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On solvability of perturbed Sobolev type equations. (English. Russian original) Zbl 1206.34080
St. Petersbg. Math. J. 20, No. 4, 645-664 (2009); translation from Algebra Anal. 20, No. 4, 189-217 (2008).
Summary: Linear Sobolev type equations \(L\dot u(t)=Mu(t)+Nu(t),\quad t\in\overline{\mathbb{R}}_+,\) are considered, with degenerate operator \( L\), strongly \( (L,p)\)-radial operator \( M\), and perturbing operator \( N\). By using methods of perturbation theory for operator semigroups and the theory of degenerate semigroups, unique solvability conditions for the Cauchy problem and Showalter problem for such equations are deduced. The abstract results obtained are applied to the study of initial boundary value problems for a class of equations, the operators in which are polynomials of elliptic selfadjoint operators, including various equations of filtration theory. Perturbed linearized systems of the phase space equations and of the Navier-Stokes equations are also considered. In all cases the perturbed operators are integral or differential.

MSC:
34G10 Linear differential equations in abstract spaces
34G25 Evolution inclusions
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
35G10 Initial value problems for linear higher-order PDEs
47D06 One-parameter semigroups and linear evolution equations
47J35 Nonlinear evolution equations
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