Qualitative properties of coupled parabolic systems of evolution equations
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 2, p. 287-312
We apply functional analytical and variational methods in order to study well-posedness and qualitative properties of evolution equations on product Hilbert spaces. To this aim we introduce an algebraic formalism for matrices of sesquilinear mappings. We apply our results to parabolic problems of different nature: a coupled diffusive system arising in neurobiology, a strongly damped wave equation, and a heat equation with dynamic boundary conditions.
Classification:  11G35,  35K45,  47D09
@article{ASNSP_2008_5_7_2_287_0,
author = {Cardanobile, Stefano and Mugnolo, Delio},
title = {Qualitative properties of coupled parabolic systems of evolution equations},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 7},
number = {2},
year = {2008},
pages = {287-312},
zbl = {1179.35181},
mrnumber = {2437029},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2008_5_7_2_287_0}
}

Cardanobile, Stefano; Mugnolo, Delio. Qualitative properties of coupled parabolic systems of evolution equations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 2, pp. 287-312. http://www.numdam.org/item/ASNSP_2008_5_7_2_287_0/

[1] F. Ali Mehmeti and S. Nicaise, Nonlinear interaction problems, Nonlinear Anal. 20 (1993), 27-61. | MR 1199063 | Zbl 0817.35035

[2] H. Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 11 (1984), 593-676. | Numdam | MR 808425 | Zbl 0625.35045

[3] W. Arendt, Semigroups and evolution equations: functional calculus, regularity and kernel estimates, In: “Handbook of Differential Equations: Evolutionary Equations”, Vol. 1, C. M. Dafermos and E. Feireisl (eds.), North Holland, Amsterdam, 2004. | MR 2103696 | Zbl 1082.35001

[4] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, “Vector-valued Laplace Transforms and Cauchy Problems", Monographs in Mathematics n. 96, Birkhäuser, Basel, 2001. | MR 1886588 | Zbl 0978.34001

[5] S. Binczak, J. C. Eilbeck and A. C. Scott, Ephaptic coupling of myelinated nerve fibres, Phys. D 148 (2001), 159-174. | MR 1811390 | Zbl 0961.92007

[6] H. Bokil, N. Laaris, K. Blinder, M. Ennis and A. Keller, Ephaptic interactions in the mammalian olfactory system, J. Neurosci. 21 (2001), 21:RC173, 1-5.

[7] V. Casarino, K.-J. Engel, R. Nagel and G. Nickel, A semigroup approach to boundary feedback systems, Integral Equations Operator Theory 47 (2003), 289-306. | MR 2012840 | Zbl 1048.47054

[8] S. Cardanobile and D. Mugnolo, Analysis of a FitzHugh-Nagumo-Rall model of a neuronal network, Math. Methods Appl. Sci. 30 (2007), 2281-2308. | MR 2362954 | Zbl 1195.92007

[9] S. Cardanobile, D. Mugnolo and R. Nittka, Well-posedness and symmetries of strongly coupled network equations. J. Phys. A 41 (2008). | MR 2433422 | Zbl 1132.35389

[10] E.B. Davies, “Heat Kernels and Spectral Theory", Cambridge Tracts in Mathematics, n. 92, Cambridge University Press, Cambridge, 1990. | MR 1103113 | Zbl 0699.35006

[11] K.-J. Engel, “Operator Matrices and Systems of Evolution Equations", Book manuscript. | Zbl 0936.34044

[12] M. Haase, “The Functional Calculus for Sectorial Operators”, Oper. Theory Adv. Appl., Vol. 169, Birkhäuser, Basel, 2006. | MR 2244037 | Zbl 1101.47010

[13] G.R. Holt and C. Koch, Electrical interaction via the extracellular potential near cell bodies, J. Comput. Neurosci. 2 (1999), 169-184. | Zbl 0927.92007

[14] D. Mugnolo, Matrix methods for wave equations, Math. Z. 253 (2006), 667-680. | MR 2221094 | Zbl 1112.47036

[15] D. Mugnolo, Gaussian estimates for a heat equation on a network, Netw. Heter. Media 2 (2007), 55-79. | MR 2291812 | Zbl 1142.35349

[16] D. Mugnolo, A variational approach to strongly damped wave equations, In: “Functional Analysis and Evolution Equations: Dedicated to Gunter Lumer", H. Amann et al. (eds.), Birkhäuser, Basel, 2007, 503-514. | MR 2402747 | Zbl 1171.35439

[17] R. Nagel, Towards a “matrix theory” for unbounded operator matrices, Math. Z. 201 (1989), 57-68. | MR 990188 | Zbl 0672.47001

[18] E. M. Ouhabaz, ${L}^{p}$-contraction semigroups for vector-valued functions, Positivity 3 (1999), 83-93. | MR 1675466 | Zbl 0935.47030

[19] E. M. Ouhabaz, “Analysis of Heat Equations on Domains", LMS Monograph Series, n. 30, Princeton University Press, Princeton, 2004. | MR 2124040 | Zbl 1082.35003