This work is concerned with positive, blowing-up solutions of the semilinear heat equation ut -Au = u p in Rn. Our main contribution is a sort of center manifold analysis for the equation in similarity variables, leading to refined asymptotics for u in a backward space-time parabola near any blowup point. We also explore a connection between the asymptotics of u and the local geometry of the blowup set. 1 1 w, -' v * ( p V w ) + 2w = w p , P P -1

Abstract. We present a unified approach to improved L p Hardy inequalities in R N . We consider Hardy potentials that involve either the distance from a point, or the distance from the boundary, or even the intermediate case where the distance is taken from a surface of codimension 1 < k < N. In our main result, we add to the right hand side of the classical Hardy inequality a weighted L p norm with optimal weight and best constant. We also prove nonhomogeneous improved Hardy inequalities, where the right hand side involves weighted L q norms, q = p.

Let O be a bounded domain in R N ; N 53, containing the origin. Motivated by a question of Brezis and V! a azquez, we consider an Improved Hardy Inequality with best constant b, that we formally write as: ÀD5ð NÀ22 Þ 2 1 jxj 2 þ bV ðxÞ. We first give necessary conditions on the potential V , under which the previous inequality can or cannot be further improved. We show that the best constant b is never achieved in H 1 0 ðOÞ, and in particular that the existence or not of further correction terms is not connected to the nonachievement of b in H 1 0 ðOÞ. Our analysis reveals that the original inequality can be repeatedly improved by adding on the right-hand side specific potentials. This leads to an infinite series expansion of Hardy's inequality. The series obtained is in some sense optimal. In establishing these results we derive various sharp improved Hardy-Sobolev Inequalities. # 2002 Elsevier Science (USA)

We consider the semilinear heat equation with critical power nonlinearity. Using formal arguments based on matched asymptotic expansion techniques, we give a detailed description of radially symmetric sign-changing solutions, which blow-up at x = 0 and t = T < ∞, for space dimension N = 3, 4, 5, 6. These solutions exhibit fast blow-up; i.e. they satisfy lim t↑T