Generalizations of logarithmic Sobolev inequalities

Abstract

We generalize logarithmic Sobolev inequalities to logarithmic Gagliardo-Nirenberg inequalities, and apply these inequalities to prove ultracontractivity of the semigroup generated by the doubly nonlinear p-Laplacian u = ∆pu m .Our proof does not use Moser iteration, but shows that the time-dependent Lebesgue norm u(t) r(t) stays bounded for a variable exponent r(t) blowing up in arbitrary short time.

Fractional logarithmic inequalities and blow-up results with logarithmic nonlinearity on homogeneous groups

“…Then this inequality was extended to homogeneous groups in [14]. Also, the logarithmic Gagliardo-Nirenberg inequality with s = 1 was proved in [15] and its fractional version was proved in [9]. In this paper we obtain the fractional logarithmic Gagliardo-Nirenberg inequality on the homogeneous groups.…”

“…In the homogeneous group setting, the fractional Sobolev inequality was obtained in [12]. In [15], the author proved the logarithmic Sobolev inequality in the following form:…”

“…In the Abelian (Euclidean) case G = (R N , +), we have Q = N and | • | = | • | E (| • | E is the Euclidean distance), if s → 1 − and from (3.19) we get the logarithmic Sobolev inequality in[15]. ([14], Fractional CKN inequality).…”

Fractional Gagliardo–Nirenberg and Hardy inequalities under Lorentz norms

“…Merker [16,17] has studied (1.4) for s = 1 and applied in doubly nonlinear diffusion equations. Cotsiolis and Tavoularis in [18] established (1.5) for p = 2.…”

Existence of weak solutions to doubly degenerate diffusion equations

“…In [7], [5], [12] it is shown that weak solutions of doubly nonlinear diffusion equations with initial values u(0) ∈ L m ′ are instantly regularized to functions u(t) ∈ L ∞ for t > 0. Using boundedness of u(t), in the next step Hölder continuity of weak solutions in space can be verified (see e.g.…”