Global and non-global solutions of a nonlinear parabolic equation (Q2568562)

The aim of this paper is to consider the existence of global and nonglobal positive solutions of the nonlinear parabolic equation \[ u_t=u^{\sigma}(\Delta u+u^p),\quad x\in{\mathbb R}^n,\quad t>0, \] with \(\sigma,p>1\) and \(n\geq 1\). For this purpose, the authors investigate the positive solution of the suitably defined problem \[ \varphi ''+(n-1)/\xi \varphi '+\varphi^p+\alpha\varphi^{1-\sigma}+ \beta\xi\varphi^{-\sigma}\varphi '=0,\quad \xi>0, \] with the initial conditions \[ \varphi '(0)=0,\quad \varphi(0)=\eta, \] where \(\xi, \alpha, \beta\) are defined and \(\eta>0\) is given. It is proved that the solution of the above problem exists globally for any \(\eta>0\). Next, the asymptotic behaviour of a positive solution to this problem, as \(\xi\to\infty\), is considered. It is proved that \(\lim_{\xi\to\infty}\{\xi^{\alpha/\beta}\varphi(\xi)\}\) exists and is positive. Moreover, these global solutions tend to zero as \(t\to\infty\). Finally, the authors give main conclusions concerning a symmetric positive monotone self-similar solution to the parabolic equation.