The identification of an unknown coefficient in the lower term of pseudoparabolic differential equation of diffusion (u + ηM u)t + M u + ku = f and elliptic second order differential equation M u + ku = f with the Dirichlet boundary condition is considered. The identification of k is based on an integral boundary data. The local existence and uniqueness of generalized strong solutions for the inverse problems are proved. The stability estimates are exposed.
We consider the inverse problem for a quasilinear second order elliptic equation with the unknown coefficient at the lower-order term and the first kind boundary condition and the integral overedetermination condition on the boundary. We prove a theorem on the local existence and uniqueness of a strong solution to the problem. The result is illustrated by an example of a nonlinear equation satisfying all the assumptions of the theorem. Bibliography: 6 titles.
The asymptotic behavior of the strong solution to the inverse problem on recovering an unknown coefficient k(t) in a pseudoparabolic equation (u + ηMu) t + Mu + k(t)u = f is investigated. The differential operator M of the second order with respect spacial variables is supposed to be elliptic and selfajoint. It is proved that the solution of the inverse problem stabilizes to the solution of the appropriate stationary inverse problem as t → + ∞.
We establish the stabilization of the strong solution ( u ( t , x ) , k ( t ) ) {(u(t,x),k(t))} to the inverse problem for a pseudoparabolic equation ( u + η M u ) t + M u + k ( t ) u = f {(u+\eta Mu)_{t}+Mu+k(t)u=f} with an unknown coefficient k ( t ) {k(t)} to the solution ( u ∞ , k ∞ ) {(u^{\infty},k^{\infty})} of the appropriate stationary inverse problem. The operator M = - div ( ℳ ( x ) ∇ ) + m ( x ) I {M=-\operatorname{div}(\mathcal{M}(x)\nabla)+m(x)I} is supposed to be elliptic and self-adjoint. The asymptotic behavior of the solution ( u ( t , x ) , k ( t ) ) {(u(t,x),k(t))} is investigated as t → + ∞ {t\to+\infty} .
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