Asymptotic behavior of the nonlinear diffusion equation n t = (n−1n x)x

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Abstract
The asymptotic behavior of the equation nt = (ln n)xx is studied on the finite interval 0⩽x⩽1 with the boundary conditions n(0,t) = n(1,t) = n0 and initial data n(x,0)⩽n0. We prove that asymptotically ln[n(x,t)/n0]→A exp(−π2t/n0)21/2 sin πx and also provide rigorous upper and lower bounds on the asymptotic amplitude A in terms of integrals of nonlinear functions of the initial data. The rigorous bounds are compared to values of A obtained from computer experiments. The lower bound L = (23/2/π)exp[li(1+Q)−γ], where li is the logarithmic integral, γ is Euler’s constant, and Q = (π/2)F[n(x,0)/n0−1]sin πx dx, is found to be the best known estimate of A.