Existence of solution to parabolic equations with mixed boundary condition on non-cylindrical domains

Mixed boundary value problems for parabolic equations in Sobolev spaces with mixed-norms

“…For other previous results on the mixed boundary value problems for parabolic equations in unmixed-norm spaces, we refer the reader to [2,3,14] and the references therein. We note that in these work, either p is assumed to be 2 or an implicit condition is imposed on the operator so that p needs to be sufficiently close to 2.…”

Optimal regularity of mixed Dirichlet-conormal boundary value problems for parabolic operators

“…See also [12] for a further result about equations with nonhomogeneous boundary data as well as an earlier result in [11] about the elliptic mixed boundary value problem. We also mention the work [25,16] for parabolic equations in non-cylindrical domains. In [25] Savaré considered parabolic equations in a non-cylindrical domain with C 1,1 boundary and separation Γ.…”

“…Under certain condition on the excess of Γ with respect to t, he introduced an approximation approach to general abstract evolution equations in L 2 -based Sobolev spaces and obtained optimal regularity under quite weak assumptions on the data. Recently in [16], Kim and Cao studied linear and semilinear parabolic equations in non-cylindrical domains with the mixed Dirichlet, Neumann, and Robin conditions and Lipschitz leading coefficients. They considered smooth domains which are C 1 in t and C 2 in x and general D T and N T , and proved the solvability in L 2 -based Sobolev spaces.…”