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Second-order hyperbolic equations with strong characteristic degeneracy at the initial hypersurface
A. V. Deryabina State Academy of Consumer Services
Abstract:
Equations of the following form are considered:
\begin{equation}
\psi^2(t,x)u_{tt}+\varphi(t,x)u_t-\sum_{i,j}\bigl(a^{ij}(t,x)u_{x_i}\bigr)_{x_j}+\sum_ib^i(t,x)u_{x_i}+c(t,x)u=f(t,x),
\tag{1}
\end{equation}
where
\begin{gather*}
(t,x)\in H=(0,T)\times\mathbb R^n, \qquad \psi(t,x)\geqslant 0, \qquad \varphi(t,x)\geqslant0;
\\
\sum_{i,j}a^{ij}(t,x)\xi_i\xi_j\geqslant0 \quad \forall\,(t,x)\in H, \quad \forall\,\xi=(\xi_1,\dots,\xi_n)\in\mathbb R^n.
\end{gather*}
In place of the Cauchy problem for (1), a problem without initial data but with constraints on the admissible growth of the solution as
$t\to0$ and as
$|x|\to\infty$ is discussed. The unique solubility of (1) in certain Sobolev-type weighted spaces is proved. The smoothness properties of generalized solutions are studied.
UDC:
517.956
MSC: Primary
35L80; Secondary
35D10 Received: 12.05.1998 and 17.09.1999
DOI:
10.4213/sm469
Fulltext:
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