## Rocky Mountain Journal of Mathematics

### A dynamical system for a nonlocal parabolic equation with exponential nonlinearity

Tosiya Miyasita

#### Abstract

We consider a nonlocal parabolic and the corresponding elliptic equation with exponential nonlinearity. At first, we study the set of a stationary solution and compute its Morse index. Next, we obtain the time-global solution in the use of the Lyapunov function and define the dynamical system. Finally, we construct an exponential attractor by a squeezing property.

#### Article information

Source
Rocky Mountain J. Math., Volume 45, Number 6 (2015), 1897-1917.

Dates
First available in Project Euclid: 14 March 2016

https://projecteuclid.org/euclid.rmjm/1457960341

Digital Object Identifier
doi:10.1216/RMJ-2015-45-6-1897

Mathematical Reviews number (MathSciNet)
MR3473161

Zentralblatt MATH identifier
1338.35234

#### Citation

Miyasita, Tosiya. A dynamical system for a nonlocal parabolic equation with exponential nonlinearity. Rocky Mountain J. Math. 45 (2015), no. 6, 1897--1917. doi:10.1216/RMJ-2015-45-6-1897. https://projecteuclid.org/euclid.rmjm/1457960341

#### References

• S.B. Angenent, The Morse-Smale property for a semi-linear parabolic equation, J. Diff. Equat. 62 (1986), 427–442.
• ––––, The zeroset of a solution of a parabolic equation, J. reine angew. Math. 390 (1988), 79–96.
• H. Brezis, Analyse fonctionnelle, Dunod, 1999.
• H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of $-\Delta u = V(x) e^u$ in two dimensions, Comm. Partial Diff. Equat. 16 (1991), 1223–1253.
• E. Caglioti, P.L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description, Comm. Math. Phys. 143 (1992), 501–525.
• ––––, A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description, II, Comm. Math. Phys. 174 (1995), 229–260.
• C.-C. Chen and C.-S. Lin, On the symmetry of blowup solutions to a mean field equation, Ann. Inst. H. Poincaré Anal. Non Lin. 18 (2001), 271–296.
• M.G. Crandall and P.H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321–340.
• A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential attractors for dissipative evolution equations, John-Wiley and Sons, New-York, 1994.
• M. Fila and H. Matano, Connecting equilibria by blow-up solutions, Discr. Contin. Dynam. Syst. 6 (2000), 155–164.
• M. Fila, H. Matano and P. Poláčik, Existence of $L^1$-connections between equilibria of a semilinear parabolic equation, J. Dyn. Diff. Equat. 14 (2002), 463–491.
• M. Fila and P. Poláčik, Global solutions of a semilinear parabolic equation, Adv. Diff. Equat. 4 (1999), 163–196.
• I.M. Gel'fand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Trans. 29 (1963), 295–381.
• B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximal priciple, Comm. Math. Phys. 68 (1979), 209–243.
• J.K. Hale, Asymptotic behavior of dissipative systems, Math. Surv. Mono. 25, American Mathematical Society, Providence, RI, 1988.
• D. Henry, Geometric theory of semilinear parabolic equations, Lect. Notes Math. 840, Springer-Verlag, New York, 1981.
• D.D. Joseph and T.S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rat. Mech. Anal. 49 (1972/73), 241–269.
• T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1966.
• M.K.H. Kiessling, Statistical mechanics of classical particles with logarithmic interaction, Comm. Pure Appl. Math. 46 (1993), 229–260.
• Y.Y. Li, Harnack type inequality: The method of moving planes, Comm. Math. Phys. 200 (1999), 421–444.
• Y.Y. Li and I. Shafrir, Blow-up analysis for solutions of $-\Delta u = V e^u$ in dimension two, Indiana Univ. Math. Journal 43 (1994), 1255–1270.
• C.-S. Lin, Topological degree for the mean field equation on $S^2$, Duke Math. Journal 104 (2000), 501–536.
• H. Matano, Nonincrease of the lap-number of a solution for a one-dimensinal semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Math. 29 (1982), 401–441.
• T. Miyasita, Non-local elliptic problem in higher dimension, Osaka J. Math. 44 (2007), 159–172.
• ––––, On the dynamical system of a parabolic equation with non-local term, Adv. Stud. Pure Math. 47, Math. Soc. Japan, Tokyo, 2007.
• T. Miyasita and T. Suzuki, Non-local Gel'fand problem in higher dimensions, Banach Center Publ. 66, Polish Acad. Sci., Warsaw, 2004.
• J. Moser, A sharp form of an inequality by N.Trudinger, Indiana Univ. Math. Journal 20 (1970/71), 1077–1091.
• M. Nolasco and G. Tarantello, On a sharp Sobolev-type inequality on two-dimensional compact manifolds, Arch. Rat. Mech. Anal. 145 (1998), 161–195.
• K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlin. Anal. 51 (2002), 119–144.
• T. Ricciardi and G. Tarantello, On a periodic boundary value problem with exponential nonlinearities, Diff. Int. Equat. 11 (1998), 745–753.
• J. Spruck and Y. Yang, Topological solutions in the self-dual Chern-Simons gauge theory: existence and approximation, Ann. Inst. H. Poin. Anal. Non Lin. 12 (1995), 75–97.
• M. Struwe and G. Tarantello, On multivortex solutions in Chern-Simons gauge theory, Boll. Un. Mat. Ital. Mat. 1 (1998), 109–121.
• T. Suzuki, Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity, Ann. Inst. Poin. Anal. Non Lin. 9 (1992), 367–398.
• G. Tarantello, Multiple condensate solutions for the Chern-Simons-Higgs theory, J. Math. Phys. 37 (1996), 3769–3796.
• R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Appl. Math. Sci. 68, Springer-Verlag, New York, 1997.
• S. Zheng, Nonlinear parabolic equations and hyperbolic-parabolic coupled systems, Pitman Mono. Surv. Pure Appl. Math. 76, John Wiley & Sons, Inc., New York, 1995.