Maximal $L_{p} $-regularity for a second-order differential equation with unbounded intermediate coefficient

• Main Author: Kordan Ospanov
• Format: Article
• Language: English
• Published: University of Szeged 2019-08-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: differential equation; unbounded coefficients; well-posedness; maximal regularity; separability
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=7311
• ISSN: 1417-3875; 1417-3875

We consider the following equation \[-y''+r\left(x\right)y'+q\left(x\right)y=f(x), \] where the intermediate coefficient $r$ is not controlled by $q$ and it is can be strong oscillate. We give the conditions of well-posedness in $L_{p} \left(-\infty ,\, +\infty \right)$ of this equation. For the solution $y$, we obtained the following maximal regularity estimate: \[\left\| y''\right\| _{p} +\left\| ry'\right\| _{p} +\left\| qy\right\| _{p} \le C\left\| f\right\| _{p} ,\] where $\left\| \, \cdot \, \right\| _{p} $ is the norm of $L_{p} \left(-\infty ,\, +\infty \right)$.