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Full Description
Unlike the classical Sturm theorems on the zeros of solutions of second-order ODEs, Sturm's evolution zero set analysis for parabolic PDEs did not attract much attention in the 19th century, and, in fact, it was lost or forgotten for almost a century. Briefly revived by Pólya in the 1930's and rediscovered in part several times since, it was not until the 1980's that the Sturmian argument for PDEs began to penetrate into the theory of parabolic equations and was found to have several fundamental applications.
Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications focuses on geometric aspects of the intersection comparison for nonlinear models creating finite-time singularities. After introducing the original Sturm zero set results for linear parabolic equations and the basic concepts of geometric analysis, the author presents the main concepts and regularity results of the geometric intersection theory (G-theory). Here he considers the general singular equation and presents the geometric notions related to the regularity and interface propagation of solutions. In the general setting, the author describes the main aspects of the ODE-PDE duality, proves existence and nonexistence theorems, establishes uniqueness and optimal Bernstein-type estimates, and derives interface equations, including higher-order equations. The final two chapters explore some special aspects of discontinuous and continuous limit semigroups generated by singular parabolic equations.
Much of the information presented here has never before been published in book form. Readable and self-contained, this book forms a unique and outstanding reference on second-order parabolic PDEs used as models for a wide range of physical problems.
Contents
Sturm Theorems for Linear Parabolic Equations and Intersection Comparison. Transversality, Concavity and Sign-Invariants. B-Concavity and Transversality on Nonlinear Subsets for Quasilinear Heat Equations. Eventual B-convexity: a Criterion of Complete Blow-up and Extinction for Quasilinear Heat Equations. Blow-up Interfaces for Quasilinear Heat Equations. Complete and Incomplete Blow-up in Several Space Dimensions. Geometric Theory of Nonlinear Singular Parabolic Equations. Intersection Techniques in Generalized Free-Boundary Problems. Intersection Techniques for Solutions Changing Sign. Discontinuous Limit Semigroups for the Singular Zhang Equation and Generalizations. Further Examples of Discontinuous and Continuous Limit Semigroups.
