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Register a new userSobolev estimates for fractional parabolic equations with space-time non-local operators
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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.
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We consider time fractional parabolic equations in both divergence and non-divergence form when the leading coefficients $a^{ij}$ are measurable functions of $(t,x_1)$ except for $a^{11}$ which is a measurable function of either $t$ or $x_1$. We obtain the solvability in Sobolev spaces of the equations in the whole space, on a half space, or on a partially bounded domain. The proofs use a level set argument, a scaling argument, and embeddings in fractional parabolic Sobolev spaces for which we give a direct and elementary proof.
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.
In this note, we use the non-homogeneous Poisson stochastic process to show how knowing Schauder and Sobolev estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs. The method is probability. We generalize the result of Krylov-Priola [7].
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.
In this paper, we establish $L_p$ estimates and solvability for time fractional divergence form parabolic equations in the whole space when leading coefficients are merely measurable in one spatial variable and locally have small mean oscillations with respect to the other variables. The corresponding results for equations on a half space are also derived.
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