# 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)
A single point blow-up for solutions of semilinear parabolic systems. (English) Zbl 0648.35042
Consider the system $$(1.1)\quad u\sb t-\alpha u\sb{xx}=f(v)\ (-a<x<a,\ t>0),$$ $$(1.2)\quad v\sb t-\beta v\sb{xx}=g(u)\ (-a<x<a,\ t>0)$$ with $$(1.3)\quad u(\pm a,t)=0\ (t>0),\ u(x,0)=\phi (x)\ (-a<x<a),$$ $$(1.4)\quad v(\pm a,t)=0\quad (t>0),\ v(x,0)=\psi (x)\ (-a<x<a),$$ where $\alpha >0$, $\beta >0$, and assume: (1.5) $\phi(x)=\phi(-x)$, $\phi(x)\ge 0$, $\phi\in C\sp 1[- a,a];\phi'(x)\le 0$ if $0<x<a$, $\phi (a)=0$; $\psi (x)=\psi (-x)$, $\psi(x)\ge 0$, $\psi\in C\sp 1[-a,a];\psi'(x)\le 0$ if $0<x<a$, $\psi(a)=0,$ (1.6) $f,g\in C\sp 1(R\sp 1),$ $f(s)>0$, $g(s)>0$ if $s>0$; $f'(s)>0$, $g'(s)>0$ if $s>0.$ Set $$H\sb{\alpha}w=w\sb t-\alpha w\sb{xx},\quad Q\sb{\sigma}=\{(x,t);\quad -a<x<a,\quad 0<t<\sigma \}.$$ Then there exists a unique classical solution of (1.1)-(1.4) in some $Q\sb{t\sb 0}$, and $u\ge 0$, $v\ge 0$ by the maximum principle. Let $T=\sup t\sb 0$, for all $t\sb 0$ as above. We claim $$(1.7)\quad \sup\sb{Q\sb{\sigma}} u\to \infty \ if\ \sigma \to T.$$ Further we assume that, for some $M>1$, $$(2.1)\quad pf(v)\le vf'(v)\ if\ v>M,\ p>1;\ qg(u)\le ug'(u)\ if\ u>M,\ q>1$$ and that the solution (u,v) satisfies the estimates: $$(2.2)\quad u\le C(v\sp{\gamma}+1);\quad v\le C(u\sp{1/\gamma}+1),\ C>0,\ \gamma >0,\ p>\gamma,\ q>1/\gamma.$$ Then we see: Suppose that u and v solves (1.1), (1.2) with (1.3)-(1.6). If the conditions (2.1), (2.2) are satisfied, then there is a single blow-up point.
Reviewer: Y.Ebihara

##### MSC:
 35K55 Nonlinear parabolic equations 35B40 Asymptotic behavior of solutions of PDE 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35K15 Second order parabolic equations, initial value problems 35K45 Systems of second-order parabolic equations, initial value problems