During my research I performed some numerics and videos. These simulations train our intuition about the dynamics of the PDE. Also in Youtube one can find some interesting real experiments and their results.

eqmuskat

Example 1:

Example 2:

Example 3:

eqmuskat
kspe

KSPE11 corresponds to β identically a constant; KSPE12 corresponds to β(x)=x; KSPE21 corresponds to β(x)=x2; KSPE22 corresponds to β(x)=log(1+x)

kspp

KSPP11 corresponds to β = α =1; KSPP12 corresponds to β= 2, α=1.5; KSPP21 corresponds to α=β=1.5; KSPP22 corresponds to α=1.75, β=2

bl

BL1 corresponds to ν=0, μ=0.5 and β=1; BL2 corresponds to ν=0.5, α=0.5, μ=0.5 and β=0.5

ks

KS1 corresponds to γ=1, δ=1, ε=0.01; KS2 corresponds to ε=0.8, δ=0.5, γ=1.45