2014/04/01 by Aleksander Ćwiszewski, Cwiszewski, Aleksander, Renata Łukasiak +1 · 1 citation
Computer Science · Engineering · Mathematics · #35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1404.0256
openalex publication_date 2014/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Criteria for the existence of T-periodic solutions of nonautonomous parabolic equation ut = Δu + f(t,x,u), x∈ℝN, t>0 with asymptotically linear f will be provided. It is expressed in terms of time average function f of the nonlinear term f and the spectrum of the Laplace operator Δ on ℝN. One of them says that if the derivative f_∞ of f at infinity does not interact with the spectrum of Δ, i.e. Ker (-Δ+ f_∞)=\0\, then the parabolic equation admits a T-periodic solution. Another theorem is derived in the situation, where the linearization at 0 and infinity differ topologically, i.e. the total multiplicities of negative eigenvalues of the averaged linearizations at 0 and ∞ are different mod 2.