J. Janno and L. V. Wolfersdorf given by the values of two functionals over the temperature. As in our papers [10, 11] we obtain global in time existence of solutions besides uniqueness and stability of solutions. For simplicity, as in  we restrict ourselves to the case of continuous memory kernels whereas in  (as in the mentioned papers by Lorentzi and Sinestrari) a weakly singular memory kernel of flux is dealt with. But we also deal with the case of a constitutive relation for the heat flux of purely integral type as in the theory of Gurtin and Pipkin  (cf. also [)2. FORMULATION OF PROBLEM In the general linear theory of heat flow in a rigid isotropic body consisting of material with thermal memory, the following system of constitutive relations holds (cf. [1, 3, 6, 7, 9, 10, 18-22]) e(x, t) = (x) (u(x, t)+ f n(t - }(, r) dr\ (2.1) q(x, t) = -7(x) ( XVu(x, t) - i m(t - r)Vu(x, t) dr\ J -00 (2.2) together with the heat balance equation I^OM) + div £(*,«) - /(x,f), x D (2.3) in bounded domain D of the Euclidian space R p ,p >
Journal of Inverse and Ill-Posed Problems – de Gruyter
Published: Jan 1, 1998
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