A note on the difference schemes for hyperbolic-elliptic equations.

Ashyralyev, A.; Judakova, G.; Sobolevskii, P.E.

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In the case of initial data belonging to a wide class of functions including distributions of Gelfand-Shilov type we establish the correct solvability of the Cauchy problem for a new class of Shilov parabolic systems of equations with partial derivatives with bounded smooth variable lower coefficients and nonnegative genus. We also investigate the conditions of local improvement of the convergence of a solution of this problem to its limiting value when the time variable tends to zero. ...

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Let $S \subset ℝ^{n + 1}$ be the graph of the function $ϕ : {[- 1, 1]}^{n} \to ℝ$ defined by $ϕ (x_{1}, \dots, x_{n}) = \sum_{j = 1}^{n} {| x_{j} |}^{β_{j}},$ with 1< $β_{1} \leq \dots \leq β_{n},$ and let $μ$ the measure on $ℝ^{n + 1}$ induced by the Euclidean area measure on S. In this paper we characterize the set of pairs (p,q) such that the convolution operator with $μ$ is $L^{p}$ - $L^{q}$ bounded.