Abstract
In this paper, we study the degenerate parabolic system
$$ u_{t}^{i} + X_{\alpha }^{*} \bigl(a_{ij}^{\alpha \beta }(z){X_{\beta }} {u^{j}}\bigr) = {g_{i}}(z,u,Xu) + X_{\alpha }^{*} f_{i}^{\alpha }(z,u,Xu), $$
u
t
i
+
X
α
∗
(
a
i
j
α
β
(
z
)
X
β
u
j
)
=
g
i
(
z
,
u
,
X
u
)
+
X
α
∗
f
i
α
(
z
,
u
,
X
u
)
,
where $X=\{X_{1},\ldots,X_{m} \}$
X
=
{
X
1
,
…
,
X
m
}
is a system of smooth real vector fields satisfying Hörmander’s condition and the coefficients $a_{ij}^{\alpha \beta }$
a
i
j
α
β
are measurable functions and their skew-symmetric part can be unbounded. After proving the $L^{2}$
L
2
estimates for the weak solutions, the higher integrability is proved by establishing a reverse Hölder inequality for weak solutions.
Existence Theory for a Strongly Degenerate Parabolic System