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Some nonclassical boundary value problems for linear mixed-type equations. (Russian, English) Zbl 1049.35144

Sib. Mat. Zh. 44, No. 1, 44-52 (2003); translation in Sib. Math. J. 44, No. 1, 37-43 (2003).
The author studies boundary value problems in a cylinder \(Q = D \times (0,T)\), \(D \subset \mathbb R^n\), for the mixed-type equation \[ k(x,t)u_{tt} + a(x,t) u_t-\sum\limits^{n}_{i,j=1} {\partial\over \partial x_i} (a_{ij}(x)u_{x_j})+ b(x)u=f(x,t) \] and the parabolic equation with varying time direction \[ k(x,t)u_{t} -\sum\limits^{n}_{i,j=1} {\partial\over \partial x_i} (a_{ij}(x)u_{x_j})+ a(x,t)u=f(x,t) \] (\(\sum\nolimits^{n}_{i,j=1}a_{ij}(x)\xi_i\xi_j \geq a_0| \xi| ^2\), \(a_0 > 0\), the function \(k(x,t)\) cannot change sign) under time-periodic conditions. Assuming that \(k(x,0)=k(x,T)\), \(x\in D\), and imposing some additional conditions on the coefficients and right-hand side, he proves some existence and uniqueness theorems of regular solutions.

MSC:

35M10 PDEs of mixed type
35B10 Periodic solutions to PDEs
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