Abstract Cauchy Problems

Although the theory of well-posed Cauchy problems is reasonably understood, ill-posed problems involved in numerous mathematical models in physics, engineering, and finance can be approached in a variety of ways. The authors structured these approaches, identifying three main strategies for dealing with ill‑posed problems: semigroup methods, abstract distribution methods, and regularization methods. Semigroup and distribution methods restore well-posedness, in a modern weak sense. Regularization methods provide approximate solutions to ill-posed problems. Although these approaches have been extensively developed by many researchers, a comprehensive treatment of all three approaches has hitherto been unforthcoming.

Abstract Cauchy Problems: Three Approaches, now in its second edition, provides an innovative, self-contained account of these methods and, furthermore, demonstrates and studies some of the profound connections between them. The authors discuss the application of different methods not only to the Cauchy problem that is not well-posed in the classical sense, but also to important generalizations: the Cauchy problem for inclusion and the Cauchy problem for second order equations.

The second edition addresses several recent developments not included in the original book, including new applications of semigroup, distribution and regularization methods to partial differential equations, to pseudo-differential equations and to stochastic problems.

The second edition continues to be useful for mathematical modellers in physics, biology, engineering and finance, and to be accessible to non­specialists and graduate students.

Irina V. Melnikova is a professor at the Ural Federal University, Ekaterinburg, Russia. She teaches both core mathematical courses and advanced specialised electives. Her research interests include ill-posed problems, generalized functions, operator semigroup theory, and stochastic analysis. She has over 150 research publications, including two monographs published by CHAPMAN & HALL/CRC.

Alexei Filinkov has a PhD in Mathematics (Differential Equations and Mathematical Physics) from the Ural State University, Russia. From 1995 to 2002 he undertook research and leadership roles in a variety of research projects at the University of Adelaide in the areas of risk analysis, financial modelling, ill-posed problems, stochastic analysis and generalised functions. From 2003 to 2024 he was a research scientist at the Defence Science and Technology Group (Australian Department of Defence) where he led projects focussed on risk-based approaches to defence and national security problems.

Currently he is an Honorary Principal Research Fellow at Torrens Resilience Initiative (Flinders University) and an Adjunct Senior Research Fellow at Adelaide University. Here, his research interests include investigation of risk and resilience paradigms for complex societal systems, and development of innovative mathematical tools for studying quantum phenomena.