Maximal regularity estimates for higher order differential equations with fluctuating coefficients

K. N. Ospanov ; Zh. B. Yeskabylova ; D. R. Beisenova

Eurasian mathematical journal, Tome 10 (2019) no. 2, pp. 65-74

Résumé

We give the well-posedness conditions in $L_2(-\infty,+\infty)$ 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 $f\in L_2(R)$. Moreover, we prove for the solution y of this equation the following maximal regularity estimate: $$ ||y'''||_2+||py'||_2+||qy||_2\leqslant C||f||_2 $$ (here $||\cdot||_2$ is the norm in $L_2(-\infty,+\infty)$). 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.