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Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms. (English) Zbl 1133.35052
The author studies the Cauchy problem for second-order divergence and non-divergence type equations:
\begin{aligned} L u(t,x)&= u_t(t,x) +a^{ij}(t,x)u_{x^ix^j}(t,x)+ b^i(t,x)u_{x^i}(t,x)+c(t,x) u(t,x),\\ {\mathcal L } u(t,x)&= u_t(t,x) +\big(a^{ij}(t,x)u_{x^i}(t,x)+ \bar b^j(t,x)u(t,x)\big)u_{x^j} + b^i(t,x)u_{x^i}(t,x)+c(t,x) u(t,x) \end{aligned} in the stripe $$(0,T)\times{\mathbb R}^d,$$ $$d\geq 1.$$ The main coefficients supposed to be in $$\text{VMO}_x$$, i.e. measurable in time and with vanishing mean oscillation in $$x.$$ The unique solvability in Sobolev spaces with mixed $$L_q(L_p)$$-norms for $$L$$ is proved if $$q\geq p$$ and for $${\mathcal L}$$ without this restriction. The technique is based on a priori pointwise estimates of the sharp functions of the second-order spatial derivatives of solutions.

##### MSC:
 35K15 Initial value problems for second-order parabolic equations 35B45 A priori estimates in context of PDEs
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##### References:
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