scholarly journals Semilinear Parabolic Equations on the Heisenberg Group with a Singular Potential

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Houda Mokrani ◽  
Fatimetou Mint Aghrabatt
Keyword(s):  
Asymptotic Behavior ◽  
Heisenberg Group ◽  
Parabolic Equations ◽  
Singular Potential ◽  
Steady States ◽  
Asymptotic Behavior Of Solutions ◽  
Semilinear Parabolic Equations ◽  
Sobolev’S Inequality ◽  
Semilinear Parabolic ◽  
Global Branch

We discuss the asymptotic behavior of solutions for semilinear parabolic equations on the Heisenberg group with a singular potential. The singularity is controlled by Hardy's inequality, and the nonlinearity is controlled by Sobolev's inequality. We also establish the existence of a global branch of the corresponding steady states via the classical Rabinowitz theorem.

scholarly journals Steady States, Global Existence and Blow-up for Fourth-Order Semilinear Parabolic Equations of Cahn–Hilliard Type

2012 ◽  
Vol 12 (2) ◽  
Author(s):  
Pablo Álvarez-Caudevilla ◽  
Victor A. Galaktionov
Keyword(s):  
Global Existence ◽  
Fourth Order ◽  
Parabolic Equations ◽  
Blow Up ◽  
Steady States ◽  
Type I ◽  
Stable Model ◽  
Semilinear Parabolic Equations ◽  
Smooth Bounded Domain ◽  
Semilinear Parabolic

AbstractFourth-order semilinear parabolic equations of the Cahn-Hilliard-typeuare considered in a smooth bounded domain Ω ⊂ ℝuThe following three main problems are studied:(i) for the unstable model (0.1), with the −Δ(|u|(ii) for the stable model (0.2), global existence of smooth solutions u(x, t) in ℝ(iii) for the unstable model (0.2), a relation between finite time blow-up and structure of regular and singular steady states in the supercritical range. In particular, three distinct families of Type I and II blow-up patterns are introduced in the unstable case.

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