Partial existence result for Homogeneous Quasilinear parabolic problems beyond the duality pairing

In this paper, we study the existence of distributional solutions solving \cref{main-3} on a bounded domain $\Omega$ satisfying a uniform capacity density condition where the nonlinear structure $\mathcal{A}(x,t,\nabla u)$ is modelled after the standard parabolic $p$-Laplace operator. In this regard, we need to prove a priori estimates for the gradient of the solution below the natural exponent and a higher integrability result for very weak solutions at the initial boundary. The elliptic counterpart to these two estimates are fairly well developed over the past few decades, but no analogous theory exists in the quasilinear parabolic setting. Two important features of the estimates proved here are that they are non-perturbative in nature and we are able to take non-zero boundary data. \emph{As a consequence, our estimates are new even for the heat equation on bounded domains.} This partial existence result is a nontrivial extension of the existence theory of very weak solutions from the elliptic setting to the quasilinear parabolic setting. Even though we only prove partial existence result, nevertheless we establish the necessary framework that when proved would lead to obtaining the full result for the homogeneous problem.