On an Anisotropic Parabolic Equation on the Domain with a Disjoint Boundary

Abstract EN

Consider the anisotropic parabolic equation with the variable exponents v t = ∑ i = 1 n ( b i ( x ) v x i q i x - 2 v x i ) x i , where b i ( x ) , q i ( x ) ∈ C 1 ( Ω ¯ ) , q i ( x ) > 1 , and b i ( x ) ≥ 0 .

If { b i ( x ) } is not degenerate on Σ p ⊂ ∂ Ω , a part of the boundary, but is degenerate on the remained part ∂ Ω ∖ Σ p , then the boundary value condition is imposed on Σ p , but there is no boundary value condition required on ∂ Ω ∖ Σ p .

The stability of the weak solutions can be proved based on the partial boundary value condition v x ∈ Σ p = 0 .