha {u_t} - \frac{\partial }{{\partial x}}\sigma \left( {{u_x}} \right) = 0$

. It is shown that the problem is related to a theory of heat flow in materials with memory.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/qam/452184
Article copyright: © Copyright 1977 American Mathematical Society


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