A new generation criterion theorem for \(C_0\)-semigroups implying a generalization of Kaiser-Weis-Batty’s perturbation theorem. (English) Zbl 07785489
Summary: By proving existence, regularity and uniqueness of solutions to Cauchy problems governed by abstract unbounded operators with finite pseudo-spectral bounds as an alternative and a serious enhancement of results by Melnikova and Filinkov, we establish a new generation criterion theorem for \(C_0\)-semigroups in general Banach spaces. A generalization of Kaiser-Weis-Batty’s perturbation generation theorem in reflexive Banach spaces is therefore derived. We apply our last theoretical result to a singular transport model in \(L^p\)-spaces, \(p\in]1,+\infty[\), where the streaming (unperturbed) semigroup cannot be explicit.
MSC:
| 47Dxx | Groups and semigroups of linear operators, their generalizations and applications |
| 47Axx | General theory of linear operators |
| 82Cxx | Time-dependent statistical mechanics (dynamic and nonequilibrium) |
Keywords:
principal dynamics; semigroup; generator; perturbation; Banach space; transport operator; boundary conditionsReferences:
| [1] | Arendt, W.; Rhandi, A., Perturbations of positive semigroups, Arch. Math., 56, 107-119 (1991) · Zbl 0687.47031 · doi:10.1007/BF01200341 |
| [2] | Batty, CJK, On a perturbation theorem of Kaiser and Weis, Semigroup Forum, 70, 471-474 (2005) · Zbl 1098.47035 · doi:10.1007/s00233-005-0504-2 |
| [3] | Boumhamdi, M.; Latrach, K.; Zeghal, A., Existence results for a nonlinear version of Rotenberg model with infinite maturation velocities, Math. Methods Appl. Sci., 38, 1795-1807 (2015) · Zbl 1354.92026 · doi:10.1002/mma.3187 |
| [4] | Chabi, M.; Latrach, K., On singular mono-energetic transport equations in slab geometry, Math. Methods Appl. Sci., 25, 1121-1147 (2002) · Zbl 1021.47051 · doi:10.1002/mma.330 |
| [5] | Chabi, M., Latrach, K.: Singular one-dimensional transport equations on \(L^p\) spaces. J. Math. Anal. Appl. 283, 319-336 (2003) · Zbl 1043.82033 |
| [6] | Engel, KJ; Nagel, R., One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics (2000), New York: Springer, New York · Zbl 0952.47036 |
| [7] | Greenberg, W.; van der Mee, C.; Protopopescu, V., Boundary Value Problems in Abstract Kinetic Theory (1987), Basel: Birkhäuser, Basel · Zbl 0624.35003 · doi:10.1007/978-3-0348-5478-8 |
| [8] | Kaiser, C.; Weis, L., A perturbation theorem for operator semigroups in Hilbert spaces, Semigroup Forum, 67, 63-75 (2003) · Zbl 1046.47037 · doi:10.1007/s002330010166 |
| [9] | Kato, T., Perturbation Theory for Linear Operators, Classics in Mathematics (1995), Berlin: Springer, Berlin · Zbl 0836.47009 · doi:10.1007/978-3-642-66282-9 |
| [10] | Lods, B., On linear kinetic equations involving unbounded cross-sections, Math. Methods Appl. Sci., 27, 1049-1075 (2004) · Zbl 1072.37062 · doi:10.1002/mma.485 |
| [11] | Lods, B., Semigroup generation properties of streaming operators with non-contractive boundary conditions, Math. Comput. Modell., 42, 1441-1462 (2005) · Zbl 1103.47033 · doi:10.1016/j.mcm.2004.12.007 |
| [12] | Lumer, G.; Phillips, RS, Dissipative operators in a Banach space, Pacif. J. Math., 11, 679-698 (1961) · Zbl 0101.09503 · doi:10.2140/pjm.1961.11.679 |
| [13] | Melnikova, IV; Filinkov, A., Abstract Cauchy Problems: Three Approaches, Monographs and Surveys in Pure and Applied Mathematics (2001), New York: Chapman and Hall/CRC, New York · Zbl 0982.34001 |
| [14] | Mokhtar-Kharroubi, M., Mathematical Topics in Neutron Transport Theory: New Aspects, Series on Advances in Mathematics for Applied Sciences (1997), Singapore: World Scientific, Singapore · Zbl 0997.82047 |
| [15] | Mokhtar-Kharroubi, M., Optimal spectral theory of the linear Boltzmann equation, J. Funct. Anal., 226, 21-47 (2005) · Zbl 1088.47033 · doi:10.1016/j.jfa.2005.02.014 |
| [16] | Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44. Springer, New York (1983) · Zbl 0516.47023 |
| [17] | Rotenberg, M., Transport theory for growing cell populations, J. Theor. Biol., 103, 181-199 (1983) · doi:10.1016/0022-5193(83)90024-3 |
| [18] | Shi, D.-K., Feng, D.-X.: Characteristic conditions of the generation of \(C_0\) semigroups in a Hilbert space. J. Math. Anal. Appl. 247, 356-377 (2000) · Zbl 1004.47026 |
| [19] | Voigt, J., On the perturbation theory for strongly continuous semigroups, Math. Ann., 229, 163-17l (1977) · Zbl 0338.47018 · doi:10.1007/BF01351602 |
| [20] | Voigt, J., On substochastic \(C_0\)-semigroups and their generators, Transp. Theory Stat. Phys., 16, 4-6, 453-466 (1987) · Zbl 0634.47040 · doi:10.1080/00411458708204302 |
| [21] | Weis, L., The stability of positive semigroups on \(L_p\) spaces, Proc. Am. Math. Soc., 123, 3089-3094 (1995) · Zbl 0851.47028 |
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