Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given $\alpha \in [0,2)$ and $f \in L^2 ((0,T)\times(0,1))$,  derive new Carleman estimates for the degenerate parabolic problem $w_t + (x^\alpha w_x) _x =f$, where $(t,x) \in (0,T) \times (0,1)$, associated to 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 ighted functions and Hardy-type inequalities. As a consequence, for all $0 \leq \alpha <2$ and $\omega\subset\subset(0,1)$,  deduce null controllability results for the degenerate one-dimensional heat equation $u_t - (x^\alpha u_x)_x = h \chi _\omega$ with the same boundary conditions as above.
['P. Cannarsa', 'Patrick Martinez', 'Judith Vancostenoble']
I am writing a research paper on a idea Carleman Estimates for a Class of Degenerate Parabolic Operators
[{"from": "human", "value": "I am writing a research paper on a idea Carleman Estimates for a Class of Degenerate Parabolic Operators"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given $\\alpha \\in [0,2)$ and $f \\in L^2 ((0,T)\\times(0,1))$,  derive new Carleman estimates for the degenerate parabolic problem $w_t + (x^\\alpha w_x) _x =f$, where $(t,x) \\in (0,T) \\times (0,1)$, associated to 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 ighted functions and Hardy-type inequalities. As a consequence, for all $0 \\leq \\alpha <2$ and $\\omega\\subset\\subset(0,1)$,  deduce null controllability results for the degenerate one-dimensional heat equation $u_t - (x^\\alpha u_x)_x = h \\chi _\\omega$ with the same boundary conditions as above."}]