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Volume 23, Issue 2
Regional, Single Point, and Global Blow-up for the Fourth-order Porous Medium Type Equation with Source

V. A. Galaktionov

DOI: 10.4208/jpde.v23.n2.1

J. Part. Diff. Eq., 23 (2010), pp. 105-146.

Published online: 2010-05

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  • 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.

  • Keywords

Higher-order quasilinear porous medium parabolic equation finite propagation blow-up similarity solutions variational operators branching

  • AMS Subject Headings

35K55 35K65

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-23-105, author = {V. A. Galaktionov }, title = {Regional, Single Point, and Global Blow-up for the Fourth-order Porous Medium Type Equation with Source}, journal = {Journal of Partial Differential Equations}, year = {2010}, volume = {23}, number = {2}, pages = {105--146}, 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.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v23.n2.1}, url = {http://global-sci.org/intro/article_detail/jpde/5224.html} }
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 -

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.

V. A. Galaktionov . (2010). Regional, Single Point, and Global Blow-up for the Fourth-order Porous Medium Type Equation with Source. Journal of Partial Differential Equations. 23 (2). 105-146. doi:10.4208/jpde.v23.n2.1
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