Explicit estimates on a mixed Neumann-Robin-Cauchy problem

TL;DR

This paper establishes the existence of weak solutions for a complex parabolic PDE with mixed boundary conditions, providing explicit estimates and handling discontinuous coefficients and nonlinear boundary conditions.

Contribution

It introduces explicit estimates for solutions of parabolic equations with nonhomogeneous boundary conditions and nonlinear boundary conditions, advancing understanding of such PDEs.

Findings

Existence of weak solutions proven using compactness techniques.

Explicit estimates derived for solutions with discontinuous coefficients.

Analysis includes nonlinear boundary conditions and nonhomogeneous data.

Abstract

We deal with the existence of weak solutions for a mixed Neumann-Robin-Cauchy problem. The existence results are based on global-in-time estimates of approximating solutions, and the passage to the limit exploits compactness techniques. We investigate explicit estimates for solutions of the parabolic equations with nonhomogeneous boundary conditions and distributional right hand sides. The parabolic equation is of divergence form with discontinuous coefficients. We consider a nonlinear condition on a part of the boundary that the power laws (and the Robin boundary condition) appear as particular cases.