Convergence and blow-up of solutions for a complex-valued heat equation with a quadratic nonlinearity
HTML articles powered by AMS MathViewer
- by Jong-Shenq Guo, Hirokazu Ninomiya, Masahiko Shimojo and Eiji Yanagida PDF
- Trans. Amer. Math. Soc. 365 (2013), 2447-2467 Request permission
Abstract:
This paper is concerned with the Cauchy problem for a system of parabolic equations which is derived from a complex-valued equation with a quadratic nonlinearity. First we show that if the convex hull of the image of initial data does not intersect the positive real axis, then the solution exists globally in time and converges to the trivial steady state. Next, on the one-dimensional space, we provide some solutions with nontrivial imaginary parts that blow up simultaneously. Finally, we consider the case of asymptotically constant initial data and show that, depending on the limit, the solution blows up nonsimultaneously at space infinity or exists globally in time and converges to the trivial steady state.References
- Keng Deng, Blow-up rates for parabolic systems, Z. Angew. Math. Phys. 47 (1996), no. 1, 132–143. MR 1408675, DOI 10.1007/BF00917578
- Keng Deng and Howard A. Levine, The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl. 243 (2000), no. 1, 85–126. MR 1742850, DOI 10.1006/jmaa.1999.6663
- S. D. Èĭdel′man, Parabolic systems, North-Holland Publishing Co., Amsterdam-London; Wolters-Noordhoff Publishing, Groningen, 1969. Translated from the Russian by Scripta Technica, London. MR 0252806
- M. Escobedo and M. A. Herrero, Boundedness and blow up for a semilinear reaction-diffusion system, J. Differential Equations 89 (1991), no. 1, 176–202. MR 1088342, DOI 10.1016/0022-0396(91)90118-S
- Marek Fila, Hirokazu Ninomiya, and Juan Luis Vázquez, Dirichlet boundary conditions can prevent blow-up reaction-diffusion equations and systems, Discrete Contin. Dyn. Syst. 14 (2006), no. 1, 63–74. MR 2170313, DOI 10.3934/dcds.2006.14.63
- Marek Fila and Philippe Souplet, The blow-up rate for semilinear parabolic problems on general domains, NoDEA Nonlinear Differential Equations Appl. 8 (2001), no. 4, 473–480. MR 1867324, DOI 10.1007/PL00001459
- Avner Friedman and Yoshikazu Giga, A single point blow-up for solutions of semilinear parabolic systems, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 1, 65–79. MR 882125
- Avner Friedman and Bryce McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), no. 2, 425–447. MR 783924, DOI 10.1512/iumj.1985.34.34025
- Hiroshi Fujita, On the blowing up of solutions of the Cauchy problem for $u_{t}=\Delta u+u^{1+\alpha }$, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109–124 (1966). MR 214914
- Yoshikazu Giga and Robert V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), no. 3, 297–319. MR 784476, DOI 10.1002/cpa.3160380304
- Yoshikazu Giga and Robert V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 (1987), no. 1, 1–40. MR 876989, DOI 10.1512/iumj.1987.36.36001
- Yoshikazu Giga and Robert V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math. 42 (1989), no. 6, 845–884. MR 1003437, DOI 10.1002/cpa.3160420607
- Yoshikazu Giga and Noriaki Umeda, On blow-up at space infinity for semilinear heat equations, J. Math. Anal. Appl. 316 (2006), no. 2, 538–555. MR 2206688, DOI 10.1016/j.jmaa.2005.05.007
- Jong-Shenq Guo, Satoshi Sasayama, and Chi-Jen Wang, Blowup rate estimate for a system of semilinear parabolic equations, Commun. Pure Appl. Anal. 8 (2009), no. 2, 711–718. MR 2461571, DOI 10.3934/cpaa.2009.8.711
- Miguel A. Herrero and Juan J. L. Velázquez, Explosion de solutions d’équations paraboliques semilinéaires supercritiques, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 2, 141–145 (French, with English and French summaries). MR 1288393
- O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). Translated from the Russian by S. Smith. MR 0241822
- Hiroshi Matano and Frank Merle, On nonexistence of type II blowup for a supercritical nonlinear heat equation, Comm. Pure Appl. Math. 57 (2004), no. 11, 1494–1541. MR 2077706, DOI 10.1002/cpa.20044
- Hiroshi Matano and Frank Merle, Classification of type I and type II behaviors for a supercritical nonlinear heat equation, J. Funct. Anal. 256 (2009), no. 4, 992–1064. MR 2488333, DOI 10.1016/j.jfa.2008.05.021
- Noriko Mizoguchi, Rate of type II blowup for a semilinear heat equation, Math. Ann. 339 (2007), no. 4, 839–877. MR 2341904, DOI 10.1007/s00208-007-0133-z
- Kiyoshi Mochizuki and Qing Huang, Existence and behavior of solutions for a weakly coupled system of reaction-diffusion equations, Methods Appl. Anal. 5 (1998), no. 2, 109–124. MR 1636542, DOI 10.4310/MAA.1998.v5.n2.a1
- Nejla Nouaili and Hatem Zaag, A Liouville theorem for vector valued semilinear heat equations with no gradient structure and applications to blow-up, Trans. Amer. Math. Soc. 362 (2010), no. 7, 3391–3434. MR 2601595, DOI 10.1090/S0002-9947-10-04902-0
- Hisashi Okamoto, Takashi Sakajo, and Marcus Wunsch, On a generalization of the Constantin-Lax-Majda equation, Nonlinearity 21 (2008), no. 10, 2447–2461. MR 2439488, DOI 10.1088/0951-7715/21/10/013
- Bob Palais, Blowup for nonlinear equations using a comparison principle in Fourier space, Comm. Pure Appl. Math. 41 (1988), no. 2, 165–196. MR 924683, DOI 10.1002/cpa.3160410204
- Fernando Quirós and Julio D. Rossi, Non-simultaneous blow-up in a semilinear parabolic system, Z. Angew. Math. Phys. 52 (2001), no. 2, 342–346. MR 1834531, DOI 10.1007/PL00001549
- Pavol Quittner and Philippe Souplet, Superlinear parabolic problems, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2007. Blow-up, global existence and steady states. MR 2346798
- Julio D. Rossi and Philippe Souplet, Coexistence of simultaneous and nonsimultaneous blow-up in a semilinear parabolic system, Differential Integral Equations 18 (2005), no. 4, 405–418. MR 2122706
- Takashi Sakajo, Blow-up solutions of the Constantin-Lax-Majda equation with a generalized viscosity term, J. Math. Sci. Univ. Tokyo 10 (2003), no. 1, 187–207. MR 1963803
- Takashi Sakajo, On global solutions for the Constantin-Lax-Majda equation with a generalized viscosity term, Nonlinearity 16 (2003), no. 4, 1319–1328. MR 1986297, DOI 10.1088/0951-7715/16/4/307
- Steven Schochet, Explicit solutions of the viscous model vorticity equation, Comm. Pure Appl. Math. 39 (1986), no. 4, 531–537. MR 840339, DOI 10.1002/cpa.3160390404
- Yukihiro Seki, On directional blow-up for quasilinear parabolic equations with fast diffusion, J. Math. Anal. Appl. 338 (2008), no. 1, 572–587. MR 2386440, DOI 10.1016/j.jmaa.2007.05.033
- Masahiko Shimoj\B{o}, The global profile of blow-up at space infinity in semilinear heat equations, J. Math. Kyoto Univ. 48 (2008), no. 2, 339–361. MR 2436740, DOI 10.1215/kjm/1250271415
- Masahiko Shimojo and Noriaki Umeda, Blow-up at space infinity for solutions of cooperative reaction-diffusion systems, Funkcial. Ekvac. 54 (2011), no. 2, 315–334. MR 2867018, DOI 10.1619/fesi.54.315
- Philippe Souplet, Single-point blow-up for a semilinear parabolic system, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 1, 169–188. MR 2471135, DOI 10.4171/JEMS/145
- Philippe Souplet and Slim Tayachi, Optimal condition for non-simultaneous blow-up in a reaction-diffusion system, J. Math. Soc. Japan 56 (2004), no. 2, 571–584. MR 2048475, DOI 10.2969/jmsj/1191418646
- Ray Redheffer and Wolfgang Walter, Invariant sets for systems of partial differential equations. I. Parabolic equations, Arch. Rational Mech. Anal. 67 (1978), no. 1, 41–52. MR 473317, DOI 10.1007/BF00280826
- Mingxin Wang, Blow-up rate estimates for semilinear parabolic systems, J. Differential Equations 170 (2001), no. 2, 317–324. MR 1815186, DOI 10.1006/jdeq.2000.3823
- Hans F. Weinberger, Invariant sets for weakly coupled parabolic and elliptic systems, Rend. Mat. (6) 8 (1975), 295–310 (English, with Italian summary). MR 397126
Additional Information
- Jong-Shenq Guo
- Affiliation: Department of Mathematics, Tamkang University, Tamsui, Taipei County 25137, Taiwan
- Email: jsguo@mail.tku.edu.tw
- Hirokazu Ninomiya
- Affiliation: Department of Mathematics, Meiji University, 1-1-1 Higashimita, Tamaku, Kawasaki 214-8571, Japan
- MR Author ID: 330408
- ORCID: 0000-0001-7081-6564
- Email: ninomiya@math.meiji.ac.jp
- Masahiko Shimojo
- Affiliation: Department of Mathematics, Meiji University, 1-1-1 Higashimita, Tamaku, Kawasaki 214-8571, Japan
- Email: shimojotw@gmail.com
- Eiji Yanagida
- Affiliation: Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan
- Email: yanagida@math.titech.ac.jp
- Received by editor(s): August 30, 2011
- Published electronically: October 9, 2012
- © Copyright 2012 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 365 (2013), 2447-2467
- MSC (2010): Primary 35K57; Secondary 35K40, 35B44
- DOI: https://doi.org/10.1090/S0002-9947-2012-05797-7
- MathSciNet review: 3020104