restart:

with(PDEtools):with(plots):

declare(u(x,y,t)):

PDE1:=diff(u(x,y,t),t$1)+a*u(x,y,t)*diff(u(x,y,t),x$1)+diff(u(x,y,t),x$3)+

b*diff(u(x,y,t),y$2)=0;

Sol1:=pdsolve(PDE1);

Test1:=pdetest(Sol1,PDE1);

params:={a=6,b=1,_C1=1,_C2=1,_C3=1,_C4=1};

Sol2:=subs(params,Sol1);

plot3d(subs(t=1,rhs(Sol2)),x=-10..10,y=-10..10,axes=boxed,grid=[50,50],

style=patchnogrid,shading=Z,orientation=[-40,50]);