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 the reaction–diffusion type are discussed. For the semilinear heat equation ut = Δu + up, 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: 1. Type I(ss): 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; 2. Type I(log): self-similar non-radial blow-up with angular logarithmic TW swirl; 3. Type I(Her): non-self-similar blow-up close to stable/centre subspaces of Hermitian operators obtained via linearization about constant uniform blow-up pattern; 4. 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; 5. 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. 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