کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
9495944 1335202 2005 30 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Asymptotics of the porous media equation via Sobolev inequalities
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات اعداد جبر و تئوری
پیش نمایش صفحه اول مقاله
Asymptotics of the porous media equation via Sobolev inequalities
چکیده انگلیسی
Let 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.
ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Functional Analysis - Volume 225, Issue 1, 1 August 2005, Pages 33-62
نویسندگان
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