Global existence and blow-up for a nonlinear reaction-diffusion system. (English) Zbl 0884.35068

Chen, Hongwei

J. Math. Anal. Appl. 212, No. 2, 481-492 (1997).

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.

Reviewer: R.Guenther (Corvallis)

MSC:

35K60	Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40	Asymptotic behavior of solutions to PDEs
35K57	Reaction-diffusion equations

Keywords:

blow-up; classical solutions

References:

[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