Krzysztof P. Rybakowski

The homotopy index and partial differential equations

I The homotopy index theory.- 1.1 Local semiflows.- 1.2 The no blow-up condition. Convergence of semiflows.- 1.3 Isolated invariant sets and isolating blocks.- 1.4 Admissibility.- 1.5 Existence of isolating blocks.- 1.6 Homotopies and inclusion induced maps.- 1.7 Index and quasi-index pairs.- 1.8 Some special maps used in the construction of the Morse index.- 1.9 The Categorial Morse index.- 1.10 The homotopy index and its basic properties.- 1.11 Linear semiflows. Irreducibility.- 1.12 Continuation of the homotopy index.- II Applications to partial differential equations.- 2.1 Sectorial operators generated by partial differential operators.- 2.2 Center manifolds and their approximation.- 2.3 The index product formula.- 2.4 A one-dimensional example.- 2.5 Asymptotically linear systems.- 2.6 Estimates at zero and nontrivial solution of elliptic equations.- 2.7 Positive heteroclinic orbits of second-order parabolic equations.- 2.8 A homotopy index continuation method and periodic solutions of second-order gradient systems.- III Selected topics.- 3.1 Repeller-attractor pairs and Morse decompositions.- 3.2 Block pairs and index triples.- 3.3 A Morse equation.- 3.4 The homotopy index and Morse theory on Hilbert manifolds.- 3.5 Continuation of the categorial Morse index along paths.- Bibliographical notes and comments.

Ważewski's Principle for retarded functional differential equations☆

Abstract Wazewski Principle is an important tool in the study of the asymptotic behavior of solutions of ordinary differential equations. A direct extension of this principle to retarded functional differential equations (RFDEs) can be obtained by noticing that solutions of RFDEs generate processes on C = C ([− r , 0], R n ) and by using the general version of Wazewski Principle for flows on topological spaces. The resulting method is of little use in applications, due to the infinite-dimensionality of the space C . This paper presents a “Razumikhin-type” extension of Wazewskis Principle, which is widely applicable to concrete examples. The main results are Corollaries 3.1 and 3.2. Also, an extension of the method to RFDEs with a merely continuous right-hand side is given, and a few examples illustrate the use of the method. Throughout the paper, a standard notation is used.

The Morse index, repeller-attractor pairs and the connection index for semiflows on noncompact spaces☆

Abstract This paper extends the Morse index theory of C. C. Conley to semiflows π on a noncompact meric space X. π is assumed to satisfy a hypothesis related to conditional α-contraction. We collect background material, define quasi-index pairs and the Morse index of a compact, isolated invariant set K, and prove that the Morse index is a connected simple system. We study repeller-attractor pairs in K, define index triples, and prove their existence and several properties leading to the concepts of the connection index, the connection map and the splitting class. Finally, we consider paths (continuous families) of pairs (π, K) and study continuations of the Morse and the connection indices along such paths. The present paper is a sequel to the authors previous work: On the homotopy index for infinite-dimensional semiflows (Trans. Amer. Math. Soc. 269 (1982), 351–382).

A Morse equation in Conley's index theory for semiflows on metric spaces

Given a compact (two-sided) flow, an isolated invariant set S and a Morse-decomposition ( M 1 , …, M n ) of S , there is a generalized Morse equation, proved by Conley and Zehnder, which relates the Alexander-Spanier cohomology groups of the Conley indices of the sets M i and S with each other. Recently, Rybakowski developed the technique of isolating blocks and extended Conleys index theory to a class of one-sided semiflows on non-necessarily compact spaces, including e.g. semiflows generated by parabolic equations. Using these results, we discuss in this paper Morse decompositions and prove the above-mentioned Morse equation not only for arbitrary homology and cohomology groups, but also in this more general semiflow setting.

Attractors for reaction-diffusion equations on arbitrary unbounded domains

We prove existence of global attractors for parabolic equations of the form

Discretized ordinary differential equations and the Conley index

u_t+\beta(x)u-\sum_{ij}\partial_i(a_{ij}(x)\partial_j u)=f(x,u)

Trajectories joining critical points of nonlinear parabolic and hyperbolic partial differential equations

with Dirichlet boundary condition on an arbitrary unbounded domain

Attractors for semilinear damped wave equations on arbitrary unbounded domains

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