Carleman estimates for one-dimensional degenerate heat equations View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

2006-05

AUTHORS

P. Martinez, J. Vancostenoble

ABSTRACT

In this paper, we are interested in controllability properties of parabolic equations degenerating at the boundary of the space domain. We derive new Carleman estimates for the degenerate parabolic equation $$ w_t + \left( {a\left( x \right)w_x } \right)_x = f,\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1} \right), $$ where the function a mainly satisfies $$ a \in \mathcal{C}^0 \left( {\left[ {0,1} \right]} \right) \cap \mathcal{C}^1 \left( {\left( {0,1} \right)} \right),a \gt 0 \hbox{on }\left( {0,1} \right) \hbox{and }\frac{1} {{\sqrt a }} \in L^1 \left( {0,1} \right). $$ We are mainly interested in the situation of a degenerate equation at the boundary i.e. in the case where a(0)=0 and / or a(1)=0. A typical example is a(x)=xα (1 − x)β with α, β ∈ [0, 2). As a consequence, we deduce null controllability results for the degenerate one dimensional heat equation $$ u_t - (a(x)u_x )_x = h\chi _w ,\quad (t,x) \in (0,T) \times (0,1),\quad \omega \subset \subset (0,1). $$ The present paper completes and improves previous works [7, 8] where this problem was solved in the case a(x)=xα with α ∈[0, 2). More... »

PAGES

325-362

References to SciGraph publications

Journal

TITLE

Journal of Evolution Equations

ISSUE

2

VOLUME

6

Author Affiliations

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s00028-006-0214-6

DOI

http://dx.doi.org/10.1007/s00028-006-0214-6

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1036618667


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