Periodic-parabolic boundary value problems and positivity. (English) Zbl 0731.35050

Pitman Research Notes in Mathematics Series, 247. Harlow: Longman Scientific & Technical; New York: John Wiley & Sons, Inc. 139 p. £17.00 (1991).
In this interesting and well written book, the author discusses monotone discrete time dynamical systems on Banach spaces (monotone in the sense of order) and their applications to linear and nonlinear parabolic problems where the coefficients depend periodically on time. More specifically, he looks at the problem of the form \[ \partial u/\partial t+A(t)u=f(x,t,u,\nabla u)\text{ on } \Omega \times (0,\infty) \] on a bounded domain \(\Omega\) with an appropriate boundary condition. Here A(t) is a second order linear elliptic operator. Moreover the coefficients of A and f are T-periodic in t. He also studies some systems. Throughout he tries to introduce carefully mathematical techniques the reader may not be familiar with.
In Chapter 1, he discusses abstract discrete (and occasionally continuous) monotone dynamical systems. In particular, he considers the existence of stable fixed points. In Chapter 2, he studies the scalar linear problem including the eigenvalue problem (possibly with weight changing sign). In Chapter 3, the nonlinear scalar problem is examined, including a discussion of the existence of stable T-periodic solutions. In particular he studies Fisher’s equation when the coefficients are periodic in t. In Chapters 4 and 5, he studies Lotka-Volterra competition and predator-prey type systems of two equations when the coefficients are time periodic.
Since the book was written, a number of results have been proved. Firstly, P. Polacik and I. Terescak [preprint, Comenius Univ. (1991)] have shown that there is generic convergence to cycles (that is periodic solutions) for a class of monotone discrete dynamical systems which include those coming from time periodic parabolic equations. Takc [preprint, Vanderbilt Univ. (1990)] showed that stable cycles can occur for the scalar parabolic problem when the linear part is not self- adjoint. More recently, the author and the reviewer (work in preparation) have shown that stable cycles can still occur if the linear part is self- adjoint and f does not depend on the gradient. In addition, the reviewer has shown that uniqueness fails for the time periodic predator-prey problem with Neumann boundary conditions [preprint, Univ. New England (1990)].


35K55 Nonlinear parabolic equations
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
34G20 Nonlinear differential equations in abstract spaces
35B10 Periodic solutions to PDEs
47J05 Equations involving nonlinear operators (general)