On an Anisotropic Parabolic Equation on the Domain with a Disjoint Boundary - Huashui Zhan (Journal of Function Spaces)
Consider the anisotropic parabolic equation with the variable exponents vt=∑i=1n(bi(x)vxiqix-2vxi)xi, where bi(x),qi(x)∈C1(Ω¯), qi(x)>1, and bi(x)≥0. If {bi(x)} is not degenerate on Σp⊂∂Ω, a part of the boundary, but is degenerate on the remained part ∂Ω∖Σp, then the boundary value condition is imposed on Σp, but there is no boundary value condition required on ∂Ω∖Σp. The stability of the weak solutions can be proved based on the partial boundary value condition vx∈Σp=0.