Complutense University Library

Large solutions for a system of elliptic equations arising from fluid dynamics

Díaz Díaz, Jesús Ildefonso and Lazzo, M. and Schmidt, Paul G. (2005) Large solutions for a system of elliptic equations arising from fluid dynamics. Siam Journal on Mathematical Analysis , 37 (2). pp. 490-513. ISSN 0036-1410

[img] PDF
Restricted to Repository staff only until 31 December 2020.

352kB

Official URL: http://epubs.siam.org/simax/resource/1/sjmaah/v37/i2/p490_s1?isAuthorized=no

View download statistics for this eprint

==>>> Export to other formats

Abstract

This paper is concerned with the elliptic system (0.1) Delta upsilon=phi, Delta phi=vertical bar del upsilon vertical bar(2) posed in a bounded domain Omega subset of R-N, N is an element of N. Specifically, we are interested in the existence and uniqueness or multiplicity of "large solutions," that is, classical solutions of (0.1) that approach infinity at the boundary of Omega. Assuming that Omega is a ball, we prove that the system (0.1) has a unique radially symmetric and nonnegative large solution with v(0) = 0 (obviously, v is determined only up to an additive constant). Moreover, if the space dimension N is sufficiently small, there exists exactly one additional radially symmetric large solution with v(0) = 0 (which, of course, fails to be nonnegative). We also study the asymptotic behavior of these solutions near the boundary of Omega and determine the exact blow-up rates; those are the same for all radial large solutions and independent of the space dimension. Our investigation is motivated by a problem in fluid dynamics. Under certain assumptions, the unidirectional flow of a viscous, heat-conducting fluid is governed by a pair of parabolic equations of the form (0.2) upsilon(t) -Delta upsilon=theta, theta t-Delta theta=vertical bar del upsilon vertical bar(2), where v and theta represent the fluid velocity and temperature, respectively. The system (0.1), with phi = -theta, is the stationary version of (0.2).


Item Type:Article
Uncontrolled Keywords:boundary blow-up; differential-equations; 3-dimensional systems; diffusion equations; positive solutions; existence; uniqueness; elliptic system; boundary blow-up; large solutions; radial solutions; existence and multiplicity; asymptotic behavior
Subjects:Sciences > Mathematics > Differential equations
ID Code:15428
References:

C. Azizieh, P. Cl'ement, and E. Mitidieri, Existence and a priori estimates for positive solutions of p-Laplace equations, J. Differential Equations, 184 (2002), pp. 422–442.

C. Bandle and H. Brunner, Blowup in diffusion equations: A survey, J. Comput. Appl. Math., 97 (1998), pp. 3–22.

C. Bandle, G. D'ıaz, and J. I. D'ıaz, Solutions d''equations de r'eaction-diffusion non lin'eaires explosant au bord parabolique, C. R. Acad. Sci. Paris S'er. I Math., 318 (1994), pp. 455–460.

C. Bandle and M. Marcus, "Large'' solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behaviour, J. Anal. Math., 58 (1992), pp. 9–24.

F.-C. S\c t. Cîrstea and V. D. Rǎdulescu, Entire solutions blowing up at infinity for semilinear elliptic systems, J. Math. Pures Appl., 81 (2002), pp. 827–846.

M. G. Crandall, P. L. Lions, and P. E. Souganidis, Maximal solutions and universal bounds for some partial differential equations of evolution, Arch. Rational Mech. Anal., 105 (1989), pp. 163–190.

J. I. D'ıaz, Obstruction and some approximate controllability results for the Burgers equation and related problems, in Control of Partial Differential Equations and Applications (Proc. Int. Conf., Laredo, 1994), Lecture Notes in Pure and Appl. Math. 174, E. Casas, ed., Marcel Dekker, New York, 1996, pp. 63–76.

G. D'ıaz and R. Letelier, Explosive solutions of quasilinear elliptic equations: Existence and uniqueness, Nonlinear Anal., 20 (1993), pp. 97–125.

J. Garc'ıa-Meli'an and J. D. Rossi, Boundary blow-up solutions to elliptic systems of com- petitive type, J. Differential Equations, 206 (2004), pp. 156–181.

J. Garc'ıa-Meli'an and A. Su'arez, Existence and uniqueness of positive large solutions to some cooperative elliptic systems, Adv. Nonlinear Stud., 3 (2003), pp. 193–206.

M. Ghergu and V. D. Rǎdulescu, Explosive solutions of semilinear elliptic systems with gradient term, RACSAM Rev. R. Acad. Cien. Exactas Fis. Nat. Ser. A. Mat., 97 (2003), pp. 467–475.

M. W. Hirsch, Systems of differential equation that are competitive or cooperative. V. Con- vergence in 3-dimensional systems, J. Differential Equations, 80 (1989), pp. 94–106.

M. W. Hirsch, Systems of differential equations that are competitive or cooperative. IV: Struc- tural stability in three-dimensional systems, SIAM J. Math. Anal., 21 (1990), pp. 1225– 1234.

S. Kamin and L. V'eron, Existence and uniqueness of the very singular solution of the porous media equation with absorption, J. Anal. Math., 51 (1988), pp. 245–258.

J. B. Keller, On solutions of Δu = f(u), Comm. Pure Appl. Math., 10 (1957), pp. 503–510.

A. V. Lair and A. W. Wood, Existence of entire large positive solutions of semilinear elliptic systems, J. Differential Equations, 164 (2000), pp. 380–394.

P. J. McKenna, W. Reichel, and W. Walter, Symmetry and multiplicity for nonlinear elliptic differential equations with boundary blow-up, Nonlinear Anal., 28 (1997), pp. 1213– 1225.

L. Markus, Asymptotically autonomous differential systems, in Contributions to the Theory of Nonlinear Oscillations 3, Ann. of Math. Stud. 36, S. Lefschetz, ed., Princeton University Press, Princeton, NJ, 1956, pp. 17–29.

K. Mischaikow, H. Smith, and H. R. Thieme, Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions, Trans. Amer. Math. Soc., 347 (1995), pp. 1669–1685.

R. Osserman, On the inequality Δ u ≥ f(u), Pacific J. Math., 7 (1957), pp. 1641–1647.

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), pp. 487–513.

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr. 41, AMS, Providence, RI, 1995.

J. L. V'azquez and M. Wal'ıas, Existence and uniqueness of solutions of diffusion-absorption equations with general data, Differential Integral Equations, 7 (1994), pp. 15–36.

L. V'eron, Semilinear elliptic equations with uniform blow-up on the boundary, J. Anal. Math., 59 (1992), pp. 231–250.

W. Walter, Differential and Integral Inequalities, Springer-Verlag, Berlin, 1970.

Deposited On:30 May 2012 07:39
Last Modified:30 May 2012 07:39

Repository Staff Only: item control page