Behaviour of Solutions of Parabolic Boundary Value Problems for Large Values of Time

Abstract

In ℝ ^{n +1}₊₊ we consider the parabolic boundary value problem:

$$\begin{array}{*{20}{c}} {\sum\limits_{{j = 1}}^{m} {{{l}_{{lkj}}}(D,{{D}_{t}}){{u}_{j}}(x,t) = 0,} } & {(x,t) \in \mathbb{R}_{{ + + }}^{{n + 1}},} & {k = 1, \ldots ,m;} \\ \end{array}$$

(1.1)

$$\begin{array}{*{20}{c}} {\sum\limits_{{j = 1}}^{m} {{{b}_{{qj}}}(D,{{D}_{t}}){{u}_{j}}(x,t){{|}_{{{{x}_{n}} = + 0}}} = {{\varphi }_{q}}(x\prime ,t),} } & {(x\prime ,t) \in \mathbb{R}_{ + }^{n},q = 1, \ldots ,br,} \\ \end{array}$$

(1.2)

in spaces of smooth bounded Hölder functions that vanish at t = 0 together with all their derivatives that appear in (1.1) and (1.2). Here l _kj (D, D _t ) and b _qj (D, D _t ) are quasihomogeneous operators with constant coefficients of orders s _k + t _j and σ _q + t _j , respectively, $\sum\limits_{{k = 1}}^{m} {({{s}_{k}} + {{t}_{k}}) = 2br}$.