Geodesic
Maximal regularity estimates for higher order differential equations with fluctuating coefficients
Eurasian mathematical journal, Tome 10 (2019) no. 2, pp. 65-74

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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. 
@article{EMJ_2019_10_2_a4,
     author = {K. N. Ospanov and Zh. B. Yeskabylova and D. R. Beisenova},
     title = {Maximal regularity estimates for higher order differential equations with fluctuating coefficients},
     journal = {Eurasian mathematical journal},
     pages = {65--74},
     publisher = {mathdoc},
     volume = {10},
     number = {2},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EMJ_2019_10_2_a4/}
}
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K. N. Ospanov; Zh. B. Yeskabylova; D. R. Beisenova. Maximal regularity estimates for higher order differential equations with fluctuating coefficients. Eurasian mathematical journal, Tome 10 (2019) no. 2, pp. 65-74. http://geodesic.mathdoc.fr/item/EMJ_2019_10_2_a4/