The asymptotic behavior of solutions of some doubly degenerate nonlinear parabolic equations. (English) Zbl 0848.35067

Yang, Jinshun; Zhao, Junning

Northeast. Math. J. 11, No. 2, 241-252 (1995).

We study the large time behavior of nonnegative solutions for the Cauchy problem \[ u_t= \text{div} (|Du^m|^{p- 2} Du^m)- u^q\quad \text{in } S= \mathbb{R}^N\times (0, \infty),\quad u(x, 0)= \varphi(x)\quad \text{on } \mathbb{R}^n. \] Here \(p> 1\), \(m> 0\), \(q> 1\), \(N\geq 1\) and \(\varphi\in L^1(\mathbb{R}^N)\) is a nonnegative function. The existence of a solution, defined in some weak sense, is well established. In this paper, we are interested in the behavior of solutions as \(t\to \infty\).

MSC:

35K65	Degenerate parabolic equations
35B40	Asymptotic behavior of solutions to PDEs
35K15	Initial value problems for second-order parabolic equations

Keywords:

large time behavior