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Cauchy problem for a semilinear Éidel’man parabolic equation

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We establish conditions for the existence and uniqueness of a generalized solution of the Cauchy problem for the equation

$$ \begin{aligned} & u_{t} + {\sum\limits_{{\left| \alpha \right|} = {\left| \beta \right|} = 2} {{\left( { - 1} \right)}^{{{\left| \alpha \right|}}} \;D^{\alpha }_{x} {\left( {a_{{\alpha \beta }} {\left( {z,\;t} \right)}D^{\beta }_{x} u} \right)}} } \\ & \;\;\; - {\sum\limits_{{\left| \alpha \right|} = {\left| \beta \right|} = 1} {{\left( { - 1} \right)}^{{{\left| \alpha \right|}}} \;D^{\alpha }_{y} {\left( {b_{{\alpha \beta }} {\left( {z,\;t} \right)}D^{\beta }_{y} u} \right)}} } + {\sum\limits_{{\left| \alpha \right|} = 1} {c_{\alpha } {\left( {z,\;t} \right)}D^{\alpha }_{z} u + c{\left( {z,\,t,\,u} \right)}} } \\ & \quad \quad \quad \quad \quad = {\sum\limits_{{\left| \alpha \right|} \leq 2} {{\left( { - 1} \right)}^{{{\left| \alpha \right|}}} D^{\alpha }_{x} f_{\alpha } {\left( {z,\;t} \right)} - {\sum\limits_{{\left| \alpha \right|} = 1} {D^{\alpha }_{y} g_{\alpha } {\left( {z,\;t} \right)}} }} } \\ \end{aligned} $$

in a Tikhonov-type class.

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Correspondence to O. E. Korkuna.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 5, pp. 586–602, May, 2008.

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Korkuna, O.E. Cauchy problem for a semilinear Éidel’man parabolic equation. Ukr Math J 60, 671–691 (2008). https://doi.org/10.1007/s11253-008-0081-0

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Keywords

  • Generalize Solution
  • Cauchy Problem
  • Parabolic Equation
  • Initial Function
  • Parabolic System