Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

6
The categories of this paper are: astro-ph
35
The categories of this paper are: mtrl-th
17
The categories of this paper are: q-fin
7
The categories of this paper are: hep-th
11
The categories of this paper are: hep-lat
31
The categories of this paper are: ao-sci
24
The categories of this paper are: dg-ga
4
The categories of this paper are: cond-mat
21
The categories of this paper are: adap-org
15
The categories of this paper are: nucl-ex
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
32
The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas
8
The categories of this paper are: hep-ex
26
The categories of this paper are: bayes-an