Abstract
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.
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