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Application of monotone type operators to parabolic and functional parabolic PDE’s. (English) Zbl 1291.35006
Dafermos, C.M.(ed.) et al., Handbook of differential equations: Evolutionary equations. Vol. IV. Amsterdam: Elsevier/North-Holland (ISBN 978-0-444-53034-9/hbk). Handbook of Differential Equations, 267-321 (2008).
From the introduction: “The aim of these lecture notes is to give a short introduction to the theory of monotone type operators, and by using this theory to consider abstract stationary and evolution equations with operators of this type. Then the abstract theory is applied to “weak” solutions of nonlinear elliptic, parabolic, functional parabolic, hyperbolic and functional hyperbolic equations of “divergence type”. By using the theory of monotone type operators, it is possible to treat several types of nonlinear partial differential equations (not only semilinear PDEs) and to prove global existence of solutions of time dependent problems. However, there are a lot of problems in physics, chemistry, biology etc. the mathematical models of which are nonlinear PDEs but the monotone type operators can not be applied to them. These equations need particular treatment.” \(\ldots\)
“In Chapter 1 we consider nonlinear stationary problems and as particular cases nonlinear elliptic differential equations, functional equations and variational inequalities. In Chapter 2 first order evolution equations and as particular cases nonlinear parabolic differential equations, functional parabolic equations are considered. Finally, in Chapter 3 second order nonlinear evolution equations and certain nonlinear hyperbolic equations are treated. In each chapter the “general” results are illustrated by examples.”
For the entire collection see [Zbl 1173.35002].

35A16 Topological and monotonicity methods applied to PDEs
34G20 Nonlinear differential equations in abstract spaces
35J60 Nonlinear elliptic equations
35K55 Nonlinear parabolic equations
35K90 Abstract parabolic equations
35R10 Partial functional-differential equations
35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
47H05 Monotone operators and generalizations