Abstract
We consider an abstract fourth order boundary value problem where the coefficients are accretive operators in Hilbert space. We show existence, uniqueness and maximal regularity of the solution under some necessary and sufficient conditions on the data. To this end, we give an explicit representation formula, using analytic semigroups, sectorial operators with Bounded Imaginary Powers, the theory of strongly continuous cosine operator functions and the perturbation theory of m-accretive operators. An illustrative example is also given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems. Birkhäuser, Basel (2001)
Crouzeix, M.: Operators with numerical range in a parabola. Arch. Math. 82, 517–527 (2004)
Davies, E.B.: One Parameter Semigroups. Academic Press, London (1980)
Engel, K.J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (1999)
Fattorini, H.O.: Ordinary differential equations in linear topological spaces I. J. Differ. Equ. 5, 72–105 (1969)
Fattorini, H.O.: Second Order Linear Differential Equations in Banach Spaces. North-Holland, Amsterdam (1985)
Favini, A., Labbas, R., Maingot, S., Tanabe, H., Yagi, A.: Complete abstract differential equations of elliptic type in UMD spaces. Funkc. Ekv. 49, 193–214 (2006)
Favini, A., Labbas, R., Maingot, S., Tanabe, H., Yagi, A.: A simplified approach in the study of elliptic differential equations in UMD Spaces and new applications. Funkc. Ekv. 51, 165–187 (2008)
Grisvard, P.: Spazi di tracce e applicazioni. Rend. Mat. 5, 657–729 (1972)
Gustafson, K.: A perturbation lemma. Bull. Am. Math. Soc. 72, 334–338 (1966)
Gustafson, K., Rao, D.: Numerical range and accretivity of operator products. J. Math. Anal. Appl. 60(3), 693–702 (1977)
Hayashi, M., Ozawa, T.: On Landau–Kolmogorov inequalities for dissipative operators. Proc. Amer. Math. Soc. 145, 847–852 (2017)
Kato, T.: Fractional powers of dissipative operators. Proc. Jpn. Acad. 13(3), 246–274 (1961)
Kato, T.: Perturbation Theory for Linear Operators. Springer, New York (1995)
Krein, S.G.: Linear Differential Equations in Banach Spaces. Birkhäuser, Boston (1982)
Labbas, R., Lemrabet, K., Maingot, S., Thorel, A.: Generalized linear models for population dynamics in two juxtaposed habitats. Discrete Contin. Dyn. Syst. Ser A. 39(5), 2933–2960 (2019)
Labbas, R., Maingot, S., Manceau, D., Thorel, A.: On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain. J. Math. Anal. Appl. 450, 351–376 (2017)
Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. II. Springer, Berlin-Heidelberg (1972)
Lions, J.-L., Peetre, J.: Sur une classe d’espaces d’interpolation. Publ. Math. l’IHÉS 19, 5–68 (1964)
Okazawa, N.: Perturbations of linear m-accretive operators. Proc. Am. Math. Soc. 37(1), 169–174 (1973)
Okazawa, N.: Two perturbation theorems for contraction semigroups in a Hilbert space. Proc. Jpn. Acad. 45, 850–853 (1969)
Okazawa, N.: On the perturbation of linear operators in Banach and Hilbert spaces. J. Math. Soc. Jpn. 34, 677–701 (1982)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin-Heidelberg-New York (1983)
Phillips, R.S.: Dissipative operators and hyperbolic systems of partial differential equations. Proc. Am. Math. Soc. 90(2), 193–254 (1959)
Prüss, J.: Evolutionary Integral Equations and Applications. Birkhäuser, Basel (1993)
Prüss, J., Sohr, H.: On operators with bounded imaginary powers in Banach spaces. Math. Z. 203, 429–452 (1990)
Prüss, J., Sohr, H.: Imaginary powers of elliptic second order differential operators in \(L^p\)-spaces. Hiroshima Math. J. 23(1), 161–192 (1993)
Takenaka, T., Okazawa, N.: Abstract Cauchy problems for second order linear differential equations in a Banach space. Hiroshima Math. J. 17, 591–612 (1987)
Thorel, A.: Operational approach for biharmonic equations in \(L^p\)-spaces. J. Evol. Equ. 20, 631–657 (2020)
Travis, C.C., Webb, G.F.: Cosine families and abstract nonlinear second order differential equations. Acta Math. Hungar. 32(3–4), 75–96 (1978)
Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978)
Yoshikawa, A.: On perturbation of closed operators in a Banach space. J. Fac. Sci. Hokkaido Univ. 22, 50–61 (1972)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Abdelaziz Rhandi.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported by the Laboratory of Fundamental and Applicable Mathematics of Oran (LMFAO) and the Algerian research project: PRFU, no. C00L03ES310120220003 (DGRSDT).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Benharrat, M., Bouchelaghem, F. & Thorel, A. On the solvability of fourth-order boundary value problems with accretive operators. Semigroup Forum 107, 17–39 (2023). https://doi.org/10.1007/s00233-023-10365-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-023-10365-y