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)
Approximate controllability of a semilinear heat equation in N . (English) Zbl 0935.93011

This note treats the approximate controllability of the semilinear heat equation in unbounded domains Ω when the control acts on the interior of Ω.

When Ω is a bounded set, C. Fabre, J.-P. Puel and E. Zuazua [IMA Vol. Math. Appl. 70, 73-91 (1995; Zbl 0822.35075)] proved the approximate controllability of the semilinear heat equation in L p (Ω), 1p<. Their proof is divided into two parts: a) approximate controllability of the linearized systems, b) a fixed point technique. This method cannot be applied when Ω is an unbounded set since the compactness of Sobolev’s embeddings is one of the main ingredients used in b). In a note, L. de Teresa and E. Zuazua [Nonlinear Anal. (to appear)] prove the approximate controllability of the semilinear heat equation in unbounded domains by an approximation method.

Also the control problem in bounded domains is studied. In this note previously introduced techniques are adapted to unbounded domains by introducing the weighted Sobolev spaces of M. Escobedo and O. Kavian [Nonlinear Anal., Theory Methods Appl. 11, 1103-1133 (1987; Zbl 0639.35038)] which permit (guarantee) the compactness of Sobolev embeddings.


MSC:
93B05Controllability
35K05Heat equation
35K55Nonlinear parabolic equations