Solutions of evolutionary equation based on the anisotropic variable exponent Sobolev space
In this paper, we are concerned with the equation
ut=∑i=1N∂∂xi(ai(x)|uxi|pi(x)−2uxi)+∑i=1N∂bi(u)∂xi, (x,t)∈Ω×(0,T), where ai(x)|x∈∂Ω=0 and ai(x)|x∈Ω>0 . By the theory of anisotropic variable exponent Sobolev spaces, we study the well-posedness of weak solutions of this equation. Since ai(x) is degenerate on the boundary, the stability of weak solutions may be established without any boundary value condition. The main feature which distinguishes this paper from other related works lies in the fact that we propose a novel analytical method to deal with the stability of weak solutions.
Zhan, H., Feng, Z. Solutions of evolutionary equation based on the anisotropic variable exponent Sobolev space. Z. Angew. Math. Phys. 70, 110 (2019). https://doi.org/10.1007/s00033-019-1150-y
Z. Angew. Math. Phys.