# 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