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