Parabolic Equations of a Quasi-Homogeneous Structure

Abstract

We start from the case of a $\overrightarrow {2b}$-parabolic equation of the form (1.1.10), of the first order with respect to the variable t, with bounded coefficients, that is the equation

$$ \begin{gathered} L_N (t,x,\partial _t ,\partial _x )u(t,x): = \left( {I\partial _t - \sum\limits_{\left\| k \right\| \leqslant 2b} {a_k (t,x)\partial _x^k } } \right)u(t,x) \hfill \\ = f(t,x), (t,x) \in \Pi _{(0,T]} , \hfill \\ \end{gathered} $$

((2.1.1))

in which N ∈ ℕ, the coefficients T] → ℂ _N,N , ‖k‖ ≤ 2b, are bounded functions, f : П_[0,T] → ℂ _N,1 is a given function, u : П_[0,T] → ℂ _N,1 is an unknown function.