no. 2
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.

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 
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     title = {Qualitative properties of coupled parabolic systems of evolution equations},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {287--312},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 7},
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     year = {2008},
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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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

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

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

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

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

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

[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 | Zbl

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

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

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