| Abstract: |
| Let $j\in {\mathbb Z}$. Motivated by Swiderski's result [4], the following Cauchy problem
$$ \left\{ \begin{array}{l}
u_t^{\prime}=xu_{xx}^{\prime\prime}-(j-1)u_x^{\prime},\, x\geq 0,\, t>0,
\nonumber \
\displaystyle\lim_{t\to 0^{+}}u(t,x)=f(x),\,\, x\geq 0, \nonumber\end{array}\right.$$
was considered in the paper [2].
A semigroup of positive operators was investigated in [3] and it was proved that it provides a solution of the Cauchy problem. Direct approaches to find solutions were also considered in [3].
The Cauchy problem
$$ \left\{ \begin{array}{l}
u_t^{\prime}=\dfrac{x}{2}u_{xx}^{\prime\prime},\, x\geq 0,\, t>0,
\nonumber \
\displaystyle\lim_{t\to 0^{+}}u(t,x)=f(x),\,\, x\geq 0, \nonumber\end{array}\right.$$
was investigated in [1] using the theory of $C_0$-semigroups. We will present the connections between the solutions of the two problems.
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References
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[1] F. Altomare, I. Carbone, On Some Degenerate Differential Operators on
Weighted Function Spaces, J. Math. Anal. Appl. 213 (1997), 308-333.
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[2] U. Abel, A.M. Acu, M. Heilmann, I. Ra\c sa, On some Cauchy problems and positive linear operators (manuscript)
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[3] U. Abel, A.M. Acu, M. Heilmann, I. Ra\c sa, Commutativity and spectral properties for a general class of Sz\`asz-Mirakjan-Durrmeyer operators, arXiv:2407.21722
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[4] T. Swiderski, Global approximation theorems for the
generalized modified Sz\`asz-Mirakyan operators in polynomial weight spaces,
Demo. Math., 36(2), 2003, 383-392. |
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