## 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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