Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type
OUP Oxford, 3 ago 2006 - 234 páginas
This text is concerned with the quantitative aspects of the theory of nonlinear diffusion equations; equations which can be seen as nonlinear variations of the classical heat equation. They appear as mathematical models in different branches of Physics, Chemistry, Biology, and Engineering, and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on estimates and functional analysis.Concentrating on a class of equations with nonlinearities of power type that lead to degenerate or singular parabolicity ("equations of porous medium type"), the aim of this text is to obtain sharp a priori estimates and decay rates for general classes of solutions in terms of estimates of particular problems. These estimates are the building blocks in understanding the qualitative theory, and the decay rates pave the way to the fine study of asymptotics. Many technically relevant questionsare presented and analyzed in detail. A systematic picture of the most relevant phenomena is obtained for the equations under study, including time decay, smoothing, extinction in finite time, and delayed regularity.
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Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations ...
Juan Luis Vázquez
Vista previa restringida - 2006
analysis asymptotic behaviour backward effect best constant blow-up bounded boundedness Cauchy problem Chapter comparison convergence critical exponent critical line decay rate defined diffusion equation dimension Dirac mass estimates evolution Exercise existence exponent fast diffusion equation finite flux formula function Harnack inequality heat equation infinity initial data u0 integrable L1loc Rn Lemma limit logarithmic diffusion Lp norm Lp spaces Lp(Rn Marcinkiewicz spaces maximal solutions monotone Moreover Mp Rn Mp(Rn non-negative solutions nonlinear nonlinear diffusion Note obtain orbit p(Rn parameters phase plane phase-plane PME/FDE porous medium positive precise proof properties proved radially symmetric range result Ricci flow scaling Section self-similar solutions semigroup singular singular solution solutions with data source solution strong smoothing effect Subsection Theorem theory transformation unique variable weak smoothing effect Yamabe problem zero ZKB solutions