Abstract
The initial-boundary value problem for a class of linear and nonlinear equations in Hilbert space is considers. We prove the existence and uniqueness of solution of this problem. The results of this investigation are applied to solvability of initial-boundary value problems for quasilinear impulsive hyperbolic equations with non-stationary transmission and boundary conditions.
Highlights
Biνui t ui t Ai t ui t fi t, hyperbolic equations, Cki ν t uk t k1 giν t, non stationary boundary and transmission conditions, m
We offer the analogues abstract model of investigation of mixed problem with non stationary boundary and transmission conditions for impulsive linear and semilinear hyperbolic equations
In the Hilbert space Hi, it was defined the system of the inner products ·, · Hi t, which generate uniform equivalent norms, that is, c1−1 u
Summary
Biνui t ui t Ai t ui t fi t , hyperbolic equations , Cki ν t uk t k1 giν t , non stationary boundary and transmission conditions , m. 0, T , the linear operators Cki ν t , that act from Hk to Xνi , are is strongly continuously differentiable ν 1, . Let conditions (i)–(xi) are satisfied, the problem 1.1 - 1.2 has a unique solution. We define the operator A t in the Hilbert space H in the following way: Atw
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