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The large-time asymptotic behaviour of solutions of the Cauchy problem is investigated for a non-linear Sobolev-type equation with dissipation. For small initial data the approach taken is based on a detailed analysis of the Green's function of the linear problem and the use of the contraction mapping method. The case of large initial data is also closely considered. In the supercritical case the asymptotic formulae are quasi-linear. The asymptotic behaviour of solutions of a non-linear Sobolev-type equation with a critical non-linearity of the non-convective kind differs by a logarithmic correction term from the behaviour of solutions of the corresponding linear equation. For a critical convective non-linearity, as well as for a subcritical non-convective non-linearity it is proved that the leading term of the asymptotic expression for large times is a self-similar solution. For Sobolev equations with convective non-linearity the asymptotic behaviour of solutions in the subcritical case is the product of a rarefaction wave and a shock wave.

Bibliography: 84 titles.

Steenrod homotopy theory is a natural framework for doing algebraic topology on general spaces in terms of algebraic topology of polyhedra; or from a different viewpoint, it studies the topology of the functor (for inverse sequences of groups). This paper is primarily concerned with the case of compacta, in which Steenrod homotopy coincides with strong shape. An attempt is made to simplify the foundations of the theory and to clarify and improve some of its major results. With geometric tools such as Milnor's telescope compactification, comanifolds (=mock bundles), and the Pontryagin-Thom construction, new simple proofs are obtained for results by Barratt-Milnor, Geoghegan-Krasinkiewicz, Dydak, Dydak-Segal, Krasinkiewicz-Minc, Cathey, Mittag-Leffler-Bourbaki, Fox, Eda-Kawamura, Edwards-Geoghegan, Jussila, and for three unpublished results by Shchepin. An error in Lisitsa's proof of the `Hurewicz theorem in Steenrod homotopy' is corrected. It is shown that over compacta, R.H. Fox's overlayings are equivalent to I.M. James' uniform covering maps. Other results include:

A morphism between inverse sequences of countable (possibly non-Abelian) groups that induces isomorphisms on and is invertible in the pro-category. This implies the `Whitehead theorem in Steenrod homotopy', thereby answering two questions of Koyama.

If is an -compactum, , then its -dimensional Steenrod homotopy classes are representable by maps , provided that is simply connected. The assumption of simple connectedness cannot be dropped, by a well-known result of Dydak and Zdravkovska.

A connected compactum is Steenrod connected (=pointed 1-movable), if and only if every uniform covering space of it has countably many uniform connected components.

Bibliography: 117 titles.