# zbMATH — the first resource for mathematics

Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations. (English) Zbl 0822.35068
Summary: Let $$D\subset \mathbb{R}^ N$$ be either all of $$\mathbb{R}^ N$$ or else a cone in $$\mathbb{R}^ N$$ 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 $$0<p_ 1+ q_ 1\leq p_ 2+ q_ 2$$. We consider the long-time behavior of nonnegative solutions of the system $u_ t= \Delta u+ u^{p_ 1} v^{q_ 1}, \qquad v_ t= \Delta v+ u^{p_ 2} v^{q_ 2} \tag{S}$ in $$D\times [0,\infty)$$ with $$u_ 0= v_ 0=0$$ on $$\partial D$$, $$(u, v)^ t (x,0)= (v_ 0, v_ 0)^ t (x)$$, $$u_ 0, v_ 0\geq 0$$, $$u_ 0, v_ 0\in L^ \infty (D)$$.
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 $$u_ t= \Delta u+ u^ p$$, $$u(x,0)= u_ 0 (x)\geq 0$$. The principal result in the case $$D= \mathbb{R}^ N$$ and $$p_ 2 q_ 1>0$$ is that when $$p_ 1\geq 1$$, the system behaves like the single equation $$u_ t= \Delta u+ u^{p_ 1+ q_ 1}$$ 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 $$D= \mathbb{R}^ N$$ 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.

##### MSC:
 35K57 Reaction-diffusion equations 35B40 Asymptotic behavior of solutions to PDEs 35K40 Second-order parabolic systems
Full Text:
##### References:
 [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. · Zbl 0693.35081 · 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. · Zbl 0697.76099 [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. · Zbl 0777.35012 [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 =$$\Delta$$u+u 1+$$\alpha$$, 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. · Zbl 0602.35054 [14] M. H. Protter & H. F. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, New York, 1967. · Zbl 0153.13602 [15] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd Ed., Cambridge University Press, London, New York, 1944. · Zbl 0063.08184 [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
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.