Regional, Single Point, and Global Blow-up for the Fourth-order Porous Medium Type Equation with Source

Blow-up behaviour for the fourth-order quasilinear porous medium equation with source, (0.1) ut = −(|u|u)xxxx + |u|u in R × R+, n > 0, p > 1, is studied. Countable and finite families of similarity blow-up patterns of the form uS(x, t) = (T − t)− 1 p−1 f(y), where y = x/(T − t) , β = p−(n+1) 4(p−1) , which blow-up as t → T− < ∞, are described. These solutions explain key features of regional (for p = n+1), single point (p > n+1), and global (p ∈ (1, n+1)) blow-up. The concepts and various variational, bifurcation, and numerical approaches for revealing the structure and multiplicities of such blow-up patterns are presented.