×

zbMATH — the first resource for mathematics

Life span of solutions with large initial data for a semilinear parabolic system. (English) Zbl 1219.35027
The author investigates the initial-boundary value problem \(u_t=\Delta u-v^q\), \(v_t=\Delta v-u^q\) in \(\Omega\times (0,T)\), \(u(x,t)=v(x,t)=0\) on \(\partial\Omega\times (0,T)\), \(u(x,0)= \rho\varphi\), \(v(x,0)= \rho\psi\) in \(\Omega\), where \(p,q \geq 1\) and \(pq>1\), \(\Omega\) is a bounded domain in \( \mathbb R^n\) with a smooth boundary \(\partial \Omega\), \(\rho>0\) is a parameter, and \(\varphi (x)\) and \(\psi (x)\) are nonnegative continuous functions on \(\overline{\Omega}\).
Reviewer: Jiaqi Mo (Wuhu)

MSC:
35B44 Blow-up in context of PDEs
35K57 Reaction-diffusion equations
35K51 Initial-boundary value problems for second-order parabolic systems
35K58 Semilinear parabolic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Deng, K.; Levine, H.A., The role of critical exponents in blow-up theorems: the sequel, J. math. anal. appl., 243, 85-126, (2000) · Zbl 0942.35025
[2] Escobedo, M.; Herrero, M.A., Boundedness and blow-up for a semilinear reaction-diffusion system, J. differential equations, 89, 176-202, (1991) · Zbl 0735.35013
[3] Escobedo, M.; Herrero, M.A., A semilinear parabolic system in a bounded domain, Ann. mat. pura appl., 165, 315-336, (1993) · Zbl 0806.35088
[4] Friedman, A.; Lacey, A., The blow-up time of solutions of nonlinear heat equations with small diffusion, SIAM J. math. anal., 18, 711-721, (1987) · Zbl 0643.35013
[5] Fujita, H., On the blowing up of solutions of the Cauchy problem for \(u_t = \operatorname{\Delta} u + u^{1 + \alpha}\), J. fac. sci. univ. Tokyo sect. IA math., 16, 105-113, (1966)
[6] Gui, C.; Wang, X., Life span of solutions of the Cauchy problem for a semilinear heat equation, J. differential equations, 115, 166-172, (1995) · Zbl 0813.35034
[7] Huang, Q.; Mochizuki, K.; Mukai, K., Life span and asymptotic behavior for a semilinear parabolic system with slowly decaying initial values, Hokkaido math. J., 27, 393-407, (1998) · Zbl 0906.35044
[8] Kaplan, S., On the growth of solutions of quasi-linear parabolic equations, Comm. pure anal. math., 16, 305-330, (1963) · Zbl 0156.33503
[9] Kobayashi, Y., Behavior of the life span for solutions to the system of reaction-diffusion equations, Hiroshima math. J., 33, 167-187, (2003) · Zbl 1050.35031
[10] Lee, T.-Y.; Ni, W.-M., Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Trans. amer. math. soc., 333, 1434-1446, (1992)
[11] Mizoguchi, N.; Yanagida, E., Life span of solutions with large initial data in a semilinear parabolic equation, Indiana univ. math. J., 50, 1, 591-610, (2001) · Zbl 0996.35006
[12] Mizoguchi, N.; Yanagida, E., Life span of solutions for a semilinear parabolic problem with small diffusion, J. math. anal. appl., 261, 350-368, (2001) · Zbl 0993.35011
[13] Mochizuki, K., Blow-up, life span and large time behavior of solutions of a weakly coupled system of reaction-diffusion equations, (), 175-198 · Zbl 0932.35028
[14] Quittner, P.; Souplet, Ph., Superlinear parabolic problems. blow-up, global existence and steady states, Birkhäuser adv. texts, (2007), Birkhäuser Basel · Zbl 1128.35003
[15] Sato, S., Life span of solutions with large initial data for a superlinear heat equation, J. math. anal. appl., 343, 1061-1074, (2008) · Zbl 1154.35060
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.