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

Abstract

Blow-up behaviour for the fourth-order quasilinear porous medium equation with source, u_t=(|u|^nu)_{xxxx}+|u|^{p-1}u in R×R_+, where n > 0, p > 1, is studied. Countable and finite families of similarity blow-up patterns of the form u_S(x,t)=(T-t)^{-\frac{1}{p-1}}f(y), where y=\frac{x}{T-t}^β, β=\frac{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 (for p > n+1), and global (for p∈(1,n+1)) blowup. The concepts and various variational, bifurcation, and numerical approaches for revealing the structure and multiplicities of such blow-up patterns are presented.