The trace of $u\in W^{1,1}_{\mathrm{loc}}(\Omega )\bigcap L^{\infty}(\Omega )$ and its applications

Abstract This paper is concerned with the well-posedness problem of a doubly degenerate parabolic equation variable exponents. By parabolically regularized method, existence local solution proved. Moreover, trace $u\in W^{1,1}_{0}(\Omega )$ u ∈ W 0 1 , ( Ω ) generalized to W^{1,1}_{\mathrm{loc}}(\Omega )\bigcap L^{\infty}(\Omega loc ⋂ L ∞ in rational way. Then, partial boundary value condition matching up stability theorem found.