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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.

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
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