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Memory effects and homogenization. (English) Zbl 0725.45012

For each \(\epsilon >0\) consider problem \(\frac{\partial u^{\epsilon}(x,t)}{\partial t}+a^{\epsilon}(x)u^{\epsilon}(x,t)=f(x,t),\) \(x\in \Omega\), \(t\in (0,T)\); \(u^{\epsilon}(x,0)=0\), \(x\in \Omega\), and suppose that \(0\leq \alpha \leq a^{\epsilon}(x)\leq \beta\), a.e. and f is bounded and measurable. The author proved in another paper [Partial differential equations and the calculus of variations. Essays in Honor of Ennio de Giorgi, 925-938 (1989; Zbl 0682.35028)] that if \(a^{\epsilon}\) converges weakly to a function \(a^ 0\) then a subsequence of \(u^{\epsilon}\) converges weakly to a \(u^ 0\) which satisfies the integral equation \(\frac{\partial u^ 0(x,t)}{\partial t}+a^ 0(x)u^ 0(x,t)-\int^{t}_{0}k(x,t-s)u^ 0(x,s)ds=f(x,t),\) \(x\in \Omega\), \(t\in (0,T)\) \(u^ 0(x,0)=0\), \(x\in \Omega\), where k has a suitable representation. In this paper two different approaches are given to the time-dependent linear case and one of this is used to investigate the nonlinear case.
Reviewer: G.Di Blasio (Roma)

MSC:

45K05 Integro-partial differential equations

Citations:

Zbl 0682.35028
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References:

[1] Coleman, B. D., & V. J. Mizel, ?Norms and semigroups in the theory of fading memory,? Arch. Rational Mech. Anal. 23 (1967), 87-123.
[2] Coleman, B. D., & V. J. Mizel, ?On the general theory of fading memory,? Arch. Rational Mech. Anal. 29 (1968), 18-31. · Zbl 0167.24704
[3] Coleman, B. D., & W. Noll, ?An approximation theorem for functionals with applications in continuum mechanics,? Arch. Rational Mech. Anal. 6 (1960), 355-370. · Zbl 0097.16403
[4] Coleman, B. D., & W. Noll, ?Foundations of linear viscoelasticity,? Reviews Mod. Phys. 33 (1961), 239-249; erratum: ibid. 36 (1964), 1103. · Zbl 0103.40804
[5] Mascarenhas, L., ?A linear homogenization problem with time dependent coefficient,? Trans. A.M.S. 281, 1 (1984), 179-195. · Zbl 0536.45003
[6] Tartar, L., ?Nonlinear constitutive relations and homogenization,? Contemporary developments in continuum mechanics and partial differential equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), 472-484. North-Holland Math. Studies, 30, North-Holland, Amsterdam, 1978.
[7] Tartar, L., ?Nonlocal effects induced by homogenization,? Partial Differential Equations and the Calculus of Variations, Essays in Honor of Ennio De Giorgi, II, 925-938, Birkhäuser, Boston, 1989.
[8] Tartar, L., ?H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations,? to appear in Proc. Roy. Soc. Edinburgh. · Zbl 0774.35008
[9] Tartar, L., ?H-measures and small amplitude homogenization,? R. V. Kohn & G. Milton, eds. Random Media and Composites, 89-99, SIAM, Philadelphia, 1989. · Zbl 0790.73009
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