On the Boundary Value Condition of an Isotropic Parabolic Equation

The well-posedness problem of anisotropic parabolic equation with variable exponents is studied in this paper. The weak solutions and the strong solutions are introduced, respectively. By a generalized Gronwall inequality, the stability of strong solutions to this equation is established, and the uniqueness of weak solutions is proved. Compared with the related works, a new boundary value condition, ∏ i = 1 N a i x , t = 0 , x , t ∈ ∂ Ω × 0 , T , is introduced the first time and has been proved that it can take place of the Dirichlet boundary value condition in some way.

The Partial Second Boundary Value Problem of an Anisotropic Parabolic Equation

Consider an anisotropic parabolic equation with the variable exponents vt=∑i=1n(bi(x,t)vxipi(x)-2vxi)xi+f(v,x,t), where bi(x,t)∈C1(QT¯), pi(x)∈C1(Ω¯), pi(x)>1, bi(x,t)≥0, f(v,x,t)≥0. If {bi(x,t)} is degenerate on Γ2⊂∂Ω, then the second boundary value condition is imposed on the remaining part ∂Ω∖Γ2. The uniqueness of weak solution can be proved without the boundary value condition on Γ2.

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

Consider the anisotropic parabolic equation with the variable exponentsvt=∑i=1n(bi(x)vxiqix-2vxi)xi,wherebi(x),qi(x)∈C1(Ω¯),qi(x)>1, andbi(x)≥0. If{bi(x)}is not degenerate onΣp⊂∂Ω, a part of the boundary, but is degenerate on the remained part∂Ω∖Σp, then the boundary value condition is imposed onΣp, 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 conditionvx∈Σp=0.