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Periodic boundary value problem for the equation of small oscillation of viscous liquid, perturbed by nonlinear integro-differential operator. (Ukrainian) Zbl 1249.35054
Author’s abstract: Let us denote \(\Omega=\{ x\in R^3;\,\, 0\leq x_i\leq \omega_i, \,\,i=1,2,3\} \); \(D=\{ (t,x): \,\, 0<t<T,\,\, x\in \Omega\}\), \(\Delta= {{\partial^2}\over{\partial x_1^2}}+{{\partial^2}\over{\partial x_2^2}}+{{\partial^2}\over{\partial x_3^2}}\). It is considered the following problem \[ (\partial_t-\nu \Delta)\Delta u+{{\partial^2 u}\over{\partial x_3^2}}=f(t,x)+\int\limits_{\Omega}K(t,x,\xi)F(t,\xi,u(t,\xi))d\xi,\,\, (t,x)\in D; \] \[ u(0,x)-u(T,x)=0;\,\, u_t(0,x)-u_t(T,x)=0,\,\,\, x\in \Omega; \] \[ { {\partial^{2q-1}u} \over {\partial x_j^{2q-1}}} | _{x_j=0}= {{\partial^{2q-1}u}\over{\partial x_j^{2q-1}}} | _{x_j=\omega_j}=0,\,\, q=1,2,3; \,\, j=1,2,3. \] The main result is unique solvability of this problem.
MSC:
35G15 Boundary value problems for linear higher-order PDEs
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