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Extinction and non-extinction for viscous Hamilton-Jacobi equations in $\bbfR^N$. (English) Zbl 1046.35053
Consider the viscous Hamilton-Jacobi equation $$ u_t - \Delta u + \vert \nabla u\vert ^p = 0 $$ in $\Bbb{R}^N \times (0,\infty)$ with initial data $u_0(x)$. In previous work [Proc. Am. Math. Soc. 130, No. 4, 1103--1111 (2002; Zbl 1001.35007)], some of the authors showed that if $u_0 \geq 0$ 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 $\limsup \vert x\vert ^{p/(1-p)}u_0(x) < \infty$, then the solution vanishes identically for $t>T^*$. 2. If this $\limsup$ 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.

35K55Nonlinear parabolic equations