Abstract
In this chapter, a diffusion model system of equations is analyzed, and entire solutions are established under some conditions on its nonlinearity. The starting point of this work is raised by the following question: can one establish new results related to the existence and asymptotic behaviour of solutions for such systems as the one considered? We believe that this question deserves investigation, which can be structured in several scientific research objectives. The results achieved in the chapter, generated by the above question, are of high interest in the academic society and industry and want to convey a great variety of applications.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Alvarez, O.: A quasilinear elliptic equation in \( \mathbb{R}^{N}\). In: Proceedings of the Royal Society of Edinburgh Section A: Mathematics, vol. 126, pp. 911–921 (1996)
Arnold, L.: Stochastic Differential Equations. Wiley, New York (1974)
Bensoussan, A., Sethi, S.P., Vickson, R., Derzko, N.: Stochastic production planning with production constraints. SIAM J. Control Optim. 22, 920–935 (1984)
Covei, D.-P.: On the radial solutions of a system with weights under the Keller-Osserman condition. J. Math. Anal. Appl. 447, 167–180 (2017)
Covei, D.-P.: Boundedness and blow-up of solutions for a nonlinear elliptic system. Int. J. Math. 25(9), 1–12 (2014)
Covei, D.-P.: Existence Theorems for a Class of Systems Involving Two Quasilinear Operators, to be published in vol. 83. Izvestiya: Mathematics (2019). https://doi.org/10.1070/IM8731
Ghosh, M.K., Arapostathis, A., Marcus, S.I.: Optimal control of switching diffusions with application to flexible manufacturing systems. SIAM J. Control. Optim. 31(5), 1183–1204 (1993)
Gregorio, D.: A note on the Liouville method applied to elliptic eventually degenerate fully nonlinear equations governed by the Pucci operators and the Keller-Osserman condition. Mathematische Annalen 353(1), 145–159 (2012)
Grosse, H., Martin, A.: Particle Physics and the Schrodinger Equation. Cambridge Monographs on Particle Physic’s, Nuclear Physics and Cosmology (1997)
Franchi, B., Lanconelli, E., Serrin, J.: Existence and uniqueness of nonnegative solutions of quasilinear equations in \(\mathbb{R} ^{N}\). Adv. Math. 118, 177–243 (1996)
Fukagai, N., Narukawa, K.: On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems. Annali di Matematica 186(3), 539–564 (2007)
Hamydy, A., Massar, M., Tsouli, N.: Existence of blow-up solutions for a non-linear equation with gradient term in \(\mathbb{R }^{N}\). J. Math. Anal. Appl. 377, 161–169 (2011)
Keller, J.B.: On solution of \(\Delta u=f(u)\). Commun. Pure Appl. Math. 10, 503–510 (1957)
Krasnosel’skiĭ, M.A., Rutickiĭ, Y.B.: Convex Functions and Orlicz Spaces, Translated from the first Russian edition by Boron, L.F., Noordhoff, P., LTD. - Groningen - the Netherlands (1961)
Kon’kov, A.A.: On properties of solutions of quasilinear second-order elliptic inequalities. Nonlinear Anal.: Theory, Methods Appl. 123–124, 89–114 (2015)
Jaroŝ, J., Takaŝi, K.: On strongly decreasing solutions of cyclic systems of second-order nonlinear differential equations. In: Proceedings of the Royal Society of Edinburgh, vol. 145A, pp. 1007–1028 (2015)
Lair, A.V.: A necessary and sufficient condition for the existence of large solutions to sublinear elliptic systems. J. Math. Anal. Appl. 365(1), 103–108 (2010)
Lair, A.V.: Entire large solutions to semilinear elliptic systems. J. Math. Anal. Appl. 382, 324–333 (2011)
Lasry, J.M., Lions, P.L.: Nonlinear Elliptic Equations with Singular Boundary Conditions and Stochastic Control with State Constraints. Mathematische Annalen, vol. 283, pp. 583–630 (1989)
Lieberman, G.M.: Asymptotic behavior and uniqueness of blow-up solutions of quasilinear elliptic equations. J. d’Analyse Math ématique 115, 213–249 (2011)
Li, H., Zhang, P., Zhang, Z.: A remark on the existence of entire positive solutions for a class of semilinear elliptic systems. J. Math. Anal. Appl. 365, 338–341 (2010)
Losev, A.G., Mazepa, E.A.: On asymptotic behavior of positive solutions of some quasilinear inequalities on model Riemannian manifolds. Ufa Math. J. 5, 83–89 (2013)
Luthey, Z.A.: Piecewise Analytical Solutions Method for the Radial Schrodinger Equation, Ph.D. Thesis in Applied Mathematics, Harvard University, Cambridge (1974)
Mazepa, E.A.: The Positive Solutions to Quasilinear Elliptic Inequalities on Model Riemannian Manifolds. Russian Mathematics, vol. 59, 18–25 (2015)
Naito, Y., Usami, H.: Entire Solutions of the Inequality \({div}(A(\left|Du\right|)Du)\ge f(u)\). Mathematische Zeitschrift, vol. 225, pp. 167–175 (1997)
Naito, Y., Usami, H.: Nonexistence results of positive entire solutions for quasilinear elliptic inequalities. Can. Math. Bull. 40, 244–253 (1997)
Osserman, R.: On the inequality\(\Delta u\ge f(u)\). Pac. J. Math. 7, 1641–1647 (1957)
Smooke, M.D.: Error estimates for piecewise perturbation series solutions of the radial Schrödinger equation. SIAM J. Numer. Anal. 20, 279–295 (1983)
Soria, J.: Tent Spaces based on weighted Lorentz spaces, Carleson Measures. A dissertation presented to the Graduate School of Arts and Sciences of Washington University in partial fulfillment of the requirements for the degree of Doctor of Philosophy (1990)
Yang, H.: On the existence and asymptotic behavior of large solutions for a semilinear elliptic problem in \(\mathbb{R}^{N}\). Commun. Pure Appl. Anal. 4, 197–208 (2005)
Zhang, Z., Zhou, S.: Existence of entire positive k-convex radial solutions to Hessian equations and systems with weights. Appl. Math. Lett. 50, 48–55 (2015)
Zhang, X.: A necessary and sufficient condition for the existence of large solutions to ‘mixed’ type elliptic systems. Appl. Math. Lett. 25, 2359–2364 (2012)
Acknowledgements
The author would like to thank Professor Traian A. Pirvu for valuable comments and suggestions which further improved this chapter.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Covei, DP. (2020). Entire Solutions of a Nonlinear Diffusion System. In: Dutta, H., Peters, J. (eds) Applied Mathematical Analysis: Theory, Methods, and Applications. Studies in Systems, Decision and Control, vol 177. Springer, Cham. https://doi.org/10.1007/978-3-319-99918-0_15
Download citation
DOI: https://doi.org/10.1007/978-3-319-99918-0_15
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-99917-3
Online ISBN: 978-3-319-99918-0
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)