Vergara Caffarelli, Giorgio The dynamic problem in linear viscoelasticity: Uniqueness for unbounded solutions in the past. (Italian. English summary) Zbl 0678.73037 Rend. Semin. Mat. Univ. Padova 81, 21-29 (1989). Summary: We prove that the dynamic problem in linear viscoelasticity has a unique weak solution in a class of unbounded functions for \(t\to -\infty\). We assume, for the relaxation function of the material, the dissipativity condition considered by Gurtin-Herrera [M. E. Gurtin and I. Herrera, Q. Appl. Mat. 23, 235-245 (1965; Zbl 0173.527)] and we require the instantaneous elastic modulus to be greater than a value depending on the dissipativity condition, on the memory of the material and on the asymptotic behaviour in the past of the admissible eigensolutions. MSC: 74Hxx Dynamical problems in solid mechanics 74G30 Uniqueness of solutions of equilibrium problems in solid mechanics 74H25 Uniqueness of solutions of dynamical problems in solid mechanics 74D05 Linear constitutive equations for materials with memory 74D10 Nonlinear constitutive equations for materials with memory 45N05 Abstract integral equations, integral equations in abstract spaces 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) Keywords:unique weak solution; class of unbounded functions; dissipativity condition; asymptotic behaviour; admissible eigensolutions Citations:Zbl 0667.73027; Zbl 0173.527 PDFBibTeX XMLCite \textit{G. Vergara Caffarelli}, Rend. Semin. Mat. Univ. Padova 81, 21--29 (1989; Zbl 0678.73037) Full Text: Numdam EuDML References: [1] M.E. Gurtin - I. Herrera , On dissipation inequalities and linear viscoelasticity , Quart. Appl. Mat. , 23 ( 1965 ), pp. 235 - 245 . MR 189346 | Zbl 0173.52703 · Zbl 0173.52703 [2] M. Chipot - G. Vergara Caffarelli , Viscoelasticity without initial data, in corso di stampa sui Proceeding of the Meeting on Volterra Integrodifferential equations in Banach Spaces and Applications , Trento ( 1987 ). Zbl 0672.73038 · Zbl 0672.73038 [3] G. Capriz , Sulla impostazione di problemi dinamici in viscoelasticità, tavola rotonda su Continui con Memoria , Accademia Nazionale dei Lincei ( 1986 ). [4] G. Fichera , Avere una memoria tenace crea gravi problemi , Arch. Rational Mech. Anal. , 70 ( 1979 ), pp. 101 - 112 . MR 1553577 | Zbl 0425.73002 · Zbl 0425.73002 · doi:10.1007/BF00250347 [5] G. Fichera , Sul principio di memoria evanescente , Rend. Sem. Mat. Univ. Padova , 68 ( 1982 ), pp. 235 - 259 . Numdam | Zbl 0515.73003 · Zbl 0515.73003 [6] G. Capriz - E.G. Virga , Esempi di non unicità in viscoelasticità lineare , in corso di stampa in Atti Accad. Sc. Torino ( 1986 ). [7] G. Fichera , On linear viscoelasticity , Mech. Res. Comm. , 12 ( 1985 ), pp. 241 - 242 . MR 823436 [8] E.G. Virga - G. Capriz , Un teorema di unicità in viscoelasticità lineare , in corso di stampa in Rend. Sem. Mat. Univ. Padova ( 1986 ). [9] G. Vergara Caffarelli - E.G. Virga , Sull’unicità della soluzione del problema dinamico della viscoelasticità lineare , Rend. Acc. Naz. Lincei , 81 ( 1987 ), pp. 379 - 387 . MR 999829 | Zbl 0667.73027 · Zbl 0667.73027 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.