PUB - Publikationen an der Universität Bielefeld

We prove boundary higher integrability for the (spatial) gradient of very weak solutions of quasilinear parabolic equations of the form {u(t) - divA(x, t,del u) = 0 on Omega x (-T, T), u = 0 on partial derivative Omega x (-T, T), where the non-linear structure A(x,t, del u) is modeled after the variable exponent p(x, t)-Laplace operator given by vertical bar del u vertical bar(p(x, t)-2)del u. To this end, we prove that the gradients satisfy a reverse Holder inequality near the boundary by constructing a suitable test function which is Lipschitz continuous and preserves the boundary values. In the interior case, such a result was proved in Verena Bogelein and Qifan Li (2014) provided p(x, t) >= p(-) >= 2 holds and was then extended to the singular case 2n/n+2 < p(-) <= p(x, t) <= p(+)<= 2 in Qifan Li (2017). This restriction was necessary because the intrinsic scalings for quasilinear parabolic problems are different in the case p(+) <= 2 andp(-) >= 2. In this paper, we develop a new approach, using which, we are able to extend the results of Verena Bogelein and Qifan Li (2014), Qifan Li (2017) to the full range 2n/n+2 < p(-) <= p(x, t)<= p(+)