AbstractIn this paper we present an analytical framework for the following system of multivalued parabolic variational inequalities in a cylindrical domain $$Q=\varOmega \times (0,\tau )$$
Q
=
Ω
×
(
0
,
τ
)
: For $$k=1,\dots , m$$
k
=
1
,
⋯
,
m
, find $$u_k\in K_k$$
u
k
∈
K
k
and $$\eta _k\in L^{p'_k}(Q)$$
η
k
∈
L
p
k
′
(
Q
)
such that $$\begin{aligned}&u_k(\cdot ,0)=0\ \text{ in } \varOmega ,\ \ \eta _k(x,t)\in f_k(x,t,u_1(x,t), \dots , u_m(x,t)), \\&\langle u_{kt}+A_k u_k, v_k-u_k\rangle +\int _Q \eta _k\, (v_k-u_k)\,dxdt\ge 0,\ \ \forall \ v_k\in K_k, \end{aligned}$$
u
k
(
·
,
0
)
=
0
in
Ω
,
η
k
(
x
,
t
)
∈
f
k
(
x
,
t
,
u
1
(
x
,
t
)
,
⋯
,
u
m
(
x
,
t
)
)
,
⟨
u
kt
+
A
k
u
k
,
v
k
-
u
k
⟩
+
∫
Q
η
k
(
v
k
-
u
k
)
d
x
d
t
≥
0
,
∀
v
k
∈
K
k
,
where $$K_k $$
K
k
is a closed and convex subset of $$L^{p_k}(0,\tau ;W_0^{1,p_k}(\varOmega ))$$
L
p
k
(
0
,
τ
;
W
0
1
,
p
k
(
Ω
)
)
, $$A_k$$
A
k
is a time-dependent quasilinear elliptic operator, and $$f_k:Q\times \mathbb {R}^m\rightarrow 2^{\mathbb {R}}$$
f
k
:
Q
×
R
m
→
2
R
is an upper semicontinuous multivalued function with respect to $$s\in {\mathbb R}^m$$
s
∈
R
m
. We provide an existence theory for the above system under certain coercivity assumptions. In the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence and enclosure results. As an application, a multivalued parabolic obstacle system is treated. Moreover, under a lattice condition on the constraints $$K_k$$
K
k
, systems of evolutionary variational-hemivariational inequalities are shown to be a subclass of the above system of multivalued parabolic variational inequalities.
Plancherel's theorem and integration without the lattice condition
