s1 = NDSolve[{D[u[t, x, y], t] ==
D[u[t, x, y], x, x] + D[u[t, x, y], y, y] + 3*Tanh[u[t, x, y]],
u[0, x, y] == Sin[([Pi] x)/5] Sin[(2 [Pi] y)/5],
u[t, -5, y] == 0, u[t, 5, y] == 0, u[t, x, -5] == 0,
u[t, x, 5] == 0}, u, {t, 0, 6}, {x, -5, 5}, {y, -5, 5},
Method -> {"FixedStep", Method -> "ExplicitEuler"},
MaxStepFraction -> 10000, WorkingPrecision -> MachinePrecision]

NDSolve::eerr
Warning: scaled local spatial error estimate of 26.30473550335526` at
t = 6.` in the direction of independent variable x is much greater
than the prescribed error tolerance. Grid spacing with 15 points may
be too large to achieve the desired accuracy or precision. A
singularity may have formed or a smaller grid spacing can be
specified using the MaxStepSize or MinPoints method options.

a = Table[
Plot3D[u[t, x, y] /. s1, {x, -5, 5}, {y, -5, 5}, Mesh -> 100,
PlotRange -> All,
ColorFunction -> Function[{x, y, z}, Hue[.3 (1 - z)]]], {t, 0, 6}]