Publisher Summary
This chapter discusses the existence and stability for a class of partial functional differential equations. As a model for this class, one may take the equation wt(x, t) = wxx (x, t) + ƒ(t, w(x, t − r)), 0 ≤ x ≤ Π, t ≥ 0,

w(0, t) = w(Π, t) = 0, t ≥ 0, w(x, t) = Φ (x, t), 0 ≤ x ≤ Π, −r ≤ t ≤ 0, where ƒ is a linear or nonlinear scalar-valued function, r a positive real number, and Φ a given initial function. The second derivative term corresponds to a strongly continuous semigroup of linear operators on a Banach space of functions determined by the boundary conditions. Accordingly, the approach relies primarily on semigroup methods and the treatment of equation as an ordinary functional differential equation in a Banach space. The chapter also describes semigroup and infinitesimal generator in the autonomous case.

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Existence and stability for partial functional differential equations

The evolution problem 0∈du/dt+A(t)u(t),u(s)=x, where theA(t) are nonlinear operators acting in a Banach space, is studied. Evolution operators are constructed from theA(t) under various assumptions. Basic properties of these evolution operators are established and their relationship to the evolution equation is determined. The results obtained extend several known existence theorems and provide generalized solutions of the evolution equation in more general cases.

Nonlinear Evolution Equations in Banach Spaces.

This book gives a first systematic account on the subject of convex analysis and optimization in Hadamard spaces. It is primarily aimed at both graduate students and researchers in analysis and optimization.

Convex analysis and optimization in Hadamard spaces

Convergence of an algorithm for mean curvature motion

Aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X, d,m). On the geometric side, our new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N -dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, our new approach uses the nonlinear diffusion semigroup induced by the N -dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger’s energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong CD∗(K,N) condition of Bacher-Sturm.

Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces

Convergence and approximation of semigroups of nonlinear operators in Banach spaces

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Convergence and approximation of semigroups of nonlinear operators in Banach spaces