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Register a new userAn approach for weighted mixed-norm estimates for parabolic equations with local and non-local time derivatives
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We give a unified approach to weighted mixed-norm estimates and solvability for both the usual and time fractional parabolic equations in nondivergence form when coefficients are merely measurable in the time variable. In the spatial variables, the leading coefficients locally have small mean oscillations. Our results extend the previous result in [6] for unmixed $L_p$-estimates without weights.
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We obtain $L_p$ estimates for fractional parabolic equations with space-time non-local operators $$ partial_t^alpha u - Lu= f quad mathrm{in} quad (0,T) times mathbb{R}^d,$$ where $partial_t^alpha u$ is the Caputo fractional derivative of order $alpha in (0,1]$, $Tin (0,infty)$, and $$Lu(t,x) := int_{ mathbb{R}^d} bigg( u(t,x+y)-u(t,x) - ycdot abla_xu(t,x)chi^{(sigma)}(y)bigg)K(t,x,y),dy $$ is an integro-differential operator in the spatial variables. Here we do not impose any regularity assumption on the kernel $K$ with respect to $t$ and $y$. We also derive a weighted mixed-norm estimate for the equations with operators that are local in time, i.e., $alpha = 1$, which extend the previous results by using a quite different method.
In this paper, we study both elliptic and parabolic equations in non-divergence form with singular degenerate coefficients. Weighted and mixed-norm $L_p$-estimates and solvability are established under some suitable partially weighted BMO regularity conditions on the coefficients. When the coefficients are constants, the operators are reduced to extensional operators which arise in the study of fractional heat equations and fractional Laplace equations. Our results are new even in this setting and in the unmixed case. For the proof, we establish both interior and boundary Lipschitz estimates for solutions and for higher order derivatives of solutions to homogeneous equations. We then employ the perturbation method by using the Fefferman-Stein sharp function theorem, the Hardy-Littlewood maximum function theorem, as well as a weighted Hardys inequality.
This paper is a comprehensive study of $L_p$ estimates for time fractional wave equations of order $alpha in (1,2)$ in the whole space, a half space, or a cylindrical domain. We obtain weighted mixed-norm estimates and solvability of the equations in both non-divergence form and divergence form when the leading coefficients have small mean oscillation in small cylinders.
We study a class of elliptic and parabolic equations in non-divergence form with singular coefficients in an upper half space with the homogeneous Dirichlet boundary condition. Intrinsic weighted Sobolev spaces are found in which the existence and uniqueness of strong solutions are proved when the partial oscillations of coefficients in small parabolic cylinders are small. Our results are new even when the coefficients are constants
We establish the $L_p$-solvability for time fractional parabolic equations when coefficients are merely measurable in the time variable. In the spatial variables, the leading coefficients locally have small mean oscillations. Our results extend a recent result in [6] to a large extent.
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