Share:


A mathematical model for the use of energy resources: a singular parabolic equation

    Daniel López-García Affiliation
    ; Rosa Pardo Affiliation

Abstract

We consider a singular parabolic equation, for , arising in symmetric boundary layer flows. Here is a bounded domain with C2 boundary is bounded, and T > 0 is some fixed time. We establish sufficient conditions for the existence and uniqueness of a weak solution of this singular parabolic equation with Dirichlet boundary conditions, and we investigate its regularity.


There are two different cases depending on β. If β < 1, for any initial data, there exists a unique weak solution, which in fact is a strong solution. The singularity is removable when β < 1. While if β = 1, there exists a unique solution of the singular parabolic problem  The initial data cannot be arbitrarily chosen. In fact, if f is continuous and , as t → 0, then, this solution converges, as t → 0, to the solution of the elliptic problem, for , with Dirichlet boundary conditions. Hence, no initial data can be prescribed when β = 1, and the singularity in that case is strong.

Keyword : singular parabolic equation, degenerate parabolic equations, existence, uniqueness, symmetric boundary layer, regularity

How to Cite
López-García, D., & Pardo, R. (2020). A mathematical model for the use of energy resources: a singular parabolic equation. Mathematical Modelling and Analysis, 25(1), 88-109. https://doi.org/10.3846/mma.2020.9792
Published in Issue
Jan 13, 2020
Abstract Views
87
PDF Downloads
32
Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

References

J.R. Anderson. Local existence and uniqueness of solutions of degenerate parabolic equations. Comm. Partial Differential Equations, 16(1):105–143, 1991. https://doi.org/10.1080/03605309108820753

D. Andreucci. -estimates for local solutions of degenerate parabolic equations. SIAM J. Math. Anal., 22(1):138–145, 1991. https://doi.org/10.1137/0522008

A.K. Aziz, D.A. French, S. Jensen and R. Bruce Kellogg. Origins, analysis, numerical analysis, and numerical approximation of a forward-backward parabolic problem. M2AN Math. Model. Numer. Anal., 33(5):895–922, 1999. https://doi.org/10.1051/m2an:1999125

D. Blanchard and G.A. Francfort. A few results on a class of degenerate parabolic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 18(2):213–249, 1991.

H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.

C.Xu, Z. Wang, C. Chang and F. Sun. Energy and exergy analysis of solar power tower plants. Applied Thermal Engineering, 31:3904–3913, 2011. Available from Internet: https://doi.org/10.1016/j.applthermaleng.2011.07.038

E. DiBenedetto, Y. Kwong and V. Vespri. Local space-analyticity of solutions of certain singular parabolic equations. Indiana Univ. Math. J., 40(2):741–765, 1991. https://doi.org/10.1512/iumj.1991.40.40033

L.C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2010. https://doi.org/10.1090/gsm/019

M.S. Floater. Blow-up at the boundary for degenerate semilinear parabolic equations. Arch. Rational Mech. Anal., 114(1):57–77, 1991. https://doi.org/10.1007/BF00375685

Y. Giga, S. Goto, H. Ishii and M.-H. Sato. Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ. Math. J., 40(2):443–470, 1991. https://doi.org/10.1512/iumj.1991.40.40023

V. Giovangigli. An existence theorem for a free boundary problem of hypersonic flow theory. SIAM J. Math. Anal., 24(3):571–582, 1993. https://doi.org/10.1137/0524035

J.A. Goldstein and Q.S. Zhang. On a degenerate heat equation with a singular potential. J. Funct. Anal., 186(2):342–359, 2001. https://doi.org/10.1006/jfan.2001.3792

D. Henry. Geometric theory of semilinear parabolic equations, volume 840 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1981.

M. Ivanchov and N. Saldina. An inverse problem for strongly degenerate heat equation. Journal of Inverse and Ill-posed Problems, 14(5):465–480, 2019. https://doi.org/10.1515/156939406778247598

A.V. Ivanov. Uniform H¨older estimates for weak solutions of quasilinear doubly degenerate parabolic equations. Preprinty LOMI [LOMI Preprints], E10-89. Akad. Nauk SSSR, Mat. Inst. Leningrad. Otdel., Leningrad, 1989. https://doi.org/10.1007/BF01671935

O.V. Makhnei. Boundary problem for the singular heat equation. Carpathian Math. Publ., 9(1):86–91, 2017. https://doi.org/10.15330/cmp.9.1.86-91

I. Malyshev. On the parabolic potentials in degenerate-type heat equation. J. Appl. Math. Stochastic Anal., 4(2):147–160, 1991. https://doi.org/10.1155/S1048953391000114

K. Mochizuki and R. Suzuki. On blow-up of solutions for quasilinear degenerate parabolic equations. Publ. RIMS, Kyoto Univ., 745(27):193–201, 1991. Available from Internet: http://hdl.handle.net/2433/102185

O.A. Oleĭnik. The prandtl system of equations in boundary layer theory. Dokl. Akad. Nauk SSSR, 4(3):28–31, 1963.

O.A. Oleĭnik. On the system of boundary-layer equations for axisymmetric flows. Dokl. Akad. Nauk SSSR, 175:77–80, 1967.

N.H. Paul and J.R. Engel. Experience with the molten-salt reactor experiment. Nuclear Applications and Technology, 8(2):118–136, 1970. https://doi.org/10.13182/NT8-2-118

A. Pazy. Semigroups of linear operators and applications to partial differential equations, volume 44 of Applied Mathematical Sciences. Springer-Verlag, New York, 1983. https://doi.org/10.1007/978-1-4612-5561-1

T.N. Phuoc and L. V´eron. Initial trace of positive solutions of a class of degenerate heat equation with absorption. Discrete Contin. Dyn. Syst., 33(5):2033–2063, 2013. https://doi.org/10.3934/dcds.2013.33.2033

M.M. Porzio. -estimates for degenerate and singular parabolic equations. Nonlinear Anal., 17(11):1093–1107, 1991. https://doi.org/10.1016/0362546X(91)90194-6

P.A. Raviart and J.M. Thomas. Introduction à l’analyse numérique des équations aux dérivées partielles. Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree]. Masson, Paris, 1983.

E.C. Young. Uniqueness of solution of a singular heat equation. Internat. J. Math. Math. Sci., 7(1):201–204, 1984. https://doi.org/10.1155/S0161171284000211

J.N. Zhao. Source-type solutions of degenerate quasilinear parabolic equations. J. Differential Equations, 92(2):179–198, 1991. https://doi.org/10.1016/00220396(91)90046-C

W. Zhou and P. Lei. A one-dimensional nonlinear heat equation with a singular term. J. Math. Anal. Appl., 368(2):711–726, 2010. https://doi.org/10.1016/j.jmaa.2010.03.066