On very singular similarity solutions of a higher-order semilinear parabolic equation**Research supported by TMR networks ERB FMRX CT98-0201 and RTN network HPRN-CT-2002-00274.

Abstract

We study the large-time behaviour of solutions of a semilinear 2mth-order parabolic equation with bounded integrable initial data u0 decaying exponentially at infinity. For the semilinear heat equation (m = 1), the asymptotic behaviour was established in detail in the 1980s. Our main goal is to justify that, for any m ⩾ 1 in the subcritical range 1 < p < p0 = 1 + (2m/N), there exists a finite number, M ∼ N(p0 − p)/2(p − 1) → ∞ as p → 1+, of different very singular self-similar solutions of the form where each V is a radial, exponentially decaying solution of the elliptic equation By a perturbation technique, we establish the existence of radially symmetric very singular solution profiles Vl for p close to critical bifurcation exponents pl = 1 + (2m/(l + N)), l = 0, 2, …, where the first one, V0, is shown to be stable. Discrete and countable subsets of other self-similar and approximately self-similar patterns are introduced.