| Abstract: |
| We present parabolic equations in divergence form that include a half-order time derivative term on the right-hand side. We discuss the motivation for considering such equations, particularly their usefulness (or necessity) when dealing with parabolic equations in divergence form with highly irregular coefficients or domains. As an application, we demonstrate the $L_p$ solvability of parabolic equations with the conormal derivative boundary condition in very irregular domains, assuming the coefficients are only measurable in time. A key challenge to solvability, when obtaining necessary estimates, is the presence of the time derivative of the solution on the right-hand side, which may lack sufficient regularity to belong to $L_p$ spaces. To address this, we reformulate the term as the half-order time derivative of a function in $L_p$ spaces and employ the aforementioned results. |
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