Examples

RegionPlot3D[x y z<1,{x,-5,5},{y,-5,5},{z,-5,5},PlotStyle->Directive[Yellow,Opacity[0.5]],Mesh->None]

Show[%1,ImageSize->Small]

L = 4;
usol = NDSolveValue[{D[u[t,x,y], t, t] == D[u[t,x,y], x,x] + D[u[t, x, y], y, y] + Sin[u[t,x,y]], u[t,-L, y] == u[t, L, y], u[t, x, -L] == u[t, x, L], u[0,x, y] == Exp[-(x^2 + y^2)], Derivative[1,0,0][u][0,x,y] == 0}, u, {t,0,L/2}, {x,-L,L},{y,-L,L}]

The solution evolves in time along the z axis:

SliceContourPlot3D[usol[t,x,y],"CenterCutSphere",{x,-L,L},{y,-L,L},{t,0,L/2},AxesLabel->Automatic,PlotLegends->Automatic]

{ContourPlot[x^2+y^2,{x,-1,1},{y,-1,1}],SliceContourPlot3D[x^2+y^2,"BackPlanes",{x,-1,1},{y,-1,1},{z,-1,1}]}

RegionPlot3D[x y z<1,{x,-5,5},{y,-5,5},{z,-5,5},PlotStyle->Directive[Yellow,Opacity[0.5]],Mesh->None]

Model I
1. (SuperStar[l]) ( e.t. Cv< H(α))

k:=0.1
θ:=0.3
d:=1

cv:=(1/(256 k (((-1+α))^2)))(1+4 d+6 (d^2)+4 (d^3)+(d^4)-2 α-8 d α-12 (d^2) α-8 (d^3) α-2 (d^4) α+16 k α+32 d k α+16 (d^2) k α+(α^2)+4 d (α^2)+6 (d^2) (α^2)+4 (d^3) (α^2)+(d^4) (α^2)-16 k (α^2)-32 d k (α^2)-16 (d^2) k (α^2)-192 (k^2) (α^2)+8 θ+16 d θ+8 (d^2) θ-256 k θ-16 α θ-32 d α θ-16 (d^2) α θ+320 k α θ+8 (α^2) θ+16 d (α^2) θ+8 (d^2) (α^2) θ-64 k (α^2) θ+16 (θ^2)-32 α (θ^2)+16 (α^2) (θ^2))

D[cv,α]

Solve[cv==0,α]

Plot[cv,{α,0,0.8}]

(UnderBar[α]):=((1+2 d+(d^2)+4 θ-16k)/(1+2 d+(d^2)-8 k+4 θ))

(UnderBar[α])

(OverBar[α]):=((1+2 d+(d^2)+4 θ)/(1+2 d+(d^2)+8 k+4 θ))
(UnderBar[α])

α0:=((1+2 d+(d^2)-12 θ)/(1+2 d+(d^2)+24 k-12 θ))
α0

Profitv:=(1/(256 k (((-1+α))^2)))(1+4 (d^3) (((-1+α))^2)+(d^4) (((-1+α))^2)-256 cv k (((-1+α))^2)-2 α+16 k α+(α^2)-16 k (α^2)-192 (k^2) (α^2)+8 θ-256 k θ-16 α θ+320 k α θ+8 (α^2) θ-64 k (α^2) θ+16 (θ^2)-32 α (θ^2)+16 (α^2) (θ^2)-2 (d^2) (-1+α) (3+α (-3+8 k-4 θ)+4 θ)-4 d (-1+α) (1+α (-1+8 k-4 θ)+4 θ))
Profits:=(1/(128 k (((-1+α))^2)))(1+4 (d^3) (((-1+α))^2)+(d^4) (((-1+α))^2)-128 cs k (((-1+α))^2)-2 α-16 k α+(α^2)+16 k (α^2)+64 (k^2) (α^2)+8 θ-16 α θ+64 k α θ+8 (α^2) θ-64 k (α^2) θ+16 (θ^2)-32 α (θ^2)+16 (α^2) (θ^2)+4 d (-1+α) (-1+α+8 k α-4 θ+4 α θ)+2 (d^2) (-1+α) (-3-4 θ+α (3+8 k+4 θ)))
CS:=(1/8) (((1+d))^2)
SW:=(1/(256 k (((-1+α))^2)))(12 (d^3) (((-1+α))^2)+3 (d^4) (((-1+α))^2)-64 (k^2) (α^2)+3 (((-1+α))^2) (((1+4 θ))^2)-16 k (-1+α) (2+16 cs (-1+α)+16 cv (-1+α)-3 α-16 θ+12 α θ)+4 d (-1+α) (8 k (-2+3 α)+3 (-1+α) (1+4 θ))+2 (d^2) (-1+α) (8 k (-2+3 α)+3 (-1+α) (3+4 θ)))

k:=0.1
θ:=0.1
d:=1
cs:=1

Plot3D[Profitv,{α,0,0.82}, {cv,0,0.8},PlotStyle->Directive[Yellow,Opacity[0.5]]]

Plot3D[Profits,{α,0,0.5}, {cv,0,0.5}]

Plot3D[SW,{α,0,0.5}, {cv,0,0.5}]

α1:=((1+2 d+(d^2)-16 k+4 θ)/(1+2 d+(d^2)-8 k+4 θ))
α2:=((1+2 d+(d^2)+4 θ)/(1+2 d+(d^2)+8 k+4 θ))
α3:=(1+4 d+6 (d^2)+4 (d^3)+(d^4)-8 k-16 d k-8 (d^2) k+8 θ+16 d θ+8 (d^2) θ-160 k θ+16 (θ^2)-16 (Sqrt[(k^2)+4 d (k^2)+6 (d^2) (k^2)+4 (d^3) (k^2)+(d^4) (k^2)-192 (k^3) θ+48 (k^2) (θ^2)]))/(1+4 d+6 (d^2)+4 (d^3)+(d^4)-16 k-32 d k-16 (d^2) k-192 (k^2)+8 θ+16 d θ+8 (d^2) θ-64 k θ+16 (θ^2))
cv:=(1/(256 k (((-1+α))^2)))(1+4 d+6 (d^2)+4 (d^3)+(d^4)-2 α-8 d α-12 (d^2) α-8 (d^3) α-2 (d^4) α+16 k α+32 d k α+16 (d^2) k α+(α^2)+4 d (α^2)+6 (d^2) (α^2)+4 (d^3) (α^2)+(d^4) (α^2)-16 k (α^2)-32 d k (α^2)-16 (d^2) k (α^2)-192 (k^2) (α^2)+8 θ+16 d θ+8 (d^2) θ-256 k θ-16 α θ-32 d α θ-16 (d^2) α θ+320 k α θ+8 (α^2) θ+16 d (α^2) θ+8 (d^2) (α^2) θ-64 k (α^2) θ+16 (θ^2)-32 α (θ^2)+16 (α^2) (θ^2))
k:=0.1
θ:=0.3
d:=1
α1
α2
α3

Plot[cv,{α,0,1}]

1. (SuperStar[l])

Profits:=(1/(128 k (((-1+α))^2)))(1+4 (d^3) (((-1+α))^2)+(d^4) (((-1+α))^2)-128 cs k (((-1+α))^2)-2 α-16 k α+(α^2)+16 k (α^2)+64 (k^2) (α^2)+8 θ-16 α θ+64 k α θ+8 (α^2) θ-64 k (α^2) θ+16 (θ^2)-32 α (θ^2)+16 (α^2) (θ^2)+4 d (-1+α) (-1+α+8 k α-4 θ+4 α θ)+2 (d^2) (-1+α) (-3-4 θ+α (3+8 k+4 θ)))
Profitv:=(1/(256 k (((-1+α))^2)))(1+4 (d^3) (((-1+α))^2)+(d^4) (((-1+α))^2)-256 cv k (((-1+α))^2)-2 α+16 k α+(α^2)-16 k (α^2)-192 (k^2) (α^2)+8 θ-256 k θ-16 α θ+320 k α θ+8 (α^2) θ-64 k (α^2) θ+16 (θ^2)-32 α (θ^2)+16 (α^2) (θ^2)-2 (d^2) (-1+α) (3+α (-3+8 k-4 θ)+4 θ)-4 d (-1+α) (1+α (-1+8 k-4 θ)+4 θ))
SW:=(1/(256 k (((-1+α))^2)))(12 (d^3) (((-1+α))^2)+3 (d^4) (((-1+α))^2)-64 (k^2) (α^2)+3 (((-1+α))^2) (((1+4 θ))^2)-16 k (-1+α) (2+16 cs (-1+α)+16 cv (-1+α)-3 α-16 θ+12 α θ)+4 d (-1+α) (8 k (-2+3 α)+3 (-1+α) (1+4 θ))+2 (d^2) (-1+α) (8 k (-2+3 α)+3 (-1+α) (3+4 θ)))

k:=0.1
θ:=0.3
d:=1
cs:=0.5
Plot3D[Profits,{cv,1,2},{α,0,0.5}]
Plot3D[Profitv,{cv,1,2},{α,0,0.5}]

k:=0.1
θ:=0.1
d:=1
cs:=0.5
cv:=0.5
Plot[Profits,{α,0,1}]
Plot[Profitv,{α,0,1}]

2. (l^VA)

Profitv:=(1/(256 k (((-1+α))^2)))(1+4 (d^3) (((-1+α))^2)+(d^4) (((-1+α))^2)-256 cv k (((-1+α))^2)-2 α+16 k α+(α^2)-16 k (α^2)-192 (k^2) (α^2)+8 θ-256 k θ-16 α θ+320 k α θ+8 (α^2) θ-64 k (α^2) θ+16 (θ^2)-32 α (θ^2)+16 (α^2) (θ^2)-2 (d^2) (-1+α) (3+α (-3+8 k-4 θ)+4 θ)-4 d (-1+α) (1+α (-1+8 k-4 θ)+4 θ))
Profits:=(1/(128 k (((-1+α))^2)))(1+4 (d^3) (((-1+α))^2)+(d^4) (((-1+α))^2)-128 cs k (((-1+α))^2)-2 α-16 k α+(α^2)+16 k (α^2)+64 (k^2) (α^2)+8 θ-16 α θ+64 k α θ+8 (α^2) θ-64 k (α^2) θ+16 (θ^2)-32 α (θ^2)+16 (α^2) (θ^2)+4 d (-1+α) (-1+α+8 k α-4 θ+4 α θ)+2 (d^2) (-1+α) (-3-4 θ+α (3+8 k+4 θ)))
CS:=(1/8) (((1+d))^2)
SW:=(1/(256 k (((-1+α))^2)))(12 (d^3) (((-1+α))^2)+3 (d^4) (((-1+α))^2)-64 (k^2) (α^2)+3 (((-1+α))^2) (((1+4 θ))^2)-16 k (-1+α) (2+16 cs (-1+α)+16 cv (-1+α)-3 α-16 θ+12 α θ)+4 d (-1+α) (8 k (-2+3 α)+3 (-1+α) (1+4 θ))+2 (d^2) (-1+α) (8 k (-2+3 α)+3 (-1+α) (3+4 θ)))

k:=0.1
θ:=0.1
d:=1
cs:=0.5
cv:=1.5
Plot[Profits,{α,0,1}]
Plot[Profitv,{α,0,1}]