Asymptotic behavior of solutions of quasilinear parabolic equations with supercritical nonlinearity (Q1874488)

There is considered the Cauchy problem: \[ \begin{aligned} u_t-\Delta u^m=u^p &\quad \text{in} \quad \mathbb R^N\times (0,T),\;N\geq 1, \tag{1}\\ u(x,0)=\tau u_0(x) & \quad \text{in} \quad \mathbb R^N, \tag{2}\end{aligned} \] where \(\tau>0,\;p>m\geq 1,\;u_0(x)>0\) is a bounded continuous radially symmetric function in \(\mathbb R^N\). Parameter \(p\) from open interval \((p_s,p_p)\) with \(p_s=\frac{m(N+2)}{(N-2)_+}\) is such that a peaking solution (incomplete blow-up solution) of equation (1) exists. The main result is the following. Let \(u_0(x)\) is nonincreasing in large \(r=|x|\) function which decays slowly: \(u_0(x)=O (|x|^{-\alpha})\;\text{ as} |x|\to \infty,\;\frac 2/{(p-m)}<\alpha<N\); let \(u_{\tau}(x,t)\) is solution of problem (1), (2). Then \(u_{\tau}\) is classified into one of the next three types according to the value \(\tau\) as follows: there exists \(\tau_1 \in (0,\infty)\) such that: 1) \(u_{\tau}\) blows up completely in finite time if \(\tau>\tau_1\), 2) \(u_{\tau}\) blows up incompletely in finite time and \(\|u_{\tau}(\cdot,t)\|_{L^{\infty}(\mathbb R^N)}=O(t^{-1/(p-1)})\) as \(t\to \infty\) if \(\tau=\tau_1\), 3) \(u_{\tau}\) does not blow up in finite time and \(\|u_{\tau}(\cdot,t)\|_{L^{\infty}(\mathbb R^N)}=O(t^{-1/(p-1)})\) as \(t\to \infty\) if \(0<\tau<\tau_1\).