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Development of singularities in solutions of the equations of nonlinear thermoelasticity. (English) Zbl 0661.35009
The system of equations of one-dimensional thermoelasticity is considered in the Lagrangian formulation as \[ \partial_ tu-\partial_ xv=0;\quad \partial_ tv-\partial_ x\sigma =0;\quad \partial_ t(\epsilon +(1/2)v^ 2)-\partial_ x(\sigma v)=\partial_ xq \] where u is the deformation gradient, v is velocity and \(\sigma\) is stress, while \(\epsilon\) denotes the internal energy and q does the heat flux, and that some constitutive assumptions for \(\epsilon\), \(\sigma\) and q are made as specified functions of u, T (standing for absolute temperature) or u, T and \(\partial_ xT\), respectively. The authors consider the Cauchy problem for the above system with initial conditions \[ u(x,0)=u_ 0(x),v(x,0)=v_ 0(x),T(x,0)=T_ 0(x). \] It is known that solutions to the problem exist globally in time for small initial data. They proved under suitable assumptions on data that solutions to the problem blow up in finite time when the initial data are large. The technique for the proof is to transform the system to an equivalent one so that the standard theory of parabolic equations can be applied.
Reviewer: T.Kakita

35F25 Initial value problems for nonlinear first-order PDEs
35B35 Stability in context of PDEs
35B50 Maximum principles in context of PDEs
35L45 Initial value problems for first-order hyperbolic systems
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