On the Well-Posedness of Some Problems of Filtration in Porous Media

Using the theory of semigroups of linear transformations, we establish the uniform well-posedness of initial-boundary value problems for a class of integrodifferential equations in bounded and half-bounded regions describing the processes of nonstationary filtration of squeezing liquid in porous media. Babenko considered a particular case of these equations on the semi-infinite straight line with Dirichlet condition on the boundary. In that work it was required to find the pressure gradient on the boundary, and the answer is obtained by the formal application of fractional integro-differentiation while ignoring the question of continuous dependence on the intial data. The solution is expressed as a formal series involving an unbounded operator, whose convergence is not discussed. The theory of strongly continuous semigroups of transformations enables us to establish the uniform well-posedness of the Dirichlet and Neumann problems for both finite and infinite regions. It enables us to calculate the pressure gradient on the boundary in the case of the Dirichlet problem and the boundary value of the solution in the case of the Neumann problem. We also prove that the solution is stable with respect to the initial data.