Uniqueness of limit flow for a class of quasi-linear parabolic equations

2016/02/09 by Marco Squassina, Squassina, Marco, Tatsuya Watanabe +1
Mathematics · #35B40 (Primary) #35B44 (Secondary) #35K59 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35B40 #msc:35B44 #msc:35K59

paper · pdf · doi:10.48550/arxiv.1602.03002

37 pages

arxiv created 2016/02/09 · arxiv updated 2016/02/10

Abstract

We investigate the issue of uniqueness of the limit flow for a relevant class of quasi-linear parabolic equations defined on the whole space. More precisely, we shall investigate conditions which guarantee that the global solutions decay at infinity uniformly in time and their entire trajectory approaches a single steady state as time goes to infinity. Finally, we obtain a characterization of solutions which blow-up, vanish or converge to a stationary state for initial data of the form λφ₀ while λ>0 crosses a bifurcation value λ₀.

Related

• Large-time behavior of solutions of parabolic equations on the real line with convergent initial data II: equal limits at infinity
• Decay at infinity for parabolic equations
• On the uniqueness of the limit for an asymptotically autonomous semilinear equation on R^N
• Large-time behavior of solutions of parabolic equations on the real line with convergent initial data III:unstable limit at infinity
• Quantitative unique continuation for parabolic equations with Neumann boundary conditions