W_{[`p], \omega}^{m, 1} solvability of the first initial-boundary problem for parabolic equations.
by
R.Ch. Mustafaev
Institute of Mathematics & Mechanics of Acad. Sci. Azerbaijan, Baku

Consider the parabolic equation
Lu \equiv [(\partialu)/(\partialt)]+(-1)^[ m/2]\sum_{| \alpha| <= m}a_\alpha (x, t)D_x^\alphau=f(x, t) (m be a even natural number)
which coefficients are defined on the P_T^{a, b}=Q^{a, b}×(0, T)=(a₁, b₁) × ... (a_n, b_n) ×(0, T) and satisfy following condition
\lambda^-1| \xi| ^m <= \sum_{| \alpha| = m} a_\alpha (x, t)\xi^\alpha <= \lambda| \xi|^m,   for all(x, t) in P_T^{a, b},   for all\xi in Rⁿ.

Suppose that
(a) a_\alpha in C([`(P_T^{a, b})]), for all\alpha:| \alpha| = m; (b) | a_\alpha (x, t)| <= M for all(x, t) in P_T^{a, b}, for all\alpha:| \alpha| < m.

Let [`W] <sub>[`p], \omega</sub>^{m, 1}(P_T^{a, b}),  [`p]=(p<sub>1</sub>, ... , p<sub>n+1</sub>) be the closure in the W<sub>[`p], \omega</sub>^{m, 1}(P_T^{a, b}) norm
|| u|| _{W[`p], \omega<sup>m, 1</sup>(PT<sup>a, b</sup>)</sub>=||u|| <sub>L[`p], \omega(PT^{a, b})}+ \sum_{| \alpha| = m}|| D_x^\alpha u||_{L[`p], \omega(PT<sup>a, b</sup>)</sub>+|| D<sub>t</sub>u|| <sub>L[`p], \omega(PT^{a, b})},
|| u|| <sub>L[`p], \omega(PT<sup>a, b</sup>)</sub> = ( \int<sub>0</sub><sup>T</sup> ( \int<sub>-\infty</sub><sup>+\infty</sup> ... ( \int<sub>-\infty</sub><sup>+\infty</sup>| u(x<sub>1</sub>, ... , x<sub>n</sub>, t)|<sup>p<sub>1</sub></sup>dx<sub>1</sub>) <sup>[(p<sub>2</sub>)/(p<sub>1</sub>)]</sup> ... ) <sup>[(p<sub>n+1</sub>)/(p<sub>n</sub>)]</sup>\omega(t)  dt) <sup>[ 1/(p<sub>n+1</sub>)]</sup>,
of the space {u in C<sup>\infty</sup> ([`(P_T^{a, b})]):u(x, 0)=0 , D_x^\alpha u | _{BT^{a, b}} = 0, |\alpha| <= m-2}. Here B_T^{a, b}=\partialQ^{a, b} ×(0, T).

We consider the first initial-boundary problem

ì í î Lu=f(x, t), (x, t) in PTa, b u| t=0=0, Dx\alpha u | BTa, b = 0 , |\alpha| <= m-2

(1)

The following theorem is valid

Theorem. Let 1 < [`p] < \infty , the coefficients a<sub>\alpha</sub> (x, t) of the operator L satisfy the conditions (a), (b), the weighted function \omega satisfies the condition c) or d), and f in L<sub>[`p], \omega</sub>(P_T^{a, b}):

c) \omega is increasing functions on (0, T), and
sup_{0 < t < T}( \int_t^T\omega( \tau) \tau^-p_n+1d\tau) ( \int₀^t/2\omega( \tau) ^1-p_n+1'd\tau) ^p_n+1-1 < \infty;
( \int₀^T\omega(s)^1-p'_n+1 ( \int_s^T\omega( t ) dt ) ^p'_n+1/p_n+1ds)^1/p'_n+1 <  \infty; \omega( T-0) < \infty;

d) \omega is decreasing functions on (0, T).

Then the problem (1) has a unique solution in the space [`W]<sub>[`p], \omega</sub>^{m, 1}(Q_T). Moreover || u|| _{W[`p], \omega<sup>m, 1</sup>(PT<sup>a, b</sup>)</sub> <= C(T) || f|| <sub>L[`p], \omega(PT^{a, b})}, with constant C=C(\lambda, n, p, T, \omega) independent of f.

Date received: August 12, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caia-05.