Five Types of Blow-up in a Semilinear Fourth-order Reaction-diffusion Equation: an Analytic-numerical Approach

Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion type ut = −∆2u+ |u|p−1u in R × (0, T ), p > 1, limt→T− supx∈RN |u(x, t)| = +∞, are discussed. For the semilinear heat equation ut = ∆u+ u , various blow-up patterns were under scrutiny since 1980s, while the case of higher-order diffusion was studied much less, regardless a wide range of its application. The types of blow-up include: (i) Type I(ss): various patterns of self-similar single point blow-up, including those, for which the final time profile |u(·, T−)|N(p−1)/4 is a measure; (ii) Type I(log): self-similar non-radial blow-up with angular logarithmic TW swirl; (iii) Type I(Her): non self-similar blow-up close to stable/centre subspaces ofHermitian operators obtained via linearization about constant uniform blow-up pattern; (iv) Type II(sing): non self-similar blow-up on stable/centre manifolds of a singular steady state in the supercritical Sobolev range p ≥ pS = N+4 N−4 for N > 4; and (v) Type II(LN): non self-similar blow-up along the manifold of stationary generalized Loewner–Nirenberg type explicit solutions in the critical Sobolev case p = pS, when |u(·, T−)|N(p−1)/4 contains a measure as a singular component. All proposed types of blow-up are very difficult to justify, so formal analytic and numerical methods are key in supporting some theoretical judgements. 1. From second-order to higher-order blow-up R–D models: a PDE route from XXth to XXIst century 1.1. The RDE–4 and applications. This paper is devoted to a description of blow-up patterns for the fourth-order reaction-diffusion equation (the RDE–4 in short) (1.1) ut = −∆2u+ |u|p−1u in R × (0, T ), where p > 1, where ∆ stands for the Laplacian in R . This has the bi-harmonic diffusion −∆2 and is a higher-order counterpart of classic second-order PDEs, which we begin our discussion with. For applications of such higher-diffusion models, see short surveys and references in Date: January 27, 2009. 1991 Mathematics Subject Classification. 35K55, 35K40 .