2021/05/05 by Viorel Barbu, Michael Röckner, Barbu, Viorel +1
Mathematics · #Advanced Mathematical Physics Problems #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP #math.PR #msc:35C99 #msc:58J165 #msc:60G46 #msc:60H10 #msc:60H30
paper · pdf · doi:10.48550/arxiv.2105.02328
24 pages
arxiv created 2021/05/05 · arxiv updated 2021/05/07
One studies here, via the La Salle invariance principle for nonlinear semigroups in Banach spaces, the properties of the ω-limit set ω(u0) corresponding to the orbit γ(u0)=\u(t,u0); t≥0\, where u=u(t,u0) is the solution to the nonlinear Fokker-Planck equation ut-Δβ(u)+\rm div(Db(u)u)=0 in (0,∞)×ℝd,
u(0,x)=u0(x), x∈ℝd, u0∈ L1(ℝd), d≥3. Here, β∈ C1(ℝ) and β'(r)>0, ∀ r≠0. Moreover, β is a sublinear function, possibly degenerate in the origin, b∈ C1(ℝ), b bounded, b≥ b0∈(0,∞), D is bounded such that D=-∇Φ, where Φ∈ C(ℝd) is such that Φ≥1, Φ(x)→∞ as |x|→∞ and satisfies a condition of the form ΔΦ-α|∇Φ|2≤0, a.e. on ℝd. The main conclusion is that the equation has an equilibrium state and the set ω(u0) is a non-empty, compact subset of L1(ℝd) while, for each t≥0, the operator u0→ u(t,u0) is an isometry on ω(u0). In the nondegenerate case 0<γ0≤β'≤γ1 studied in [Barbu, Röckner: arXiv:1808.10706], it follows that limt→∞S(t)u0=u_∞ in L1(ℝd), where u_∞ is the unique bounded stationary solution to the equation.