*(English)*Zbl 1046.35053

Consider the viscous Hamilton-Jacobi equation

in ${\mathbb{R}}^{N}\times (0,\infty )$ with initial data ${u}_{0}\left(x\right)$. In previous work [Proc. Am. Math. Soc. 130, No. 4, 1103–1111 (2002; Zbl 1001.35007)], some of the authors showed that if ${u}_{0}\ge 0$ is bounded, continuous and integrable and the positive exponent $p$ satisfies $p<N/(N+1)$, the solution vanishes identically after some finite time ${T}^{*}$. In the present paper, this result is strengthened and refined, with the following results: 1. If $0<p<1$ and ${lim\; sup\left|x\right|}^{p/(1-p)}{u}_{0}\left(x\right)<\infty $, then the solution vanishes identically for $t>{T}^{*}$. 2. If this $lim\; sup$ is not finite, the solution remains positive for all positive times. 3. Solutions with ${Z}^{n}$-periodic initial data always stabilize at spatial constants after some finite time, for any $0<p<1$. 4. However, for general non-negative initial data, solutions are not expected to stabilize at spatial constants after finite times.

In addition, the paper contains a number of results on the asymptotic behavior of solutions of this problem in ${L}^{1}$ and ${L}^{\infty}$, for $0<p<1+{(N+1)}^{-1}$. The proofs rely on comparison arguments.

##### MSC:

35K55 | Nonlinear parabolic equations |