zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.


MSC:
35K57Reaction-diffusion equations
35B40Asymptotic behavior of solutions of PDE
35K40Systems of second-order parabolic equations, general
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. · 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