Sobolev space theory for the heat and time-fractional heat equations in non-smooth domains

Abstract: We present a general $L_p$-solvability result for the classical heat equation and time-fractional heat equations in non-smooth domains with the zero Dirichlet boundary condition. We consider domains $\Omega$ admitting the Hardy inequality: There exists a constant $N>0$ such that $$ \int_{\Omega}\Big|\frac{f(x)}{d(x,\partial\Omega)}\Big|^2\mathrm{d}x\leq N\int_{\Omega}|\nabla f|^2 \mathrm{d} x\quad\text{for any}\quad f\in C_c^{\infty}(\Omega)\,. $$ To illustrate the boundary behavior of solutions in a general framework, we employ a weight system composed of a superharmonic function and a distance function to the boundary. Further, we investigate applications for various non-smooth domains, including convex domains, domains with exterior cone condition, totally vanishing exterior Reifenberg domains, and domains $\Omega\subset\mathbb{R}^d$ for which the Aikawa dimension of $\Omega^c$ is less than $d-2$. By using superharmonic functions tailored to the geometric conditions of the domain, we derive weighted $L_p$-solvability results for various non-smooth domains, with specific weight ranges that vary by domain condition. In addition, we provide an application for the H\"older continuity of solutions in domains with the volume density condition, and a pointwise estimate for solutions in Lipschitz cones.