1,721,016 research outputs found
Super and ultracontractive bounds for doubly nonlinear evolution equations
No full textWe 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? / tß for any r = q Î [1,+8) and t > 0 and the exponents ß, ? are shown to be the only possible for a bound of such type
Asymptotics of the porous media equation via Sobolev inequalities
No full textAbstractLet M be a compact Riemannian manifold without boundary. Consider the porous media equation u˙=▵(um), u(0)=u0∈Lq, ▵ being the Laplace–Beltrami operator. Then, if q⩾2∨(m-1), the associated evolution is Lq-L∞ regularizing at any time t>0 and the bound ‖u(t)‖∞⩽C(u0)/tβ holds for t<1 for suitable explicit C(u0),γ. For large t it is shown that, for general initial data, u(t) approaches its time-independent mean with quantitative bounds on the rate of convergence. Similar bounds are valid when the manifold is not compact, but u(t) approaches u≡0 with different asymptotics. The case of manifolds with boundary and homogeneous Dirichlet, or Neumann, boundary conditions, is treated as well. The proof stems from a new connection between logarithmic Sobolev inequalities and the contractivity properties of the nonlinear evolutions considered, and is therefore applicable to a more abstract setting
Singular Evolution On Manifolds, Their Smoothing Properties, AND Sobolev Inequalities
No full textThe evolution equation [u_ = \Delta_pu] , posed on a Riemannian manifold, is studied in the singular range [p \in 2] (1; 2). It is shown that if the manifold supports a suitable Sobolev inequality, the smoothing effect [||u(t)||\infty\leq C ||u(0)||_q^\gamma] / [t^\alpha] holds true for suitable for [\alpha, \gamma] and that the converse holds if [p] is sufficiently close to 2, or in the degenerate range [p] > 2. In such ranges, the Sobolev inequality and the smoothing efect are then equivalent
Ultracontractive bounds for nonlinear evolution equations governed by the subcritical -Laplacian
No full textSpecial fast diffusion with slow asymptotics. Entropy method and flow on a Riemannian manifold
No full textReverse Smoothing Effects, Fine Asymptotics, and Harnack Inequalities for Fast Diffusion Equations
Full text linkWe investigate local and global properties of positive solutions to the fast diffusion equation ut=Δum in the good exponent range (d−2)+/d<m<1, corresponding to general nonnegative initial data. For the Cauchy problem posed in the whole Euclidean space â„Âd, we prove sharp local positivity estimates (weak Harnack inequalities) and elliptic Harnack inequalities; also a slight improvement of the intrinsic Harnack inequality is given. We use them to derive sharp global positivity estimates and a global Harnack principle. Consequences of these latter estimates in terms of fine asymptotics are shown. For the mixed initial and boundary value problem posed in a bounded domain of â„Âd with homogeneous Dirichlet condition, we prove weak, intrinsic, and elliptic Harnack inequalities for intermediate times. We also prove elliptic Harnack inequalities near the extinction time, as a consequence of the study of the fine asymptotic behavior near the finite extinction time
Smoothing Effects and Extinction in Finite Time for Fractional Fast Diffusions on Riemannian Manifolds
Full text linkWe study nonnegative solutions to the Cauchy problem for the Fractional Fast Diffusion Equation on a suitable class of connected, noncompact Riemannian manifolds. This parabolic equation is both singular and nonlocal: the diffusion is driven by the (spectral) fractional Laplacian on the manifold, while the nonlinearity is a concave power that makes the diffusion singular, so that solutions lose mass and may extinguish in finite time. Existence of mild solutions follows by nowadays standard nonlinear semigroups techniques, and we use these solutions as the building blocks for a more general class of so-called weak dual solutions, which allow for data both in the usual L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}L<^>1\end{document} space and in a larger weighted space, determined in terms of the fractional Green function. We focus in particular on a priori smoothing estimates (also in weighted Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}L<^>p\end{document} spaces) for a quite large class of weak dual solutions. We also show pointwise lower bounds for solutions, showing in particular that solutions have infinite speed of propagation. Finally, we start the study of how solutions extinguish in finite time, providing suitable sharp extinction rates
Positivity, local smoothing and Harnack inequalities for very fast diffusion equations
No full textAbstractWe investigate qualitative properties of local solutions u(t,x)⩾0 to the fast diffusion equation, ∂tu=Δ(um)/m with m<1, corresponding to general nonnegative initial data. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of the form [0,T]×Ω, with Ω⊆Rd. They combine into forms of new Harnack inequalities that are typical of fast diffusion equations. Such results are new for low m in the so-called very fast diffusion range, precisely for all m⩽mc=(d−2)/d. The boundedness statements are true even for m⩽0, while the positivity ones cannot be true in that range
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