TY - JOUR
T1 - Regional, Single Point, and Global Blow-up for the Fourth-order Porous Medium Type Equation with Source
AU - V. A. Galaktionov 
JO - Journal of Partial Differential Equations
VL - 2
SP - 105
EP - 146
PY - 2010
DA - 2010/05
SN - 23
DO - http://doi.org/10.4208/jpde.v23.n2.1
UR - https://global-sci.org/intro/article_detail/jpde/5224.html
KW - Higher-order quasilinear porous medium parabolic equation
KW - finite propagation
KW - blow-up
KW - similarity solutions
KW - variational operators
KW - branching
AB - <p>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 &gt; 0, p &gt; 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^- &lt; ∞, are described. These solutions explain key features of regional (for p=n+1), single point (for p &gt; 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.</p>