zbMATH — the first resource for mathematics

Global existence and blow-up for a nonlinear reaction-diffusion system. (English) Zbl 0884.35068
The author considers the system \[ u_t-\Delta u=u^{m_1} v^{n_1}, \quad v_t-\Delta v=u^{m_2} v^{n_2} \] for \(x\in\Omega\), \(t>0\), where \(\Omega\subset \mathbb{R}^n\) is a bounded domain with a smooth boundary, \(u\) and \(v\) have given nonnegative values initially and are zero when \(x\in \partial\Omega\), \(t>0\). Solutions here are classical solutions. The constants \(m_1\), \(n_1\), \(m_2\), \(n_2\) satisfy \(m_1\), \(n_2\geq 0\), and \(n_1,m_2>0\). The point of the paper is the proof of:
i) If \(m_1\leq 1\), \(n_2\leq\) and \(m_2n_1 \leq(1-m) (1-n_2)\), then all nonnegative solutions are global.
ii) If \(m_1>1\) or \(n_2>1\) or \(m_2n_1> (1-m_1) (1-n_2)\), then there are both global solutions and solutions which blow up in finite time, depending on the magnitude of the initial values.

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K57 Reaction-diffusion equations
Full Text: DOI
[1] Bebernes, J.; Eberly, D., Mathematical problems from combustion theory, (1989), Springer-Verlag New York · Zbl 0692.35001
[2] Pao, C.V., On nonlinear reaction-diffusion systems, J. math. anal. appl., 87, 165-198, (1982) · Zbl 0488.35043
[3] Levine, H.A., The role of critical exponents in blow up theorems, SIAM rev., 32, 262-288, (1990) · Zbl 0706.35008
[4] 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
[5] Lu, G., Global existence and blow up for a class of semilinear parabolic systems: A Cauchy problem, Nonlinear anal., 24, 1193-1206, (1995) · Zbl 0830.35050
[6] Friedman, A.; Giga, Y., A single blow up for solutions of nonlinear parabolic systems, J. fac. sci. univ. Tokyo sect. I, 34, 65-79, (1987) · Zbl 0648.35042
[7] Escobedo, M.; Herrero, M.A., A semilinear parabolic system in a bounded domain, Ann. mat. pura appl. (4), 165, 315-336, (1993) · Zbl 0806.35088
[8] Pao, C.V., Nonlinear parabolic and elliptic equations, (1992), Plenum New York · Zbl 0780.35044
[9] Ladyzenskaja, O.A.; Solonnikov, V.A.; Ural’ceva, N.N., Linear and quasilinear equations of parabolic type, Transl. math. monogr., (1968), Amer. Math. Soc Providence
[10] Ball, J.M., Remarks on blow up and nonexistence theorems for non-linear evolution equations, Quart. J. math. Oxford ser., 28, 473-486, (1977) · Zbl 0377.35037
[11] Weissler, F.B., Single point blow-up for a semilinear initial value problem, J. differential equations, 55, 204-224, (1984) · Zbl 0555.35061
[12] Chen, H., Remarks on Dirichlet problem of the equationu_t=δuu1+α, J. huazhong univ. sci. tech., 13, 69-72, (1985)
[13] Sattinger, D.H., On global solutions of nonlinear hyperbolic equations, Arch. rational mech. anal., 30, 148-172, (1968) · Zbl 0159.39102
[14] Rothe, F., Global solutions of reaction-diffusion systems, Lecture notes in mathematics, (1984), Springer-Verlag New York · Zbl 0546.35003
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.