Double thresholds for blowup and global existence of the solution to a system of parabolic equations

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

We considered the following parabolic system:

$\left\{\begin{array}{l} u_{t}=d_{1} \Delta u-a(x) \cdot \nabla u+f(u, v), \quad x \in \Omega, t>0, \\ v_{t}=d_{2} \Delta v-b(x) \cdot \nabla v+g(u, v), \quad x \in \Omega, t>0, \\ u(x, 0)=u_{0}(x), \quad v(x, 0)=v_{0}(x), \quad x \in \Omega, \end{array}\right.$

subject to Dirichlet (or Neumann) boundary conditions. Here $\Omega \subset \mathbb{R}^{N}(N \geq 1)$ is a bounded smooth domain. In addition to some results on blowup and global existence of the solution, we found some more interesting results as follows: (1) There exists double thresholds for blowup and global existence of the solution. Under certain conditions, if $f(u, v)=f_{1}(v) g_{1}(u) \text { and } g(u, v)=f_{2}(v) g_{2}(u)$, then the first watershed is

$\int_{c_{1}}^{+\infty} \frac{d u}{g_{1}(u)}=+\infty \quad \text { and } \quad \int_{c_{2}}^{+\infty} \frac{d v}{f_{2}(v)}=+\infty$

and the second watershed is

$\int_{\tilde{c}_{1}}^{+\infty} \frac{d U}{\tilde{f}\left(\tilde{F}^{-1}(K \tilde{G}(U))\right)}=+\infty \quad \text { and } \quad \int_{\tilde{c}_{2}}^{+\infty} \frac{d V}{\tilde{g}\left(\tilde{G}^{-1}\left(\frac{1}{\epsilon} \tilde{F}(V)\right)\right)}=+\infty .$

Here $\tilde{f}, \tilde{g}, \tilde{F} \text { and } \tilde{G}$ will be defined in Section 2.2. (2) If there exist nonnegative smooth functions h(u), l(v) and H(s) such that

$f(u, v) h^{\prime}(u) l(v)+g(u, v) h(u) l^{\prime}(v)=H[h(u) l(v)] \geq 0,$

then the watershed for blowup in finite time and global existence of the solution is

$\int_{0}^{+\infty} \frac{d s}{H(s)}=+\infty$

Keywords

system of parabolic equations / global existence / blowup / double thresholds

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] L. C. Evans, Partial Differential Equations, American Mathematical Society, 2010. https://doi.org/10.1090/gsm/019

[2] K. Chueh, C. Conley, J. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J., 26 (1977), 373-392. https://doi.org/10.1512/iumj.1977.26.26029

[3] E. Conway, D. Hoff, J. Smoller, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math., 35 (1978), 1-16. https://doi.org/10.1137/0135001

[4] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer New York, 1983. https://doi.org/10.1007/978-1-4684-0152-3

[5] P. Meier, Blow-up of solutions of semilinear parabolic differential equations, J. Appl. Math. Phys., 39 (1988), 135-149. https://doi.org/10.1007/BF00945760

[6] J. Ball, Remarks on blow-up and nonexistence theorems for non linear evolution equations, Quart. J. Math. Oxford Ser., 28 (1977), 473-486. https://doi.org/10.1093/qmath/28.4.473

[7] H. Bellout, A criterion for blow-up of solutions to semilinear heat equations, SIAM J. Math. Anal., 18 (1987), 722-727. https://doi.org/10.1137/0518055

[8] H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the Pu_t = -Au + F(u), Arch. Ration. Mech. Anal., 51 (1973), 371-386. https://doi.org/10.1007/BF00263041

[9] F. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. https://doi.org/10.1007/BF02761845

[10] J. Bebernes, D. Kassoy, A mathematical analysis of blow-up for thermal reactions- the spatially nonhomogeneous ease, SIAM J. Appl. Math., 40 (1981), 476-484. https://doi.org/10.1137/0140040

[11] A. Lacey, Mathematical analysis of thermal runaway for spatially inhomogeneous reactions, SIAM J. Appl. Math., 43 (1983), 1350-1366. https://doi.org/10.1137/0143090

[12] J. Escher, Global existence and nonexistence for semilinear parabolic systems with nonlinear boundary conditions, Math. Ann., 284 (1989), 285-305. https://doi.org/10.1007/BF01442877

[13] J. López-Gómez, V. Marquez, N. Wolanski, Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition, J. Differ. Equations, 92 (1991), 384-401. https://doi.org/10.1016/0022-0396(91)90056-F

[14] V. A. Galaktionov, J. L. Vazquez, Continuation of blowup solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math., 50 (1997), 1-67. https://doi.org/10.1002/(SICI)1097-0312(199701)50:1%3C1::AID-CPA1%3E3.0.CO;2-H

[15] D. Andreucci, M. A. Herrero, J. L. Velázquez, Liouville theorems and blow up behaviour in semilinear reaction diffusion systems, Ann. Inst. Henri Poincaré C Anal. Non Liné., 14 (1997), 1-53. https://doi.org/10.1016/s0294-1449(97)80148-5

[16] M. Escobedo, M. A. Herrero, A semilinear parabolic system in a bounded domain, Ann. Mat. Pura Appl., 165 (1993), 315-336. https://doi.org/10.1007/BF01765854

[17] L. Gang, B. D. Sleeman, Non-existence of global solutions to system of semi-linear parabolic equations, J. Differ. Equations, 104 (1993), 147-168. https://doi.org/10.1006/jdeq.1993.1066

[18] M. A. Herrero, J. L. Velázquez, Blowup of solutions of supercritical semilinear parabolic equations, Rep. Acad. Sci., Ser. I, Math., 319 (1994), 141-145.

[19] J. D. Rossi, N. Wolanski, Blow-up vs. global existence for a semilinear reaction-diffusion system in a bounded domain, Commun. Partial Differ. Equations, 20 (1995), 1991-2004. https://doi.org/10.1080/03605309508821160

[20] M. X. Wang, Global existence and finite time blow up for a reaction-diffusion system, Z. Angew. Math. Phys., 51 (2000), 160-167. https://doi.org/10.1007/PL00001504

[21] Y. Fujishima, K. Ishige, H. Maekawa, Blow-up set of type I blowing up solutions for nonlinear parabolic systems, Math. Ann., 369 (2017), 1491-1525. https://doi.org/10.1007/s00208-016-1498-7

[22] C. Bandle, I. Stakgold, The formation of the dead core in parabolic reaction-diffusion problems, Trans. Amer. Math. Soc., 286 (1984), 275-293. https://doi.org/10.1090/S0002-9947-1984-0756040-1

[23] J. Bebernes, A. Lacey, Finite time blowup for semilinear reactive-diffusive systems, J. Differ. Equations, 95 (1992), 105-129. https://doi.org/10.1016/0022-0396(92)90044-N

[24] M. A. Herrero, A. A. Lacey, J. L. Velázquez, Global existence for reaction-diffusion systems modelling ignition, Arch. Rational Mech. Anal., 142 (1998), 219-251. https://doi.org/10.1007/s002050050091

[25] F. Li, R. Peng, X. F. Song, Global existence and finite time blow-up of solutions of a Gierer-Meinhardt system, J. Differ. Equations, 262 (2017), 559-589. https://doi.org/10.1016/j.jde.2016.09.040

[26] R. Castillo, M. Loayza, C. S. Paixão, Global and nonglobal existence for a strongly coupled parabolic system on a general domain, J. Differ. Equations, 261 (2016), 3344-3365. https://doi.org/10.1016/j.jde.2016.05.024

[27] K. Ishige, T. Kawakami, M. Sierżega, Supersolutions for a class of nonlinear parabolic systems, J. Differ. Equations, 260 (2016), 6084-6107. https://doi.org/10.1016/j.jde.2015.12.031

[28] M. A. Herrero, J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuol. Norm. Super. Pisa Cl. Sci., 2 (1997), 633-683.

[29] M. Escobedo, H. A. Levine, Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations, Arch. Rational Mech. Anal., 129 (1995), 47-100. https://doi.org/10.1007/BF00375126

[30] S. Tréton, Blow-up vs. global existence for a Fujita-type heat exchanger system, SIAM J. Math. Anal., 56 (2024), 2191-2212. https://doi.org/10.1137/23M1587440

[31] J. M. Arrieta, A. Jiménez-Casas, A. Rodriguez-Bernal, Flux terms and Robin boundary conditions as limit of reactions and potentials concentrating at the boundary, Rev. Mat. Iberoam., 24 (2008), 183-211. https://doi.org/10.4171/rmi/533

[32] J. M. Arrieta, A. N. Carvalho, A. Rodriguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities, J. Differ. Equations, 156 (1999), 376-406. https://doi.org/10.1006/jdeq.1998.3612

[33] B. Carhuas-Torre, R. Castillo, K. Cea, R. Freire, A. Lira, M. Loayza, Global existence results for coupled parabolic systems with general sources and some extensions involving degenerate coefficients, J. Differ. Equations, 451 (2026), 113765. https://doi.org/10.1016/j.jde.2025.113765

[34] R. Castillo, M. Loayza, Global and blow-up solutions for a heat equation with variable reaction, Appl. Anal., 104 (2025), 2317-2333. https://doi.org/10.1080/00036811.2025.2459610

[35] S. Y. Chung, J. Hwang, A necessary and sufficient condition for the existence of global solutions to reaction-diffusion equations on bounded domains, Bound. Value Probl., 2024 (2024), 18. https://doi.org/10.1186/s13661-024-01822-w

[36] A. V. Lair, A necessary and sufficient conditions for global existence for quasilinear reaction-diffusion System, Int. J. Math. Math. Sci., 11 (2005), 1809-1818. https://doi.org/10.1155/IJMMS.2005.1809

[37] B. C. Liu, H. Y. Lin, A Cauchy problem of spatial-weighted reaction-diffusion equations, Appl. Math. Lett., 92 (2019), 128-133. https://doi.org/10.1016/j.aml.2019.01.021

[38] B. Hu, Blow-Up Theories for Semilinear Parabolic Equations. Lecture Notes in Mathematics, Springer, 2011. https://doi.org/10.1007/978-3-642-18460-4

[39] P. Quittner, P. Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States, Basel: Birkhäuser Basel, 2007. https://doi.org/10.1007/978-3-7643-8442-5

[40] G. Acosta, J. D. Rossi, Blow-up vs. global existence for quasilinear parabolic systems with a nonlinear boundary condition, Z. Angew. Math. Phys., 48 (1997), 711-724. https://doi.org/10.1007/s000330050060

[41] D. F. Rial, J. D. Rossi, Localization of blow-up points for a parabolic system with a nonlinear boundary condition, Rend. Circ. Mat. Palermo, 48 (1999), 135-152. https://doi.org/10.1007/BF02844385

[42] M. Pierre, D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass, SIAM J. Math. Anal., 28 (1997), 259-269. https://doi.org/10.1137/S0036141095295437

[43] W. W. Ao, H. Chan, A. DelaTorre, M. A. Fontelos, M. M. González, J. C. Wei, ODE methods in non-local equations, J. Math. Study, 53 (2020), 370-401. https://doi.org/10.4208/jms.v53n4.20.01

[44] W. W. Ao, H. Chan, A. DelaTorre, M. A. Fontelos, M. M. González, J. C. Wei, On higher-dimensional singularities for the fractional Yamabe problem: A nonlocal Mazzeo-Pacard program, Duke Math. J., 168 (2019), 3297-3411. https://doi.org/10.1215/00127094-2019-0034

[45] A. J. Archer, M. C. Walters, U. Thiele, E. Knobloch, Generation of defects and disorder from deeply quenching a liquid to form a solid, in Mathematical Challenges in a New Phase of Materials Science. Springer Proceedings in Mathematics & Statistics, Springer, 166 (2016), 1-26. https://doi.org/10.1007/978-4-431-56104-0

[46] J. M. Arrieta, A. Rodriguez-Bernal, P. Souplet, Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena, Ann. Sc. Norm. Super. Pisa Cl. Sci., 3 (2004), 1-15. https://doi.org/10.2422/2036-2145.2004.1.01

[47] J. M. Arrieta, A. Rodriguez-Bernal, J. W. Cholewa, T. Dlotko, Linear parabolic equations in locally uniform spaces, Math. Models Methods Appl. Sci., 14 (2004), 253-293. https://doi.org/10.1142/S0218202504003234

[48] J. M. Arrieta, J. W. Cholewa, T. Dlotko, A. Rodriguez-Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains, Nonlinear Anal., 56 (2004), 515-554. https://doi.org/10.1016/j.na.2003.09.023

[49] J. M. Arrieta, P. Quittner, A. Rodriguez-Bernal, Parabolic problems with nonlinear dynamical boundary conditions and singular initial data, Differ. Integr. Equations, 14 (2001), 1487-1510. https://doi.org/10.57262/die/1356123007

[50] J. M. Arrieta, A. N. Carvalho, A. Rodriguez-Bernal, Upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions, J. Differ. Equations, 168 (2000), 33-59. https://doi.org/10.1006/jdeq.2000.3876

[51] J. M. Arrieta, A. N. Carvalho, A. Rodriguez-Bernal, Attractors of parabolic problems with nonlinear boundary conditions, Comm. Partial Differ. Equations, 25 (2000), 1-37. https://doi.org/10.1080/03605300008821506

[52] H. Berestycki, F. Hamel, A. Kiselev, L. Ryzhik, Quenching and propagation in KPP reaction-diffusion equations with a heat loss, Arch. Ration. Mech. Anal., 178 (2005), 57-80. https://doi.org/10.1007/s00205-005-0367-4

[53] A. N. Carvalho, Contracting sets and dissipation, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 1305-1329. https://doi.org/10.1017/S0308210500030523

[54] P. Constantin, A. Kiselev, L. Ryzhik, A. Zlato, Diffusion and mixing in fluid flow, Ann. of Math., 168 (2008), 643-674. https://doi.org/10.4007/annals.2008.168.643

[55] V. A. Galaktionov, Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications, Chapman and Hall/CRC, 2004. https://doi.org/10.1201/9780203998069

[56] F. Quiros, J. D. Rossi, Non-simultaneous blow-up in a semilinear parabolic system, Z. Angew. Math. Phys., 52 (2001), 342-346. https://doi.org/10.1007/PL00001549

[57] F. Quiros, J. D. Rossi, Non-simultaneous blow-up in a nonlinear parabolic system, Adv. Nonlinear Stud., 3 (2003), 397-418. https://doi.org/10.1515/ans-2003-0304