Three types of self-similar blow-up for the fourth-order [formula omitted]-Laplacian equation with source

Abstract

Self-similar blow-up behaviour for the fourth-order quasilinear p -Laplacian equation with source, u t = − ( | u x x | n u x x ) x x + | u | p − 1 u in R × R + , where n > 0 , p > 1 , is studied. Using variational setting for p = n + 1 and branching techniques for p ⁄ = n + 1 , finite and countable families of blow-up patterns of the self-similar form u S ( x , t ) = ( T − t ) − 1 p − 1 f ( y ) , where y = x / ( T − t ) β , 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 u t = ( | u x | n u x ) 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.