THREE TYPES OF SELF-SIMILAR BLOW-UP FOR THE FOURTH-ORDER p-LAPLACIAN EQUATION WITH SOURCE: VARIATIONAL AND BRANCHING APPROACHES

Self-similar blow-up behaviour for the fourth-order quasilinear p-Laplacian equation with source, ut = −(|uxx| uxx)xx + |u| u in R × R+, where n > 0, p > 1, is studied. Using variational setting for p = n+1 and branching techniques for p 6= n+1, finite and countable families of blow-up patterns of the self-similar form uS(x, t) = (T − t) − 1 p−1 f(y), where y = x/(T − t) , β = − p−(n+1) 2(n+2)(p−1) , are described by an analytic-numerical approach. Three parameter ranges: p = n + 1 (regional), p > n+1 (single point), and 1 < p < n+1 (global blow-up) are studied. This blow-up model is motivated by the second-order reaction diffusion counterpart ut = (|ux| ux)x + u p (u ≥ 0) that was studied in the middle of the 1980s, while first results on blow-up of solutions were established by Tsutsumi in 1972. This paper is an earlier extended preprint of [22].