Behavior of critical solutions of a nonlocal hyperbolic problem in Ohmic heating of foods. (English) Zbl 1005.35021
The authors discuss the behaviour of solutions of the nonlocal hyperbolic problem
\[
\begin{cases} u_t+u_x={\lambda f(u) \over\Bigl(\int^1_0 f(u)dx \Bigr)^2}, \quad &0<x<1,\;t>0,\\ u(0,t)=0,\quad & t>0,\\ u(x,0)= \psi(x) \quad & 0<x<1\end{cases} \tag{1}
\]
at a critical value of parameter \(\lambda\), say \(\lambda_*\), where \(u=u(x,t)= u(x,t,\lambda)\) and \(u^*(x,t)= u(x,t,\lambda_*)\) is referred to a critical solution of (1). The authors study the global existence and divergence of some “critical” solutions \(u_*(x,t)\) of a nonlocal hyperbolic problem modeling Ohmic heating of foods. The authors give also some estimates of the rate of divergence.
Reviewer: Messoud Efendiev (Berlin)
MSC:
35B40 | Asymptotic behavior of solutions to PDEs |
80A20 | Heat and mass transfer, heat flow (MSC2010) |
35L45 | Initial value problems for first-order hyperbolic systems |
35L60 | First-order nonlinear hyperbolic equations |