On the growth of mass for a viscous Hamilton-Jacobi equation
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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 0∈L 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 t=Δu+u p .
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© Hebrew University of Jerusalem 2003