Global existence of smooth solution to nonlinear thermoviscoelastic system with clamped boundary conditions in solid-like materials
The authors have proved the global existence and uniqueness of a smooth solution of a system of one-dimensional nonlinear thermoviscoelasticity which describes the thermomechanical process in a class of solid like material -- such as rubber etc. -- under the assumption, -- proposed by \textit{C. D. Dafermos} in 1982 [SIAM. J. Math. Anal. 13, 397-408 (1982; Zbl 0489.73124)] --, that both the ends of the one-dimensional solid are clamped, and there is no flux of heat through the boundary. A new approach is developed to obtain all important estimates for the \(L^\infty\)-norm of the strain function \(u(x,t)\). First uniform pointwise boundedness, from both below and above, of the unknown functions and their derivatives is proved. Finally the global existence and uniqueness of a smooth solution is proved by using the framework given in the paper of \textit{C. M. Dafermos} and \textit{L. Hsiao} [Nonlinear Anal., Theory Methods Appl. 6, 435-454 (1982; Zbl 0498.35015)]. Thus apriori bounds of strain, velocity and temperature in some Banach space is proved.
scientific article; zbMATH DE number 1334928