Super and ultracontractive bounds for doubly nonlinear evolution equations

Abstract

We use logarithmic Sobolev inequalities involving the p-energy functional recently derived in [15], [21] to prove Lp-Lq smoothing and decay properties, of supercontractive and ultracontractive type, for the semigroups associated to doubly nonlinear evolution equations of the form u̇= Δp(um) (with (m(p - 1) ≥ 1) in an arbitrary euclidean domain, homogeneous Dirichlet boundary conditions being assumed. The bounds are of the form ∥u(t)∥q ≤ C∥u 0∥τγ/tβ for any r ≤ q ∈ [1, +∞] and t > 0 and the exponents β, γ are shown to be the only possible for a bound of such type.