Blow-up of solutions to the Cauchy problem in nonlinear one-dimensional thermoelasticity
The authors study the blow-up phenomena of solutions in a finite time to the following Cauchy problem with a nonautonomous forcing term and thermal memory: \[ \begin{gathered} u_{tt}= au_{xx}+ b\theta_x+ du_x- mu_t+ f(t,u),\\ c\theta_t= k\theta_{xx}+ g* \theta_{xx}+ bu_{xt}+ pu_x+ q\theta_x,\\ u(x,0)= u_0,\;u_t(x,0)= u_{1,x},\;\theta(x, 0)= \theta_0(x),\quad\forall x\in\mathbb{R},\end{gathered} \] where by \(u= u(x,t)\) and \(\theta= \theta(x,t)\) denote the displacement and the temperature difference, respectively. The function \(g= g(t)\) is the relaxation kernel and the sign \(*\) denotes the convolution product. The coefficients \(a\), \(b\), \(c\) are positive constants, while \(d\), \(k\), \(p\), \(q\), \(m\) are nonnegative constants.
scientific article; zbMATH DE number 2081789
scientific article