Assuming constants:
K1 = 1;
K2 = 1;
K3 = 2/10;
K4 = 1/10;
V = Sin[x];
W = x;
eq = K1*D[w[x, t], {x, 4}] - K2*x*D[w[x, t], {x, 4}] -
2*K2*D[w[x, t], {x, 3}] + K3*D[w[x, t], {t, 2}] -
K4*x*D[w[x, t], {t, 2}] == 0
sol = w[x, t] /.
NDSolve[{eq,
w[0, t] == 0,
Derivative[1, 0][w][0, t] == 0,
Derivative[2, 0][w][0, t] == 0,
Derivative[3, 0][w][0, t] == 0,
w[x, 0] == V,
Derivative[0, 1][w][x, 0] == W},
w, {x, 0, 1}, {t, 0, 1},
MaxSteps -> Infinity, PrecisionGoal -> 1,
AccuracyGoal -> 1,
Method -> {"MethodOfLines",
"SpatialDiscretization" -> {"TensorProductGrid",
"MinPoints" -> 32, "MaxPoints" -> 32, "DifferenceOrder" -> 2},
Method -> {"Adams", "MaxDifferenceOrder" -> 1}}] // Quiet
Plot3D[sol, {x, 0, 1}, {t, 0, 1}, AxesLabel -> Automatic]