Super and ultracontractive bounds for doubly nonlinear evolution equations

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 u· = ?p(um) (with m(p - 1) = 1) in an arbitrary euclidean domain, homogeneous Dirichlet boundary conditions being assumed. The bound are of the form ||u(t)||q = C||u0||r? / ts for any r = q I [1,+8) and t > 0 and the exponents s, ? are shown to be the only possible for a bound of such type.