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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.
UDC:
517.95
MSC: 35M05 Received: 23.05.1980 and 21.04.1981
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