  We introduce $n$-parameter $\Rd$-valued Brownian-time Brownian sheet (BTBS):
a Brownian sheet where each "time" parameter is replaced with the modulus of an
independent Brownian motion. We then connect BTBS to a new system of $n$
linear, fourth order, and interacting PDEs and to a corresponding fourth order
interacting nonlinear PDE. The coupling phenomenon is a result of the
interaction between the Brownian sheet, through its variance, and the Brownian
motions in the BTBS; and it leads to an intricate, intriguing, and random field
generalization of our earlier Brownian-time-processes (BTPs) connection to
fourth order linear PDEs. Our BTBS does not belong to the classical theory of
random fields; and to prove our new PDEs connections, we generalize our BTP
approach in \cite{Abtp1,Abtp2} and we mix it with the Brownian sheet connection
to a linear PDE system, which we also give along with its corresponding
nonlinear second order PDE and $2n$-th order linear PDE. In addition, we
introduce the $n$-parameter $d$-dimensional linear Kuramoto-Sivashinsky (KS)
sheet kernel (or "transition density"); and we link it to an intimately
connected system of new linear Kuramoto-Sivashinsky-variant interacting PDEs,
generalizing our earlier one parameter imaginary-Brownian-time-Brownian-angle
kernel and its connection to the KS PDE. The interactions here mean that our
PDEs systems are to be solved for a family of functions, a feature shared with
well known fluids dynamics models. The interacting PDEs connections established
here open up another new fundamental front in the rapidly growing field of
iterated-type processes and their connections to both new and important higher
order PDEs and to some equivalent fractional Cauchy problems. We connect the
BTBS fourth order interacting PDEs system given here with an interacting
fractional PDE system and further study it in another article.
