Potential theory and optimal convergence rates in fast nonlinear diffusion (Q864177)

The paper studies the asymptotic behavior as \(t\to \infty \) of solutions \(u(x,t)\) to the nonlinear diffusion equation \(\partial u/ \partial t=\Delta (u^m)\), \(u(x,0)= u_0(x)\) on the whole space \(x\in \mathbb R^n\), \(n\geq 1\), where \(\max (0,1- 2/n)<m<1\). The initial value \(u_0(x)\) is nonnegative and integrable. Let \(\rho (x,t)\) be a solution of \(\partial \rho /\partial t=\Delta (\rho ^m)\), \(\rho (x,0)=\delta (x)\), \(x\in \mathbb R^n\), \(t>0\). Under certain assumption on \(u_0\) it is proved that \(\limsup [t\int | u(x,t) -\rho (x,t)| \,dx ]\leq \limsup [t\, \text{ess\,sup}\{ |u(x,t)-\rho (x,t)| /| \rho (x,t)|\); \(x\in \mathbb R^n\} ]<\infty \) as \(t\to \infty \).