Thermal runaway

Remarks

In thermal runaway the solution is strictly increasing and unbounded on its maximal interval of existence. Indeed, if it stayed bounded, it would converge to a limit $T_\infty \ge T_0$ with $G(T_\infty) = L(T_\infty)$, which is excluded. Example: in the dimensionless Semenov model, $G(\theta) = \psi e^{\theta}$ and $L(\theta) = \theta$ with $\theta_0 = 0$ and a constant $\psi > 0$. For $\psi = 1$ the condition holds, since $e^{\theta} > \theta$ for all $\theta$. The temperature even blows up in finite time, because $\int_0^\infty d\theta / (e^\theta - \theta) < \infty$. Non-example: for $\psi = 1/10$, one has $G(1) = e/10 < 1 = L(1)$, so the condition fails at $\theta = 1$. Then $G - L$ changes sign on $[0, 1]$, there is a steady state there, and the temperature stays below it. The word is also used more loosely for any rapid, self-accelerating rise of temperature.