2009/01/27 by В. А. Галактионов, V. A. Galaktionov
Engineering · Mathematics · #Differential Equations and Numerical Methods #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations #math.AP #msc:35K40 #msc:35K55
paper · pdf · doi:10.1088/0951-7715/22/7/012
41 pages, 27 figures
arxiv created 2009/01/27 · openalex publication_date 2009/06/11 · arxiv updated 2015/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of the reaction–diffusion type are discussed. For the semilinear heat equation u t = Δ u + u p , various blow-up patterns were under scrutiny since the 1980s, while the case of higher order diffusion was studied much less, regardless of the wide range of its application. The types of blow-up include the following: Type I(ss): patterns of s elf- s imilar single point blow-up, including those for which the final time profile | u (·, T − )| N ( p −1)/4 is a measure; Type I(log): self-similar non-radial blow-up with angular log arithmic TW swirl; Type I(Her): non-self-similar blow-up close to stable/centre subspaces of Her mitian operators obtained via linearization about constant uniform blow-up pattern; Type II(sing): non-self-similar blow-up on stable/centre manifolds of a sing ular steady state in the supercritical Sobolev range p ⩾ p S = ( N + 4)/( N − 4) for N > 4 and Type II(LN): non-self-similar blow-up along the manifold of stationary generalized L oewner– N irenberg type explicit solutions in the critical Sobolev case p = p S , when | u (·, T − )| N ( p −1)/4 contains a measure as a singular component. There is some evidence that Type I(ss) are structurally stable (generic) patterns. However, justifying this and the existence of other proposed types of blow-up remain difficult open problems, so formal analytic and numerical methods are key in supporting some theoretical judgements.