Refined asymptotics for the blowup of u_t — δu = u^p

Abstract

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

Quenching profiles for one-dimensional semilinear heat equations

“…Our analysis depends heavily on the ideas of [7,[16][17][18]23]. Many of the results there are applicable in problem (1.1)-(1.3), either in a straightforward way or after minor modifications.…”

Similarity: generalizations, applications and open problems

“…[12][13][14][15][16][17]) and more recently for their higherorder generalizations (cf. [18][19][20]).…”

“…[16,29]) owing to the Sturm-Liouville structure of the linearized operator, leading to a countably discrete spectrum. In contrast, for high-order PDE models exhibiting singularity formation, there can be a countably infinite set of local self-similar solutions whose stability properties must be studied on a case-by-case basis (cf.…”

“…[27,28]) and their analysis requires a synthesis of similarity, dynamical systems, perturbation, and numerical methods; -rigorous analysis of the above issues (cf. [12][13][14][15][16][17]29]). In many cases such analysis builds directly on the formal results;…”

Stability and dynamics of self-similarity in evolution equations

“…Different ranges for values of the parameter q yield either diffusion-or absorption-dominated dynamics. This bifurcation was analyzed by Wayne [72] via center-manifold theory, inspired by the renormalization group ideas of Bricmont & Kupianen [21,20] and earlier center-manifold ideas of Kohn and collaborators [36,45] and Bebernes [5]. For q < 0 equation (1.2) has strong absorption that can produce finite-time extinction [39].…”