Stability of blowup for a parabolic p-Laplace equation with nonlinear source

In this paper, we mainly consider the stability of blowup of solutions for the p-Laplace equation with nonlinear source ${u_t = {div}(|\nabla u|^{p-2}\nabla u) + u^q,(x,t)\in\mathbb{R}^N \times (0,T)}$ , with the initial value ${u(x,0) = u_0(x) \geq 0}$ , where ${\|u_0 (x)\|_{L^\infty} \leq M}$ and T < ∞ is the blowup time. Under a small oscillation around the radial initial value, we can prove the solution blows up in finite time and obtain the blowup rate estimate of the form ${\|u(\cdot,t)\|_{L^\infty}\leq C(T-t)^{-\frac{1}{q-1}}}$ , where the constant C > 0 is dependent only on N, p, q, and the parameters q and p are expected to be ${p > 2, p-1 < q < \frac{Np}{(N-p)}_+ -1}$ .