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- W2037693397 abstract "It is proved that solutions of the porous medium equation $u_t = Delta (| u |^{m - 1} u),m > 1$, defined in $Q = mathbb{R}^N times (0,infty )$ with initial data $u(x,0)$ integrable, compactly supported, and with changing sign, become nonnegative in finite time if $int {u_0 (x)} dx > 0$. Precise asymptotic convergence rates then follow. The positivity result is related to finite propagation and is false for the heat equation, $m = 1$. Nevertheless, we obtain asymptotic convergence rates for ${{1 geq m > (N - 2)_ + } / N}$ as $t to infty $. The same analysis works for the equation $u_t = {operatorname{div}}(| {nabla u} |^{p - 1} nabla u)$ with $p > 1$. The case of zero mass, $int {u_0 (x)} dx > 0$, is also studied for the porous medium equation (in $N = 1$) and the solution is shown to converge to an antisymmetric profile in that case." @default.
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- W2037693397 date "1991-01-01" @default.
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- W2037693397 title "Asymptotic Behaviour of Solutions of the Porous Medium Equation with Changing Sign" @default.
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