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 Mat. Sb. (N.S.), 1982, Volume 118(160), Number 2(6), Pages 252–261 (Mi msb2251)

A nonlocal boundary value problem for a class of Petrovskii well-posed equations

S. Ya. Yakubov

Abstract: As is well known, the mixed problem for the entire class of Petrovskii well-posed partial differential equations has not been studied. In this paper, a certain subclass of Petrovskii well-posed equations for which it is possible to state and study mixed problems, is isolated. In the rectangle $[0,T]\times[0,1]$, consider the equation
$$D_t^2u+aD_tD_x^{2k}u+bD_x^{2p}u+\sum\limits_{\alpha\leqslant{2k-1}} a_\alpha(t,x)D_tD_x^\alpha+\sum\limits_{\alpha\leqslant{2p-1}}b_\alpha(t,x)D_x^\alpha u=f(t, x)$$
with boundary conditions
$$L_\nu u=\alpha_\nu u_x^{(q_\nu)}(t,0)+\beta_\nu u_x^{(q_\nu)}(t,1)+ T_\nu u(t,\cdot)=0, \qquad \nu=1\div2k,$$
for $p\leqslant k$, where $|\alpha_\nu|+|\beta_\nu|\ne 0$, $\nu=1\div2k$, $0\leqslant q_\nu\leqslant q_{\nu+1}$, $q_\nu<q_{\nu+2}$, $T_\nu$ is a continuous linear functional in $W_q^{q_\nu}(0, 1)$, $q<+\infty$, and for $k<p<2k$
$$L_{2k+s}u=L_{n_s}u^{(2k)}=\alpha_{n_s}u_x^{(q_{n_s}+2k)}(t,0)+ \beta_{n_s}u_x^{(q_{n_s}+2k)}(t,1)+T_{n_s}u_x^{(2k)}(t,\cdot)=0,$$
$s=1\div2p-2k$, $1\leqslant n_s\leqslant2k$, and with initial conditions $u(0,x)=u_0(x)$ and $u'_t(0,x)=u_1(x)$.
Well-posedness conditions are found for this problem.
Bibliography: 9 titles.

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English version:
Mathematics of the USSR-Sbornik, 1983, 46:2, 255–265

Bibliographic databases:

UDC: 517.95
MSC: 35M05

Citation: S. Ya. Yakubov, “A nonlocal boundary value problem for a class of Petrovskii well-posed equations”, Mat. Sb. (N.S.), 118(160):2(6) (1982), 252–261; Math. USSR-Sb., 46:2 (1983), 255–265

Citation in format AMSBIB
\Bibitem{Yak82} \by S.~Ya.~Yakubov \paper A~nonlocal boundary value problem for a class of Petrovskii well-posed equations \jour Mat. Sb. (N.S.) \yr 1982 \vol 118(160) \issue 2(6) \pages 252--261 \mathnet{http://mi.mathnet.ru/msb2251} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=658791} \zmath{https://zbmath.org/?q=an:0549.35053|0514.35039} \transl \jour Math. USSR-Sb. \yr 1983 \vol 46 \issue 2 \pages 255--265 \crossref{https://doi.org/10.1070/SM1983v046n02ABEH002779} 

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This publication is cited in the following articles:
1. Balalayev M., “On Correct Solvability of Arbitrary-Order Differential-Operator Equations”, 317, no. 3, 1991, 526–529
2. Favini A., “Parabolicity of 2nd-Order Differential-Equations in Hilbert-Space”, Semigr. Forum, 42:3 (1991), 303–312
3. Aliev I., “Operator-Differential Equations with Nonregular Boundary-Conditions and Applications”, Differ. Equ., 30:1 (1994), 84–93
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