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Global attractor for a nonlinear thermoviscoelastic model with a non-convex free energy density. (English) Zbl 1114.35145

The paper deals with the existence of a global attractor for a semiflow governed by the initial-boundary value problem \[ u_t=v_x,\quad v_t-(f_1(u)\theta + f_2(u)+\gamma v_x)_x=0,\;c_v\theta_t-\theta f_1(u)v_x-\gamma v_x^2-\kappa \theta_{xx}=0, \] \(\sigma=0,\;\theta_x=0,\;\text{for}\;x=0,1,\quad \text{or}\;v=0\;\text{for}\;x=0,\;\text{and}\;\sigma=0\;\text{for}\;x=1.\) \[ u=u_0(x),\quad v=v_0(x),\;\theta=\theta_0(x)\;\text{for}\;t=0, \] \(f_i=F_i',\;i=1,2,\;F_1(u)=u^2,\;F_2(u)=u^6-c_2u^4-c_3u^2,\) \(\sigma=\sigma_1+\gamma v_x,\;\sigma_1(u,\theta)=f_1(u)\theta+f_2(u).\) The authors prove the existence of the nonlinear semiflow \(\{S(t)\}\) defined by the global weak solution of the problem stated above. The semiflow possesses a compact and Lyapunov stable global attractor.

MSC:

35Q72 Other PDE from mechanics (MSC2000)
35B41 Attractors
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35B40 Asymptotic behavior of solutions to PDEs
74D10 Nonlinear constitutive equations for materials with memory
74F05 Thermal effects in solid mechanics
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