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Extinction and non-extinction for viscous Hamilton-Jacobi equations in ${ℝ}^{N}$. (English) Zbl 1046.35053

Consider the viscous Hamilton-Jacobi equation

${u}_{t}-{\Delta }u+{|\nabla u|}^{p}=0$

in ${ℝ}^{N}×\left(0,\infty \right)$ 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, 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 and ${lim sup|x|}^{p/\left(1-p\right)}{u}_{0}\left(x\right)<\infty$, 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. 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. The proofs rely on comparison arguments.

##### MSC:
 35K55 Nonlinear parabolic equations
##### Keywords:
Hamilton-Jacobi equation; extinction; stabilization