Sorry but your question doesn't make a lot of sense to me. I don't really understand the first part (problem set-up) at all, and fail to see how this relates to the second part (all the implicitplots). A few simple questions

When you say $u_[t] + uu_[x] = 0$, I assume the '$' symbols are completely meaningless. The '$' construct can do useful things in Maple, but I am reasonably sure that this is not what you want, so you actually mean u_[t] + uu_[x] = 0. Now I am pretty sure that you don't want indexed values (ie [] brackets), but I am also pretty sure that you don't want two simple functions u_(), and uu_() so I am going to make another wild guess that this is actually your differential equation, so what you mean by this "expression" would actually be represented in Maple as

diff(u(x,t),t)+u(x,t)*diff(u(x,t),x)=0

which, if memory serves, looks a bit like the Hopf equation often written as u_{t}+u*u_{x}=0 (althoughMaple does not allow the use of subscripts to indicate partial differentiation - you can "spoof" it but I wouldn't bother)

Now I have to work out what you mean by

$u(x,t) = a$ if $x < -1$

= $b$ if $ -1 < x < 1$

= $c$ if $ x > 1$

Again I'm pretty sure that you don't need/want all of those '$' symbols - so the first step is to produce

u(x,t) = a if x < -1

= b if -1 < x < 1

= c if x > 1

Defining a function like this in Maple is best done by using the piecewise construct

piecewise( x<-1, u(x,t)=a,

x>=-1 and x<1, u(x,t)=b,

x>=1, u(x,t)=c

);

I'm now guessing that in fact this expression is intended to determine your initial conditions (since you claim to have an initial value problem) - but the above does not define any "initial" values, so my next guess is that you mean to define the initial conditions

ics:= piecewise( x<-1, u(x,0)=a,

x>=-1 and x<1, u(x,0)=b,

x>=1, u(x,0)=c

);

So the summary of the problem would be that you want to solve the partial differential equation

diff(u(x,t),t)+u(x,t)*diff(u(x,t),x)=0 (whhihc I think is the Hopf equation)

subject to the initial conditions

ics:= piecewise( x<-1, u(x,0)=a,

x>=-1 and x<1, u(x,0)=b,

x>=1, u(x,0)=c

);

Before I go any further (**if** I go any further), please confirm whether the above is correct, or if I have been wasting my time piling wild guess on wild guess.

I have no idea what you are trying to do with those implicitplot commands or how they relate to the first part of the problem - I would assume that you are trying to plot the solutions to the pde but can make absolutely no sense of what is written here