Asymptotic behavior of global solutions to a class of heat equations with gradient nonlinearity

Abstract

The paper is devoted to investigating a semilinear parabolic equation with a nonlinear gradient source term: \begin{document}$ u_t = u_{xx}+x^m|u_x|^p, t>0, 0 where \begin{document}$ p>m+2 $\end{document} , \begin{document}$ m\geq0 $\end{document} . Zhang and Hu [Discrete Contin. Dyn. Syst. 26 (2010) 767-779] showed that finite time gradient blowup occurs at the boundary and the accurate blowup rate is also obtained for super-critical boundary value. Throughout this paper, we present a complete large time behavior of a classical solution \begin{document}$ u $\end{document} : \begin{document}$ u $\end{document} is global and converges to the unique stationary solution in \begin{document}$ C^1 $\end{document} norm for subcritical boundary value, and \begin{document}$ u_x $\end{document} blows up in infinite time for critical boundary value. Gradient growup rate is also established by the method of matched asymptotic expansions. In addition, gradient estimate of solutions is obtained by the Bernstein-type arguments.