Renormalized solutions to a parabolic equation with mixed boundary constraints
Volume 593 / 2024
Dissertationes Mathematicae 593 (2024), 46 pp
MSC: Primary 35K55; Secondary 35A01, 35A02, 35R05.
DOI: 10.4064/dm230316-21-3
Published online: 24 April 2024
Abstract
We establish the existence and uniqueness of a renormalized solution to the parabolic equation \[ \frac{\partial b(u)}{\partial t} - \mathop{\rm div}(a(x,t,u,\nabla u)) = f \quad\ \mbox{in } \Omega\times(0,T) \] subject to a mixed boundary condition. Here $b(u)$ is a real function of $u$, $-\mathop{\rm div}(a(x,t,u,\nabla u))$ is of Leray–Lions type and $f$ is an $L^1$-function. Then we compare the renormalized solution to two other notions of solution: distributional solution and weak solution.
