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Extinction and non-extinction for viscous Hamilton-Jacobi equations in N . (English) Zbl 1046.35053

Consider the viscous Hamilton-Jacobi equation

u t -Δu+|u| p =0

in N ×(0,) 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 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|x| p/(1-p) u 0 (x)<, 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 , for 0<p<1+(N+1) -1 . The proofs rely on comparison arguments.

MSC:
35K55Nonlinear parabolic equations