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Nonlinear PDEs: Mathematical Models in Biology, Chemistry and Population Genetics

Marius Ghergu and Vicențiu D. Rădulescu
Publication Date: 
Number of Pages: 
Springer Monographs in Mathematics
[Reviewed by
Dhruba Adhikari
, on

This is a unique treatise on the application of advanced nonlinear operator theory. The book starts with the assumption that the reader is well-versed in fundamentals of functional analysis and partial differential equations. The book has heavy emphasis on applications of the methods of nonlinear analysis in areas such as Biology, Chemistry and Population Genetics, covering an extensive amount of material in a single volume. This treatment is the only one of its kind.

Both variational and topological method are extensively utilized. Elliptic operators in divergence form, singular elliptic problems, quasilinear elliptic equations, higher order elliptic problems involving the polyharmonic operator, reaction-diffusion systems, chemical reactions are some of the topics very eloquently covered in the book. The chapter on reaction-diffusion models has challenging mathematical problems coming from various models, with the nonlinearities being derived from chemical reaction formulas and pattern-formation of spatial tissue structures in morphogenesis. The appendix includes technical results and is extremely useful. The importance of mathematics in biosciences is very well supported by a great number of detailed examples in the text.

The book can used for advanced graduate courses as well as a reference for advanced researchers in applied sciences. The authors have accomplished a wonderful mathematical journey.

Dhruba Adhikari is Assistant Professor of Mathematics at Southern Polytechnic State University, Marietta, Georgia

​Viorel Barbu: Foreword.- 1.Overview of merhods in PDEs.- 2.Liouville Type Theorems for Elliptic Operators in Divergence Form.- 3.Blow-up Boundary Solutions.- 4.Singular Lane-Emden-Fowler Equations and Systems.- 5.Singular Elliptic Inequalities in Exterior Domains.- 6.Two Quasilinear Elliptic Problems.- 7.Some Classes of Polyharmonic Problems.- 8.Large Time Behavior of Solutions for Degenerate Parabolic Equations.- 9.Rection-Diffusion Systems in Chemistry.- 10.Pattern Formation and Gierer-Meinhardt Model.- Appendices.- References.- Index.