Mixed problems for degenerate abstract parabolic equations and applications

TL;DR

This paper investigates degenerate abstract parabolic equations with nonlocal boundary conditions, establishing regularity, Strichartz estimates, and existence results relevant to fluid mechanics and environmental engineering.

Contribution

It introduces new maximal regularity results and Strichartz estimates for degenerate parabolic equations with variable coefficients and nonlocal boundary conditions.

Findings

Maximal regularity properties for solutions are established.

Strichartz type estimates in mixed Lp spaces are derived.

Existence and uniqueness of solutions for nonlinear problems are proven.

Abstract

Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems and Strichartz type estimates in mixed $L_{p}$ spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering.