Two-Sided Estimates of Solutions with a Blow-Up Mode for a Nonlinear Heat Equation with a Quadratic Source

We study compactly supported solutions \(u(x, t) \geq 0\) , \(x \in \mathbb{R}\) , \(t \geq 0\) , of a one-dimensional quasilinear heat transfer equation. The equation has a transport coefficient linear in \(u\) and a self-consistent source \(\alpha u+\beta u^{2}\) of general form. For the blow-up time of compactly supported solutions, we establish two-sided estimates functionally depending on the initial conditions \(u(x, 0)\) .