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Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations. (English) Zbl 0822.35068

Summary: Let D N be either all of N or else a cone in N whose vertex we may take to be at the origin, without loss of generality. Let p i , q j , i=1,2, be nonnegative with 0<p 1 +q 1 p 2 +q 2 . We consider the long-time behavior of nonnegative solutions of the system

u t =Δu+u p 1 v q 1 ,v t =Δv+u p 2 v q 2 (S)

in D×[0,) with u 0 =v 0 =0 on D, (u,v) t (x,0)=(v 0 ,v 0 ) t (x), u 0 ,v 0 0, u 0 ,v 0 L (D).

We obtain Fujita-type global existence-global non-existence theorems for (S) analogous to the classical result of H. Fujita [J. Fac. Sci., Univ. Tokyo, Sect. I 13, 109-124 (1966; Zbl 0163.340)] and others for the initial-value problem for u t =Δu+u p , u(x,0)=u 0 (x)0. The principal result in the case D= N and p 2 q 1 >0 is that when p 1 1, the system behaves like the single equation u t =Δu+u p 1 +q 1 with respect to Fujita- type blowup theorems, whereas if p 1 <1, the behavior of the system is more complicated. Some of the results extend those of M. Escobedo and M. A. Herrero [J. Differ. Equations 89, No. 1, 176- 202 (1991; Zbl 0735.35013)] when D= N and of H. A. Levine and P. Meier [Isr. J. Math. 67, No. 2, 129-136 (1989; Zbl 0696.35013)] when D is a cone. These authors considered (S) in the case of p 1 =q 2 =0. An example of nonuniqueness is also given.

35K57Reaction-diffusion equations
35B40Asymptotic behavior of solutions of PDE
35K40Systems of second-order parabolic equations, general
[1]D. G. Aronson & H. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Advances in Math. 30 (1978), 33–76. · Zbl 0407.92014 · doi:10.1016/0001-8708(78)90130-5
[2]C. Bandle & H. A. Levine, On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains, Trans. Amer. Math. Soc. 655 (1989), 595–624. · doi:10.1090/S0002-9947-1989-0937878-9
[3]C. Bandle & H. A. Levine, Fujita type results for convective reaction diffusion equations in exterior domains, Z. Angew. Math. Phys. 40 (1989), 655–676.
[4]M. Escobedo & H. A. Levine, Explosion et existence globale pour un système faiblement couplé d’équations de réaction diffusion, C. R. Acad. Sci. Paris, Sér. I, 134 (1992), 735–739.
[5]M. Escobedo & M. A. Herrero, Boundedness and blow up for a semilinear reaction-diffusion system, J. Diff. Eqs. 89 (1991), 176–202. · Zbl 0735.35013 · doi:10.1016/0022-0396(91)90118-S
[6]H. Fujita, On the blowing up of solutions of the Cauchy problem for u t =Δu+u 1+α, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 16 (1966), 105–113.
[7]H. Fujita & S. Watanabe, On the uniqueness and non uniqueness of solutions of initial value problems for some quasi-linear parabolic equations, Comm. Pure Appl. Math. 21 (1968), 631–652. · Zbl 0165.44301 · doi:10.1002/cpa.3160210609
[8]K. Kobayashi, T. Sirao & H. Tanaka, On the growing up problem for semilinear heat equations, J. Math. Soc. Japan 29 (1977), 407–429. · Zbl 0353.35057 · doi:10.2969/jmsj/02930407
[9]H. A. Levine, The role of critical exponents in blowup theorems, SIAM Review 32 (1990), 262–288. · Zbl 0706.35008 · doi:10.1137/1032046
[10]H. A. Levine, A Fujita type global existence-global nonexistence theorem for a weakly coupled system of reaction-diffusion equations, Z. Angew. Math. Phys. 42 (1991), 408–430. · Zbl 0786.35075 · doi:10.1007/BF00945712
[11]H. A. Levine & P. Meier, The value of the critical exponent for reaction-diffusion equations in cones, Arch. Rational Mech. Anal. 109 (1990), 73–80. · Zbl 0702.35131 · doi:10.1007/BF00377980
[12]H. A. Levine, A blowup result for the critical exponent in cones, Israel J. Math. 67 (1989), 1–7. · Zbl 0696.35013 · doi:10.1007/BF02937290
[13]P. Meier, Existence et non-existence de solutions globales d’une équation de la chaleur semi-linéaire: extension d’un théorème de Fujita, C. R. Acad. Sci. Paris, Sér. I 303 (1986), 635–637.
[14]M. H. Protter & H. F. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, New York, 1967.
[15]G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd Ed., Cambridge University Press, London, New York, 1944.
[16]F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math. 38 (1981), 29–40. · Zbl 0476.35043 · doi:10.1007/BF02761845
[17]F. B. Weissler, Local existence and nonexistence for semilinear parabolic equations in L p, Indiana Univ. Math. J. 29 (1980), 79–102. · Zbl 0443.35034 · doi:10.1512/iumj.1980.29.29007