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Some problems for linear partial differential equations with constant coefficients in the entire space and for a class of degenerate equations in a halfspace
A. S. Kalashnikov
Abstract:
In the space $\mathbf R^{n+1}=\mathbf R_t^1\times\mathbf R_x^n$ we consider a linear partial differential equation with constant coefficients which is solvable in the leading derivative with respect to $t$. We prove that two problems with limit conditions as $t\to-\infty$ which are imposed on the Fourier transform $F_{x\to\sigma}[u(t,x)]$ and contain weight factors, are uniquely solvable in the class of functions $u(t,x)$ which for every $t$ belong to $L_2(\mathbf R_x^n)$ along with the derivatives appearing in the equation and which grow at an order no faster that $t$ as $t\to+\infty$ (in $L_2$). We apply these results to a class of equations in a halfspace which degenerate on the boundary hyperplane.
Bibliography: 9 titles.
Received: 20.05.1970
Citation:
A. S. Kalashnikov, “Some problems for linear partial differential equations with constant coefficients in the entire space and for a class of degenerate equations in a halfspace”, Mat. Sb. (N.S.), 85(127):2(6) (1971), 189–200; Math. USSR-Sb., 14:2 (1971), 186–198
Linking options:
https://www.mathnet.ru/eng/sm3203https://doi.org/10.1070/SM1971v014n02ABEH002612 https://www.mathnet.ru/eng/sm/v127/i2/p189
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Abstract page: | 260 | Russian version PDF: | 84 | English version PDF: | 1 | References: | 35 |
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