On the Well-Posedness of First-Order Variable Exponent Cauchy Problems with Robin and Wentzell-Robin Boundary Conditions on Arbitrary Domains

We define the notion of relative capacity of variable exponent type, referred in this article as the relative $p(\cdot)$-capacity, and use this approach to obtain a necessary and sufficient condition for the well-posedness of the corresponding parabolic boundary value problems involving the $p(\cdot)$-Laplace operator and either Robin or Wentzell-Robin boundary conditions on arbitrary domains.