zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Existence and behavior of solutions for a weakly coupled system of reaction-diffusion equations. (English) Zbl 0913.35065

Summary: We consider the weakly coupled system of reaction-diffusion equations

u t =Δu+|x| σ 1 v p ,v t =Δv+|x| σ 2 u q ,

where x N (N1), t>0, p,q1 with pq>1 and 0σ 1 <N(p-1), 0σ 2 <N(q-1). Put

α=2(p+1) pq-1,β=2(q+1) pq-1,δ 1 =σ 2 p+σ 1 pq-1,δ 2 =σ 1 q+σ 2 pq-1,

and let I a and I a (a0) be the spaces of nonnegative, bounded continuous functions satisfying

lim sup |x| |x| a ξ(x)<andlim inf |x| |x| a ξ(x)>0,

respectively. At t=0, initial values (u 0 ,v 0 )I δ 1 ×I δ 2 are prescribed. It is proved that if max{α+δ 1 ,β+δ 2 }N or if u 0 I a with a<α+δ 1 or v 0 I b with b<β+δ 2 , then every nontrivial nonnegative solution is not global in time; whereas if max{α+δ 1 ,β+δ 2 }<N and (u 0 ,v 0 )I a ×I b with a>α+δ 1 , b>β+δ 2 , then there exist both global solutions and nonglobal solutions. Moreover, we obtain the asymptotic behavior as t of the global solutions.

35K57Reaction-diffusion equations
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
35B40Asymptotic behavior of solutions of PDE
35K45Systems of second-order parabolic equations, initial value problems