Maximal regularity estimates for higher order differential equations with fluctuating coefficients - Ospanov, Kordan; Yeskabylova, Zhuldyz; Beisenova, Danagul (Eurasian Mathematical Journal, 2019)
We give the well-posedness conditions in L2 (-∞, +∞) for the following differential equation: -y''' + p(x)y' + q(x)y = f(x), where p and q are continuously differentiable and continuous functions, respectively, and f ∈ L2(R). Moreover, we prove for the solution y of this equation the following maximal regularity estimate: ‖y'''‖2 + ‖py'‖2 + ‖qy‖2 ≤ C‖f‖2 (here ‖⋅‖2 is the norm in L2(-∞, +∞)). We assume that the intermediate coefficient p is fast oscillating and not controlled by the coefficient q. The sufficient conditions obtained by us are close to necessary ones. We give similar results for the fourth-order differential equation with singular intermediate coefficients.