I am working on recreating an image of an heat equation as shown below

I have mostly been successful, however I am still not sure how to add the squiggly lines. Is there also an easier way to make sure the labels 1,0,and 1, are properly aligned? In the figure they have the same baseline, but in my graph they vary.

Any other improvements or ways to create the displayed image are appreciated as always

Code

documentclass[tikz]standalone
usetikzlibrarypositioning

definecolorUiT-mainHTML003349
definecolorUiT-blueHTML007396
definecolorUiT-redHTMLCB333B

begindocument
begintikzpicture
 defxmax2defymax2
 % Shade the box
 shade[top color=white,bottom color=UiT-blue] (0,0) rectangle (xmax,ymax);
 % Draw the border of the box and add labels
 draw [color=UiT-blue] 
 (0,0) -- (xmax,0) node [pos=0.5,label=below:colorUiT-red$u=g$]
 (xmax,0) -- (xmax,ymax) node [pos=0.5,label=right:colorUiT-red$u=0$]
 (xmax,ymax) -- (0,ymax) node [pos=0.5,label=above:colorUiT-red$u=0$] 
 (0,ymax) -- (0,0) node [pos=0.5,label= left:colorUiT-red$u=0$];
 % Add the S in the centre
 path (0,ymax) -- (xmax,0) node [pos=0.5]$S$;
 % Draw the axis
 draw[thick,-stealth] (-1,0) -- (.5+xmax,0) node[anchor= west] $x$;
 draw[thick,-stealth] (0,-0.5) -- (0,.5+ymax) node[anchor=south east] $y$;
 % Label the axis
 node[below left] (0,0) $0$;
 node[label=left:$1$] (s2) at (0,ymax) ;
 node[label=below:$1$] (s4) at (xmax,0) ;
endtikzpicture
enddocument

Very similar to Skillmons answer (in that this uses in and out).

documentclass[tikz]standalone
usetikzlibrarypositioning,arrows.meta,bending

definecolorUiT-mainHTML003349
definecolorUiT-blueHTML007396
definecolorUiT-redHTMLCB333B

begindocument
begintikzpicture
 defxmax2defymax2
 % Shade the box
 shade[top color=white,bottom color=UiT-blue] (0,0) rectangle (xmax,ymax);
 % Draw the border of the box and add labels
 draw [color=UiT-blue] 
 (0,0) -- (xmax,0) coordinate[pos=0.6] (p1) node[pos=0.5,UiT-red,below=1em](u1)$u=g$
 (xmax,0) -- (xmax,ymax) coordinate[pos=0.6] (p2) node[pos=0.5,UiT-red,right=1em](u2)$u=0$
 (xmax,ymax) -- (0,ymax) coordinate[pos=0.6] (p3) node[pos=0.5,UiT-red,above=1em](u3)$u=0$
 (0,ymax) -- (0,0) coordinate[pos=0.6] (p4) node[pos=0.5,UiT-red,left=1em](u4)$u=0$;
 foreach X in 1,...,4
 draw[-Stealth[bend],UiT-red] (uX) to[out=X*90,in=X*90-180] (pX); 
 % Add the S in the centre
 path (0,ymax) -- (xmax,0) node [pos=0.5]$S$;
 % Draw the axis
 draw[thick,-stealth] (-1,0) -- (.5+xmax,0) node[anchor= west] $x$;
 draw[thick,-stealth] (0,-0.5) -- (0,.5+ymax) node[anchor=south east] $y$;
 % Label the axis
 path (0,0) node[below left] $0$
 (0,ymax) node[left] $1$ (xmax,0) node[below] $1$;
endtikzpicture
enddocument

Packing the lines into it with to and the out and in keys (quick and dirty).

documentclass[tikz]standalone
usetikzlibrarypositioning

definecolorUiT-mainHTML003349
definecolorUiT-blueHTML007396
definecolorUiT-redHTMLCB333B

begindocument
begintikzpicture
 defxmax2defymax2
 % Shade the box
 shade[top color=white,bottom color=UiT-blue] (0,0) rectangle (xmax,ymax);
 % Draw the border of the box and add labels
 draw [color=UiT-blue] 
 (0,0) -- (xmax,0) node
 [pos=0.5,label=below:colorUiT-red$u=g$](b)
 (xmax,0) -- (xmax,ymax) node
 [pos=0.5,label=right:colorUiT-red$u=1$](r)
 (xmax,ymax) -- (0,ymax) node
 [pos=0.5,label=above:colorUiT-red$u=2$](t) 
 (0,ymax) -- (0,0) node [pos=0.5,label=
 left:colorUiT-red$u=3$](l);
 % Add the S in the centre
 path (0,ymax) -- (xmax,0) node [pos=0.5]$S$;
 % Draw the axis
 draw[thick,-stealth] (-1,0) -- (.5+xmax,0) node[anchor= west] $x$;
 draw[thick,-stealth] (0,-0.5) -- (0,.5+ymax) node[anchor=south east] $y$;
 % Label the axis
 node[below left] (0,0) $0$;
 node[label=left:$1$] (s2) at (0,ymax) ;
 node[label=below:$1$] (s4) at (xmax,0) ;
 %
 draw[<-,UiT-red] (b.center)++(.1,-0.01) to[out=-90,in=90] ++(-0.2,-0.25);
 draw[<-,UiT-red] (r.center)++(.01,+0.1) to[out=0,in=180] ++(0.25,-0.2);
 draw[<-,UiT-red] (t.center)++(.1,0.01) to[out=90,in=-90] ++(-0.2,0.25);
 draw[<-,UiT-red] (l.center)++(-.01,+0.1) to[out=180,in=0] ++(-0.25,-0.2);
endtikzpicture
enddocument