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**1 - 2**of**2**### SHARP ASYMPTOTIC BEHAVIOR FOR THE SOLUTIONS OF DEGENERATE AND SINGULAR PARABOLIC EQUATIONS

"... Abstract. We study the asymptotic behavior, as t → ∞, of the solutions to the evolutionary p-Laplace equation vt = div(|∇v|p−2∇v) with time-independent lateral boundary values. We obtain the sharp decay rate of maxx∈Ω|v(x, t) − u(x)|, where u is the stationary solution, both in the degenerate case ..."

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Abstract. We study the asymptotic behavior, as t → ∞, of the solutions to the evolutionary p-Laplace equation vt = div(|∇v|p−2∇v) with time-independent lateral boundary values. We obtain the sharp decay rate of maxx∈Ω|v(x, t) − u(x)|, where u is the stationary solution, both in the degenerate case p> 2 and in the singular case 1 < p < 2. A key tool in the proofs is the Moser iteration, which is applied to the difference v(x, t) − u(x). In the singular case we construct an example proving that the celebrated phenomenon of finite extinction time, valid for v(x, t) when u ≡ 0, does not have a counterpart for v(x, t) − u(x). 1.