Ultracontractive Bounds for Nonlinear Evolution Equations Governed by the Subcritical p-Laplacian
We consider the equation Open image in new window = Δp(u) with 2 ≤ p < d on a compact Riemannian manifold. We prove that any solution u(t) approaches its (time-independent) mean ū with the quantitative bound Open image in new window for any q ∊ [2, +∞] and t > 0 and the exponents β, γ are shown to be the only possible for a bound of such type. The proof is based upon the connection between logarithmic Sobolev inequalities and decay properties of nonlinear semigroups.
KeywordsContractivity properties asymptotics of nonlinear evolutions p-Laplacian on manifolds
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