Super and ultracontractive bounds for doubly nonlinear evolution equations. (English) Zbl 1103.35021

Bonforte, Matteo; Grillo, Gabriele

Rev. Mat. Iberoam. 22, No. 1, 111-129 (2006).

Summary: We use logarithmic Sobolev inequalities involving the \(p\)-energy functional to prove \(L^p\) – \(L^q\) smoothing and decay properties, of supercontractive and ultracontractive type, for the semigroups associated to doubly nonlinear evolution equations of the form \(\dot u=\Delta_p (u^m)\) (with \((m(p-1)\geq 1)\) in an arbitrary Euclidean domain, homogeneous Dirichlet boundary conditions being assumed. The bounds are of the form \(\|u(t)\|_q\leq C\|u_0\|^\gamma_r/ t^\beta\) for any \(r\leq q\in[1,+\infty]\) and \(t>0\) and the exponents \(\beta, \gamma\) are shown to be the only possible for a bound of such type.

MSC:

35B45	A priori estimates in context of PDEs
35K55	Nonlinear parabolic equations
35K65	Degenerate parabolic equations
35K60	Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations

Keywords:

\(L^p-L^q\) smoothing; homogeneous Dirichlet boundary conditions