Consider the viscous Hamilton-Jacobi equation
in with initial data . In previous work [Proc. Am. Math. Soc. 130, No. 4, 1103–1111 (2002; Zbl 1001.35007)], some of the authors showed that if is bounded, continuous and integrable and the positive exponent satisfies , the solution vanishes identically after some finite time . In the present paper, this result is strengthened and refined, with the following results: 1. If and , then the solution vanishes identically for . 2. If this is not finite, the solution remains positive for all positive times. 3. Solutions with -periodic initial data always stabilize at spatial constants after some finite time, for any . 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 and , for . The proofs rely on comparison arguments.