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

Abstract

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