Pseudo-Differential Parabolic Equations with Quasi-Homogeneous Symbols

TL;DRAbstract

We shall consider the Cauchy problem (4.1.1) $$ \partial _t u(t,x) + (Au)(t,x) + \sum\limits_{k = 1}^m {(A_k u)} (t,x) = f(t,x), (t,x) \in \Pi _{(0,T]} , $$ (4.1.2) $$ u(0,x) = \phi (x), $$ where A, A 1,…A m are pseudo-differential operators with the symbols a(t, x, ξ), a 1 (t, x, ξ), … a m (t, x, ξ), that is, e.g., (4.1.3) $$ (Au)(t,x) = (2\pi )^{ - n} \int\limits_{\mathbb{R}^n } {e^{i(x,\xi )} a(t,x,\xi )} \hat u(t,\xi )d\xi , $$ where $$ \hat u(t,\xi ) = \int\limits_{\mathbb{R}^n } {e^{ - i(y,\xi )} } u(t,y)dy. $$

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