Nonlinear Systems Of Partial Differential Equations Applications To Life And Physical Sciences
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Nonlinear Systems of Partial Differential Equations
Author: Anthony W. Leung
language: en
Publisher: World Scientific Publishing Company
Release Date: 2009
1. Positive solutions for systems of two equations. 1.1. Introduction. 1.2. Strictly positive coexistence for diffusive prey-predator systems. 1.3. Strictly positive coexistence for diffusive competing systems. 1.4. Strictly positive coexistence for diffusive cooperating systems. 1.5. Stability of steady-states as time changes -- 2. Positive solutions for large systems of equations. 2.1. Introduction. 2.2. Synthesizing large (biological) diffusive systems from smaller subsystems. 2.3. Application to epidemics of many interacting infected species. 2.4. Conditions for coexistence in terms of signs of principal eigenvalues of related single equations, mixed boundary data. 2.5. Positive steady-states for large systems by index method. 2.6. Application to reactor dynamics with temperature feedback -- 3. Optimal control for nonlinear systems of partial differential equations. 3.1. Introduction and preliminary results for scalar equations. 3.2. Optimal harvesting-coefficient control of steady-state prey-predator diffusive Volterra-Lotka systems. 3.3. Time-periodic optimal control for competing parabolic systems. 3.4. Optimal control of an initial-boundary value problem for fission reactor systems. 3.5. Optimal boundary control of a parabolic problem -- 4. Persistence, upper and lower estimates, blowup, cross-diffusion and degeneracy. 4.1. Persistence. 4.2. Upper-lower estimates, attractor set, blowup. 4.3. Diffusion, self and cross-diffusion with no-flux boundary condition. 4.4. Degenerate and density-dependent diffusions, non-extinction in highly spatially heterogenous environments -- 5. Traveling waves, systems of waves, invariant manifolds, fluids and plasma. 5.1. Traveling wave solutions for competitive and monotone systems. 5.2. Positive solutions for systems of wave equations and their stabilities. 5.3. Invariant manifolds for coupled Navier-stokes and second order wave equations. 5.4. Existence and global bounds for fluid equations of plasma display technology
Nonlinear Systems Of Partial Differential Equations: Applications To Life And Physical Sciences
The book presents the theory of diffusion-reaction equations starting from the Volterra-Lotka systems developed in the eighties for Dirichlet boundary conditions. It uses the analysis of applicable systems of partial differential equations as a starting point for studying upper-lower solutions, bifurcation, degree theory and other nonlinear methods. It also illustrates the use of semigroup, stability theorems and W2ptheory. Introductory explanations are included in the appendices for non-expert readers.The first chapter covers a wide range of steady-state and stability results involving prey-predator, competing and cooperating species under strong or weak interactions. Many diagrams are included to easily understand the description of the range of parameters for coexistence. The book provides a comprehensive presentation of topics developed by numerous researchers. Large complex systems are introduced for modern research in ecology, medicine and engineering.Chapter 3 combines the theories of earlier chapters with the optimal control of systems involving resource management and fission reactors. This is the first book to present such topics at research level. Chapter 4 considers persistence, cross-diffusion, and boundary induced blow-up, etc. The book also covers traveling or systems of waves, coupled Navier-Stokes and Maxwell systems, and fluid equations of plasma display. These should be of interest to life and physical scientists.
Nonlinear PDEs
Author: Marius Ghergu
language: en
Publisher: Springer Science & Business Media
Release Date: 2011-10-21
The emphasis throughout the present volume is on the practical application of theoretical mathematical models helping to unravel the underlying mechanisms involved in processes from mathematical physics and biosciences. It has been conceived as a unique collection of abstract methods dealing especially with nonlinear partial differential equations (either stationary or evolutionary) that are applied to understand concrete processes involving some important applications related to phenomena such as: boundary layer phenomena for viscous fluids, population dynamics,, dead core phenomena, etc. It addresses researchers and post-graduate students working at the interplay between mathematics and other fields of science and technology and is a comprehensive introduction to the theory of nonlinear partial differential equations and its main principles also presents their real-life applications in various contexts: mathematical physics, chemistry, mathematical biology, and population genetics. Based on the authors' original work, this volume provides an overview of the field, with examples suitable for researchers but also for graduate students entering research. The method of presentation appeals to readers with diverse backgrounds in partial differential equations and functional analysis. Each chapter includes detailed heuristic arguments, providing thorough motivation for the material developed later in the text. The content demonstrates in a firm way that partial differential equations can be used to address a large variety of phenomena occurring in and influencing our daily lives. The extensive reference list and index make this book a valuable resource for researchers working in a variety of fields and who are interested in phenomena modeled by nonlinear partial differential equations.