to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the underlying equations. The Keller-Segel model of chemotaxis (Keller and Segel in J Theor Biol 26:399–415, 1970; 30:225–234, 1971) has provided a cornerstone for much of this work, its success being a consequence of its intuitive simplicity, analytical tractability and capacity to replicate key behaviour of chemotactic populations. One such property, the ability to display “auto-aggregation”, has led to its prominence as a mechanism for self-organisation of biological systems. This phenomenon has been shown to lead to finite-time blow-up under certain formulations of the model, and a large body of work has been devoted to determining when blow-up occurs or whether globally existing solutions exist. In this paper, we explore in detail a number of variations of the original Keller–Segel model. We review their formulation from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form. We conclude with a brief discussion and expand on some of the outstanding issues revealed as a result of this work.

/pdf/a-user-s-guide-to-pde-models-for-chemotaxis-4zxd8lr4pw.pdf

A user’s guide to PDE models for chemotaxis

This book presents methods to study the controllability and the stabilization of nonlinear control systems in finite and infinite dimensions. The emphasis is put on specific phenomena due to nonlinearities. In particular, many examples are given where nonlinearities turn out to be essential to get controllability or stabilization. Various methods are presented to study the controllability or to construct stabilizing feedback laws. The power of these methods is illustrated by numerous examples coming from such areas as celestial mechanics, fluid mechanics, and quantum mechanics. The book is addressed to graduate students in mathematics or control theory, and to mathematicians or engineers with an interest in nonlinear control systems governed by ordinary or partial differential equations.

/pdf/control-and-nonlinearity-4tu8aduvuy.pdf

Control and Nonlinearity

Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model

We consider two simple conservative systems of parabolic-elliptic and parabolic-degenerate type arising in modeling chemotaxis and angiogenesis. Both systems share the same property that when the

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 $$L^{\frac{d}
{2}} $$
 norm of initial data is small enough, where d ≥ 2 is the space dimension, then there is a global (in time) weak solution that stays in all the L
 p
 spaces with max

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 $$\left\{ {\left. {1;\frac{d}
{2} - 1} \right\} \leq p < \infty .} \right.$$
 This result is already known for the parabolic-elliptic system of chemotaxis, moreover blow-up can occur in finite time for large initial data and Dirac concentrations can occur. For the parabolic-degenerate system of angiogenesis in two dimensions, we also prove that weak solutions (which are equi-integrable in L1) exist even for large initial data. But break-down of regularity or propagation of smoothness is an open problem.

/pdf/global-solutions-of-some-chemotaxis-and-angiogenesis-systems-1oek7w6wb7.pdf

Global Solutions of Some Chemotaxis and Angiogenesis Systems in High Space Dimensions

We first describe all positive bounded solutions of ∂w ∂s = ∆w − 1 2 y.∇w − w p− 1 + w . where (y, s) ∈ RN × R, 1 < p and (N − 2)p ≤ N + 2. We then obtain for blow-up solutions u(t) of ∂u ∂t = ∆u+ u uniform estimates at the blow-up time and uniform space-time comparison with solutions of u′ = up.

Optimal estimates for blowup rate and behavior for nonlinear heat equations

Stability of the blow-up profile for equations of the type ut=Δu+|u|p−1u

In this paper, we find the optimal blow-up rate for the semilinear wave equation with a power nonlinearity. The exponent p is superlinear and less than 1 + 4 N−1 if N ≥ 2.

/pdf/determination-of-the-blow-up-rate-for-the-semilinear-wave-2ccod862mt.pdf

Determination of the blow-up rate for the semilinear wave equation

In this paper, we determine the blow-up rate for the semilinear wave equation with critical power nonlinearity related to the conformal invariance.

/pdf/determination-of-the-blow-up-rate-for-a-critical-semilinear-1fjbpdyvo9.pdf

Hatem Zaag

Papers

Global Solutions of Some Chemotaxis and Angiogenesis Systems in High Space Dimensions

Optimal estimates for blowup rate and behavior for nonlinear heat equations

Stability of the blow-up profile for equations of the type ut=Δu+|u|p−1u

Determination of the blow-up rate for the semilinear wave equation

Determination of the blow-up rate for a critical semilinear wave equation