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: The problem \[ {u_t}\left ( {x, t} \right ) = \int _0^t {} a\left ( {t - \tau } \right )\frac {\partial }{{\partial x}}\sigma \left ( {{u_x}\left ( {x,\tau } \right )} \right )d\tau + f\left ( {x, t} \right ), \qquad 0 < x < 1, \qquad t > 0, \\ u\left ( {0,t} \right ) \equiv u\left ( {1,t} \right ) \equiv 0 \qquad u\left ( {x, 0} \right ) = {u_0}\left ( x \right )\] 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, \[ \dot u = - \int _0^t a \left ( {t - \tau } \right )g\left ( {u\left ( \tau \right )} \right )d\tau + f\left ( t \right )\] 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, \[ {u_{tt}} + \alpha {u_t} - \frac {\partial }{{\partial x}}\sigma \left ( {{u_x}} \right ) = 0\]. It is shown that the problem is related to a theory of heat flow in materials with memory.
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MacCamy, R. C. and J. S. Wong, Stability theorems for some functional differential equations, Trans. Amer. Math. Soc. 164, 1–37 (1972)
Nohel, J. A. and D. F. Shea, On the global behavior of a non-linear Volterra equation, to appear in Advances of Math.
Nishida, T., Global smooth solutions for the second order quasilinear equation with first order dissipation (preprint).
Steffans, O. J., Nonlinear Volterra integral equations with positive definite kernels, to appear in Proc. of Amer. Math. Soc.
Barbu, V., Integro-differential equations in Hilbert spaces, Anal. Stunt, ale Univ. Al. I. Cuza den Iasi 19, 365–383 (1973)
Grossman, S. I. and R. K. Miller, Non-linear Volterra integro-differential systems with ${L^1}$ kernels, Journal of Diff. Eqs. 13, 458–476 (1973)
Gurtin, M. E. and A. C. Pipkin, A general theory of heat conduction with finite, wave speeds, Arch, for Rat. Mech. and Anal. 31, 113–126 (1968)
Lax, P. D., Development of singularities of solutions of non-linear hyperbolic partial differential equations, Journal Math. Phys. 5, 611–613 (1964)
Levin, J. J. and J. A. Nohel, Perturbations of a non-linear Volterra equation, Mich. Math. Journal 12, 431–447 (1965)
Londen, S. -O., The qualitative behavior of the solutions of a non-linear Volterra equation, Mich. Math. Journal 18, 321–338 (1971)
MacCamy, R. C., Stability theorems for a class of functional differential equations, to appear in SIAM Journal of Appl. Math.
MacCamy, R. C., Remarks on frequency domain methods for Volterra integral equations, to appear in Journal Math. Anal. and Appl.
MacCamy, R. C. and J. S. Wong, Stability theorems for some functional differential equations, Trans. Amer. Math. Soc. 164, 1–37 (1972)
Nohel, J. A. and D. F. Shea, On the global behavior of a non-linear Volterra equation, to appear in Advances of Math.
Nishida, T., Global smooth solutions for the second order quasilinear equation with first order dissipation (preprint).
Steffans, O. J., Nonlinear Volterra integral equations with positive definite kernels, to appear in Proc. of Amer. Math. Soc.
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© Copyright 1977
American Mathematical Society