On Asymptotic Self-Similar Behaviour for a Quasilinear Heat Equation: Single Point Blow-Up

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The author studies the asymptotic behaviour near a finite blow-up time $t = T$ solutions to the degenerate quasilinear parabolic equation \[ ut = (u^\sigma u_x )_x + u^\beta \quad {\text{in }} \mathbb{R} \times (0,T),\] where $\sigma > 0$ and $\beta > \sigma + 1$ are fixed constants. These values of parameters $\sigma $, $\beta $ correspond to single point blow-up. The initial function is assumed to be bounded, symmetric, nonincreasing in $| x |$, and compactly supported. It is proved that the resealed function $f(\xi ,t) = (T - t)^{{1 / {(\beta - 1)}}} u(\xi (T - t)^m ,t),m = {{[\beta - (\sigma + 1)]} / {2(\beta - 1) > 0}}$ behaves as $t \to T$ like a nontrivial self-similar profile $\theta (\xi ) > 0$ in $\mathbb{R}$,$\theta (\xi ) \to 0$ as $\xi \to \infty $.