Global existence and asymptotic behavior of weak solutions to nonlinear thermoviscoelastic systems with clamped boundary conditions

Abstract

This paper is concerned with global existence, uniqueness, and asymptotic behavior, as time tends to infinity, of weak solutions to nonlinear thermoviscoelastic systems with clamped boundary conditions. The constitutive assumptions for the Helmholtz free energy include the model for the study of phase transitions in shape memory alloys. To describe phase transitions between different configurations of crystal lattices, we work in a framework in which the strain u u belongs to L ∞ {L^\infty } . It is shown that for any initial data of (strain, velocity, absolute temperature) ( u 0 , v 0 , θ 0 ) ∈ L ∞ × W 0 1 , ∞ × H 1 \left ( {u_0}, {v_0}, {\theta _0} \right ) \in \\ {L^\infty } \times W_0^{1, \infty } \times {H^1} , there is a unique global solution ( u , v , θ ) ∈ C ( [ 0 , + ∞ ] ; L ∞ ) × C ( 0 , + ∞ ) ; W 0 1 , ∞ ) ∩ L ∞ ( [ 0 , + ∞ ) ; W 1 , ∞ ) × C ( [ 0 , + ∞ ) ; H 1 ) \left ( u, v, \theta \right ) \in C\left ( \left [ 0, + \infty \right ]; {L^\infty } \right ) \times C\left ( 0, + \infty \right ); \\ \left . W_0^{1, \infty } \right ) \cap {L^\infty }\left ( \left [ 0, + \infty ); {W^{1, \infty }} \right ) \times C\left ( \left [ 0, + \infty \right ); {H^1} \right ) \right . . Results concerning the asymptotic behavior as time goes to infinity are obtained.