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Blow-up for the Cauchy Problem in Nonlinear One-dimensional Thermoelasticity

Part of the Operator Theory: Advances and Applications book series (OT, volume 184)

Abstract

In this chapter we study the blow-up phenomena of solutions in a finite time to the following Cauchy problem with a non-autonomous forcing term and a thermal memory:
$$ \begin{gathered} u_{tt} = au_{xx} + b\theta _x + du_x - mu_t + f\left( {t,u} \right), \hfill \\ c\theta _t = \kappa \theta _{xx} + g*\theta _{xx} + bu_{xt} + pu_x + q\theta _x \hfill \\ \end{gathered} $$
(1)
subject to the initial conditions
$$ u\left( {x,0} \right) = u_0 \left( x \right), u_t \left( {x,0} \right) = u_1 \left( x \right), \theta \left( {x,0} \right) = \theta _0 \left( x \right) x\forall \in \mathbb{R}. $$
(2)
Here by u=u(x,1) and θ=θ(x,t) we denote the displacement and the temperature difference respectively. The function g=g(t) is the relaxation kernel and the sign* denotes the convolution product, i.e., \( g*y\left( { \cdot ,t} \right) = \smallint _0^t g\left( {t - \tau } \right)y\left( { \cdot ,\tau } \right)d\tau \) . The coefficients a, b, c are positive constants, while d, k, p, q, m are non-negative constants. The function f=f(t,u) is a non-autonomous forcing term.

Keywords

Cauchy Problem Convolution Product Cauchy Inequality Relaxation Kernel Thermoelastic Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag AG 2008

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