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Parabolic equations and thermodynamics. (English) Zbl 0794.35069

The aim of this paper is to prove two theorems about periodic solutions of parabolic equations which are suggested by considerations drawn from thermodynamics and, in particular, by two papers of Clausius. (These papers are readily accessible as items 7 and 8 in the collection [The second law of thermodynamics, Dowden, Hutchinson, and Ross, Stroudsburg, Pa. (1976)], edited by J. Kestin). The author is concerned with solutions \(u(x,t)\) of the parabolic equation \[ \bigl(a(x)u_ x\bigr)_ x=b(x)u_ t,\;0<x<1,\;-\infty<t<+\infty, \] which satisfy boundary conditions of the form \(u(0,t)=u(1,t)=\tau(t)\), \(-\infty<t<+\infty\). Both \(u\) and \(\tau\) are understood to be periodic, with period \(p\), in \(t\), and it is assumed that the coefficients \(a\) and \(b\) are positive on \([0,1]\) and that \(a\in C^ 1\), \(b\in C^ 0\), and \(\tau\in C^ 3\).

MSC:

35K20 Initial-boundary value problems for second-order parabolic equations
80A10 Classical and relativistic thermodynamics
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