Blow-up of solutions to the Cauchy problem in nonlinear one-dimensional thermoelasticity (original) (raw)

A blow-up result in a Cauchy viscoelastic problem

2008

In this work we consider a Cauchy problem for a nonlinear viscoelastic equation. Under suitable conditions on the initial data and the relaxation function, we prove a finite-time blow-up result.

Neumann problem for one-dimensional nonlinear thermoelasticity

Banach Center Publications, 1992

The global existence theorem of classical solutions for one-dimensional nonlinear thermoelasticity is proved for small and smooth initial data in the case of a bounded reference configuration for a homogeneous medium, considering the Neumann type boundary conditions: traction free and insulated. Moreover, the asymptotic behaviour of solutions is investigated.

On a two-dimensional model of generalized thermoelasticity with application

Scientific Reports

A 2D first order linear system of partial differential equations of plane strain thermoelasticity within the frame of extended thermodynamics is presented and analyzed. The system is composed of the equations of classical thermoelasticity in which displacements are replaced with velocities, complemented with Cattaneo evolution equation for heat flux. For a particular choice of the characteristic quantities and for positive thermal conductivity, it is shown that this system may be cast in a form that is symmetric t-hyperbolic without further recurrence to entropy principle. While hyperbolicity means a finite speed of propagation of heat waves, it is known that symmetric hyperbolic systems have the desirable property of well-posedness of Cauchy problems. A study of the characteristics of this system is carried out, and an energy integral is derived, that can be used to prove uniqueness of solution under some boundary conditions. A numerical application for a finite slab is considered ...

Global solvability and exponential stability in one-dimensional nonlinear thermoelasticity

1993

We are mainly concerned with the Dirichlet initial boundary value problem in one-dimensional nonlinear thermoelasticity. It is proved that if the initial data are close to the equilibrium then the problem admits a unique, global, smooth solution. Moreover, as time tends to infinity, the solution is exponentially stable. As a corollary we also obtain the existence of periodic solutions for small, periodic righthand sides.

Numerical solution for a nonlinear, one-dimensional problem of thermoelasticity

Journal of Computational and Applied Mathematics, 1998

A numerical solution for a nonlinear, one-dimensional boundary-value problem of thermoelasticity for the elastic halfspace is presented. The resulting equations are discussed and the numerical method is investigated for stability. Comparison with other existing numerical schemes is carried out. The obtained results clearly indicate the process of shock formation. The presented ÿgures show the e ects of di erent nonlinear coupling constants on the distributions of the mechanical displacement and temperature in the medium. A special case is brie y discussed.