A Landesman-Lazer type result for periodic parabolic problems on ℝ^N at resonance

2014/10/13 by Aleksander Ćwiszewski, Cwiszewski, Aleksander, Renata Łukasiak +1
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory in Mathematical Physics #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1410.3400

openalex publication_date 2014/10/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We are concerned with T-periodic solutions of nonautonomous parabolic problem of the form u_t = Δu + V(x) u + f(t,x,u), t >0, x ∈ ℝ^N, with V ∈ L^∞ (ℝ^N)+L^p(ℝ^N), p ≥ N and T-periodic continuous perturbation f:ℝ^N× ℝ → ℝ. The so-called resonant case is considered, i.e. when \cal N:=Ker (Δ+ V) ≠ \0\ and f is bounded by a square-integrable function. We derive a formula for the fixed point index of the associated translation along trajectories operator in terms of the Brouwer topological degree of the time average mapping f: \cal N→ \cal N being the restriction of f to \cal N. By use of the formula and continuation techniques we show that Landesman-Lazer type conditions imply the existence of T-periodic solutions.

Citations

• Periodic and stationary solutions of nonlinear evolution equations - translation along trajectories method

Related

• Forced periodic solutions for nonresonant parabolic equations on R^N
• Averaging principle and periodic solutions for nonlinear evolution equations at resonance
• Periodic solutions for nonlinear evolution equations at resonance
• Periodic Solutions to Dissipative Hyperbolic Systems. I: Fredholm Solvability of Linear Problems
• On Time-Periodic Solutions to Parabolic Boundary Value Problems of\n Agmon-Douglis-Nirenberg Type