Well-posedness problem of an anisotropic parabolic equation with a nonstandard growth order
Huashui Zhan · Advances in Continuous and Discrete Models · 2024
Consider the well-posedness problem of an anisotropic parabolic equation with a nonstandard growth order. The weak solution is introduced, an $L^{\infty}$ -estimate is provided, and the existence is established using the parabolically regularized method. However, the weak solution under consideration is not in $L^{1}(0,T;W_{0}^{1,\overrightarrow{p}(x)}(\Omega ))$ , and defining the trace becomes a new problem. In this paper, we give a new definition for the trace of $u\in L^{\infty}(Q_{T})$ to solve this problem. Based on the new concept of the generalized trace and the selection of suitable test functions, the stability theorems of the weak solutions are obtained.