Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium TypeOUP Oxford, 3 ago 2006 - 248 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 questions are 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. |
Índice
| 1 | |
Estimates for the PMEFDE | 7 |
Study of the subcritical FDE | 73 |
Extensions and appendices | 187 |
Appendices | 201 |
| 217 | |
| 232 | |
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Términos y frases comunes
Actually allows analysis apply approximation assume asymptotic behaviour bounded calculation called Chapter comparison complete concentration conclude condition consider constant continuous convergence corresponding critical decay defined depends derive detail diffusion dimension discussed equation estimates evolution example Exercise existence explicit exponent extended extinction fact fast diffusion finite fixed flow formula function given gives hence holds implies important inequality infinity initial data integrable interesting limit mass maximal means Moreover non-negative nonlinear norm Note obtain orbit p(Rn positive possible precise present problem proof properties proved question radially range reader refer result satisfy scaling self-similar solutions similar singular smoothing effect space standard symmetric Theorem theory transformation true unique variable weak zero ZKB solutions