# zbMATH — the first resource for mathematics

Lifespan of solutions for a weakly coupled system of semilinear heat equations. (English) Zbl 1439.35181
Summary: We introduce a direct method to analyze the blow-up of solutions to systems of ordinary differential inequalities, and apply it to study the blow-up of solutions to a weakly coupled system of semilinear heat equations. In particular, we give upper and lower estimates of the lifespan of the solution in the subcritical case.
##### MSC:
 35K05 Heat equation 35B44 Blow-up in context of PDEs
Full Text:
##### References:
 [1] Y. Aoyagi, K. Tsutaya and Y. Yamauchi, Global existence of solutions for a reaction-diffusion system, Differential Integral Equations 20 (2007), 1321-1339. · Zbl 1212.35225 [2] J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford 28 (1977), 473-486. · Zbl 0377.35037 [3] P. Baras and M. Pierre, Critère d’existence de solutions positives pour des équations semi-linéaires non monotones, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), 185-212. · Zbl 0599.35073 [4] M. Escobedo and M. A. Herrero, Boundedness and blow up for a semilinear reaction-diffusion system, J. Differential Equations 89 (1991), 176-202. · Zbl 0735.35013 [5] M. Escobedo and 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. · Zbl 0822.35068 [6] M. Fila and P. Quittner, The blow-up rate for a semilinear parabolic system, J. Math. Anal. Appl. 238 (1999), 468-476. · Zbl 0934.35062 [7] H. Fujita, On the blowing up of solutions of the Cauchy problem for $$u_t=\Delta u+u^{1+\alpha}$$, J. Fac. Sci. Univ. Tokyo Sec. I 13 (1966), 109-124. · Zbl 0163.34002 [8] K. Fujiwara, M. Ikeda and Y. Wakasugi, Blow-up of solutions for weakly coupled systems of complex Ginzburg-Landau equations, Electron. J. Differential Equations 2017 (2017), No. 196, 1-18. · Zbl 1370.35247 [9] K. Fujiwara, M. Ikeda and Y. Wakasugi, Estimates of lifespan and blow-up rates for the wave equation with a time-dependent damping and a power-type nonlinearity, to appear in Funkcial. Ekvac., arXiv:1609.01035v2. · Zbl 1426.35162 [10] K. Fujiwara and T. Ozawa, Finite time blowup of solutions to the nonlinear Schrödinger equation without gauge invariance, J. Math. Phys. 57 082103 (2016), 1-8. · Zbl 1348.35231 [11] K. Fujiwara and T. Ozawa, Lifespan of strong solutions to the periodic nonlinear Schrödinger equation without gauge invariance, J. Evol. Equ. 17 (2017), 1023-1030. · Zbl 1381.35160 [12] K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad. 49 (1973), 503-505. · Zbl 0281.35039 [13] K. Ishige, T. Kawakami and M. Sięrżzega, Supersolutions for a class of nonlinear parabolic systems, J. Differential Equations 260 (2016), 6084-6107. · Zbl 1338.35228 [14] E. Kamke, Zur Theorie der Systeme gewöhnlicher Differentialgleichungen. II, Acta Math. 58 (1932), 57-85. · JFM 58.0449.02 [15] K. Kobayashi, T. Sirao and H. Tanaka, On the growing up problem for semilinear heat equations, J. Math. Soc. Japan 29 (1977), 407-424. · Zbl 0353.35057 [16] E. Mitidieri and S. I. Pohozaev, Nonexistence of weak solutions for some degenerate elliptic and parabolic problems on $$\mathbb{R}^n$$, J. Evol. Equ., 1 (2001), 189-220. · Zbl 0988.35095 [17] K. Mochizuki, Blow-up, lifespan and large time behavior of solutions of a weakly coupled system of reaction diffusion equations, Adv. Math. Appl. Sci. 48, World Scientific (1998), 175-198. · Zbl 0932.35028 [18] K. Mochizuki and Q. Huang, Existence and behavior of solutions for a weakly coupled system of reaction-diffusion equations, Methods Appl. Anal. 5 (1998), 109-124. · Zbl 0913.35065 [19] K. Nishihara and Y. Wakasugi, Global existence of solutions for a weakly coupled system of semilinear damped wave equations, J. Differential Equations 259 (2015), 4172-4201. · Zbl 1327.35238 [20] T. Ogawa and H. Takeda, Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal. 70 (2009), 3696-3701. · Zbl 1196.35142 [21] J. Renclawowicz, Global existence and blow-up for a completely coupled Fujita type system, Appl. Math. 27 (2000), 203-218. · Zbl 0994.35055 [22] N. Umeda, Blow-up and large time behavior of solutions of a weakly coupled system of reaction-diffusion equations, Tsukuba J. Math. 27 (2003), 31-46. · Zbl 1035.35018 [23] M. Wang, Blow-up rate for a semilinear reaction diffusion system, Comput. Math. Appl. 44 (2002), 573-585. · Zbl 1030.35104 [24] F. B. Weissler, Existence and non-existence of global solutions for a semilinear heat equation, Israel J. Math. 38 (1981), 29-40. · Zbl 0476.35043 [25] Qi S.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.