# On a Semilinear Sobolev-Type Mathematical Model

E.V. BychkovThis article studies a semilinear Sobolev-type mathematical model whose operator is relatively spectrally bounded. The mathematical model consists of a semilinear Sobolev-type equation of high order and initial conditions. We apply the phase space method and the theory of relatively spectrally bounded operators developed by Sviridyuk. We use Leng's method for nondegenerate equations and extend it to higher-order equations. The two cases are considered in this article. In the first case the operator L at the highest time derivative is continuously invertible, and we prove the uniqueness of solutions to the initial value problem using the theory of Banach manifolds. In the second case L has nontrivial kernel and it is known that the initial value problem with arbitrary initial data has no solution. This raises the problem of constructing and studying the phase space for the equation as the set of admissible initial data containing solutions to the equation. We construct the local phase space for the degenerate equation.Full text

- Keywords
- phase space; Sobolev-type equation; relatively spectrally bounded operator; Banach manifold; tangent bundle.
- References
- 1. Zamyshlyaeva A.A., Bychkov E.V. The Phase Space of Modified Boussinesq Equation. Bulletin of the South Ural State University. Series 'Mathematical Modelling, Programming & Computer Software', 2012, no. 18 (277), issue 12, pp. 13-19. (in Russian)

2. Sviridyuk G.А., Sukacheva T.G. [The Phase Space of a Class of Operator Equations of Sobolev Type]. Differentsial'nye uravneniya [Differential Equation], 1990, vol. 26, no. 2, pp. 250-258. (in Russian)

3. Sviriduyk G.A., Zamyshlyaeva A.A. The Phase Space of a Class of Linear Higher-Order Sobolev Type Equations. Differential Equation, 2006, vol. 42, no 2, pp. 269-278. DOI: 10.1134/S0012266106020145

4. Sagadeeva M.A. A Existence and a Stability of Solutions for Semilinear Sobolev Type Equations in Relatively Radial Case. The Bulletin of Irkutsk State University. Series 'Mathematics', 2013, no. 1, pp. 78-88. (in Russian)

5. Manakova N.А., Dylkov A.G. On One Optimal Control Problem with a Penalty Functional in General Form. The Journal of Samara State Technical University. Series 'Physical and Mathematical Sciences', 2011, no. 4(25), pp. 18-24. (in Russian)

6. Leng S. Introduction to Differentiable Manifolds. Springer-Verlag, N. Y., 2002.

7. Nirenberg L. Topics in Nonlinear Functional Analysis. New ed. (AMS), N. Y., 2001.