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This paper deals with the existence, uniqueness and maximal regularity of a solution to the multidimensional parabolic equation of order 2 m  (withm  ∈   ℕ  *${m\in \mathbb {N} ^{\ast }}$   )∂  t   u  +    (  -  1  )   m    ∑   k  =  1   N    ∂    x  k2  m    u  =  f  ,    $ \partial _{t}u+(-1)^{m}\sum _{k=1}^{N}\partial _{x_{k}}^{2m}u=f, $    defined on a type of the prismQ  =    (  t  ,   x  1   )   ∈   ℝ  2   :  0  &lt;  t  &lt;  T  ,φ  1    (  t  )   &lt;   x  1   &lt;   φ  2    (  t  )    ×   ∏   i  =  1    N  -  1]  0  ,   b  i   [$ Q=\bigl \lbrace  ( t,x_{1}) \in \mathbb {R} ^{2}:0&amp;lt;t&amp;lt;T,\, \varphi _{1}( t) &amp;lt;x_{1}&amp;lt;\varphi _{2}( t) \bigr \rbrace  \times \prod _{i=1}^{N-1}]0,b_{i}[ $    of ℝ N+1 , which is a nonstandard domain since its base shrinks at  t  = 0 (φ  1    (  0  )   =   φ  2    (  0  )${\varphi _{1}(0)=\varphi _{2}(0)}$   ). The equation is associated with Cauchy–Dirichlet boundary conditions and the right-hand side term  f  is taken in a weighted Lebesgue space. More precisely, we look for minimal conditions on the functionsφ  1    ${\varphi _{1}}$    andφ  2    ${\varphi _{2}}$    under which the solution is regular. For this purpose, the domain decomposition method is employed. This work is an extension of the one space variable case studied in  [19]  and of the case  m  = 1 studied in  [11] .

Study of a 2 m -th order parabolic equation in a non-regular type of prism of ℝ N+1