×

Global and non-global solutions of a nonlinear parabolic equation. (English) Zbl 1086.34008

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.

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
PDFBibTeX XMLCite
Full Text: DOI