Asymptotic self-similar global blow-up for a quasilinear heat equation (Q1386267)

The paper concerns the Cauchy problem for the quasilinear parabolic equation \(u_t=(u^\sigma u_x)_x+u^\beta\), where \(\sigma >0\), \(1<\beta <\sigma +1\) are fixed constants. The initial data are supposed to be compactly supported, nonnegative and Lipschitz continuous. It is known that there exists a unique nonnegative weak solution \(u\) which is continuous and smooth. If the initial data are not identically zero then \(u\) blows up in a finite time \(T\), and the blow-up is global. It is proved that the rescaled function \[ f(\xi ,t)=(T-t)^{1/(\beta -1)}u\bigl (\xi (T-t)^m,t\bigr ),\qquad m={\beta -(\sigma +1)\over 2(\beta -1)} \] converges uniformly as \(t\to T\) to a compactly supported, symmetric self-similar profile \(\theta (\xi )\geq 0\) satisfying a nonlinear ordinary differential equation.