The Barenblatt - Zheltov - Kochina Model on The Segment with Wentzell Boundary Conditions

In terms of the theory of relative p-bounded operators, we study the Barenblatt-Zheltov-Kochina model, which describes dynamics of pressure of a filtered fluid in a fractured-porous medium with general Wentzell boundary conditions. In particular, we consider spectrum of one-dimensional Laplace operator on the segment $[0,1]$ with general Wentzell boundary conditions. We examine the relative spectrum in one-dimensional Barenblatt-Zheltov-Kochina equation, and construct the resolving group in the Cauchy-Wentzell problem with general Wentzell boundary conditions. In the paper, these problems are solved under the assumption that the initial space is a contraction of the space L^2(0,1).

Keywords

Barenblatt-Zheltov-Kochina model; relatively p-bounded operator; phase space; C0-contraction semigroups; Wentzell boundary conditions.

References

1. Wentzell A.D. Semigroups of Operators Corresponding to a Generalized Differential Operator of Second Order. Doklady Academii Nauk SSSR, 1956, vol. 111, pp. 269-272. (in Russian)
2. Feller W. Generalized Second Order Differential Operators and Their Lateral Conditions. Illinois Journal of Mathematics, 1957, vol. 1, no. 4, pp. 459-504. DOI: 10.1215/ijm/1255380673
3. Wentzell A.D. On Boundary Conditions for Multidimensional Diffusion Processes. Theory of Probability and its Applications, 1959, vol. 4, pp. 164-177. DOI: 10.1137/1104014
4. Favini A., Goldstein G.R., Goldstein J.A., Romanelli S. Classification of General Wentzell Boundary Conditions for Fourth Order Operators in One Space Dimension. Journal of Mathematical Analysis and Applications, 2007, vol. 333, no. 1, pp. 219-235. DOI: 10.1016/j.jmaa.2006.11.058
5. Coclite G.M., Favini A., Gal C.G., Goldstein G.R., Goldstein J.A. Obrecht E., Romanelli S. The Role of Wentzell Boundary Conditions in Linear and Nonlinear Analysis. Advances in Nonlinear Analysis: Theory, Methods and Applications, 2009, vol. 3, pp. 279-292.
6. Gal C.G. Sturm-Liouville Operator with General Boundary Conditions. Electronic Journal of Differential Equations, 2005, vol. 2005, no. 120, pp. 1-17.
7. Favini A., Goldstein G.R., Goldstein J.A. The Laplacian with Generalized Wentzell Boundary Conditions. Progress in Nonlinear Differential Equations and Their Applications, 2003, vol. 55, pp. 169-180. DOI: 10.1007/978-3-0348-8085-5_13
8. Favini A., Goldstein G.R., Goldstein J.A., Romanelli S. The Heat Equation with Generalized Wentzell Boundary Condition. Journal of Evolution Equations, 2002, vol. 2, pp. 1-19. DOI: 10.1007/s00028-002-8077-y
9. Sviridyuk G.A., Manakova N.A. The Barenblatt-Zheltov-Kochina Model with Additive White Noise in Quasi-Sobolev Spaces. Journal of Computational and Engineering Mathematics, 2016, vol. 3, no 1, pp. 61-67. DOI: 10.14529/jcem160107
10. Banasiak J. Mathematical Properties of Inelastic Scattering Models in Linear Kinetic Theory. Mathematical Models and Methods in Applied Sciences, 2000, vol. 10, no 2, pp. 163-186. DOI: 10.1142/S0218202500000112
11. Banasiak J., Lachowicz M., Moszynski M. Chaotic Behavior of Semigroups Related to the Process of Gene Amplification-Deamplification with Cell Proliferation. Mathematical Biosciences, 2007, vol. 206, no 2, pp. 200-215. DOI: 10.1016/j.mbs.2005.08.004