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Carleman estimates for a class of degenerate parabolic operators. (English) Zbl 1168.35025
The authors derived new Carleman estimates for the degenerate parabolic problem \(w_t+(x^\alpha w_x)_x=f,\;(t,x)\in (0,T)\times (0,1)\) with the boundary conditions \(w(t,1)=0\) and \(w(t,0)=0,\) if \(0\leq \alpha<1\) or \((x^\alpha w_x)(t,0)=0\) if \(1\leq \alpha<2.\) The proof is based on the choice of suitable weighted functions and Hardy-type inequalities. As a consequence, for all \(0\leq \alpha<2\) and \(\omega\subset\subset (0,1)\) the null controllability results for the heat equation \(u_t-(x^\alpha u_x)_x=h\chi_\omega\) with the same boundary conditions are achieved.

MSC:
35K65 Degenerate parabolic equations
93B05 Controllability
93B07 Observability
35B45 A priori estimates in context of PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
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