In this paper, we study the solvability of the first boundary value problem for a class of second order linear non-uniformly parabolic equations with \(L^1\) data: 0.1 \(\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial u}{\partial t}-\frac{\partial }{\partial z_{i}}\left( a_{ij}\left( t,z \right) \frac{\partial u}{\partial z_{j}}\right) = f\left( t,z \right) , \, (t, z) \in Q_T, \\ u\left( 0,z \right) =g\left( z \right) , z\in D, \\ u \Big \vert _{S_{T}}=0, \end{array}\right. } \end{aligned}\) where \(f \in L^1(Q_T)\) , \(g \in L^1(D)\) , \(Q_T = (0, T) \times D\) , and \(S_T = (0, T) \times \partial D\) , with \(D \subset \mathbb {R}^N\) being a bounded Lipschitz domain. Problem (0.1) is studied under the assumption: 0.2 \(\begin{aligned} c_{1}(\omega (x)|\xi |^{2}+|\eta |^{2}) \le A(t,z)\zeta \cdot \zeta \le c_{2}(\omega (x)|\xi |^{2}+|\eta |^{2}) \end{aligned}\) for almost every \((t, z) \in Q_T\) and for all \(\zeta = (\xi , \eta ) \in \mathbb {R}^{N}\) such that \(\xi \in \mathbb {R}^n\) and \(\eta \in \mathbb {R}^{N-n}\) , where \(N \ge 2\) and \(1 \le n < N\) . By applying recent results on non-uniform gradient Poincaré–Sobolev type inequalities and establishing new a priori estimates, we investigate the very weak solvability of problem (0.1) under condition (0.2).
