, Volume 89, Issue 1, pp 367-383

On the growth of mass for a viscous Hamilton-Jacobi equation

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Abstract

We investigate the large time behavior of positive solutions with finite mass for the viscous Hamilton-Jacobi equationu t = Δu + |Δu| p ,t>0,x ∈ ℝ N , wherep≥1 andu(0,.)=u 0≥0,u 0≢0,u 0L 1. DenotingI =lim t→∞u(t)1≤∞, we show that the asymptotic behavior of the mass can be classified along three cases as follows:

  • • ifp≤(N+2)/(N+1), thenI =∞ for allu 0;

  • • if (N+2)/(N+1)<p<2, then bothI =∞ andI <∞ occur;

  • • ifp≥2, thenI <∞ for allu 0.

  • We also consider a similar question for the equationu tu+u p .