Steady state solution (1D) of nonlinear dispersal equation


Now I’m interested in the equation $ $ \frac{\partial }{\partial x}\Bigl(\text{sgn}(x) u \Big) +\frac{\partial}{\partial x} \Bigl[ u^2 \frac{\partial u}{\partial x} \Bigr] =0$ $ with boundary conditions $ u(-5)=u(5)=0$

Since $ \text{sgn}(x)$ is not differentiable at $ x=0$ , I expectd ND solve to have some problems. I tried

sol = NDSolveValue[{   0 == D[Sign[x]*u[x],x] + D[u[x]^2 D[u[x], x], x],    u[-6] == 0, u[6] == 0}   , u, {x, -7, 7}] 

but I can’t even plot it and I think that I’m writing it in the wrong way. Could someone confirm I wrote the right snippet and show the plot I should obtain?

  • I asked a related question three days ago, where the equation was the PDE $ \partial_t u = \partial_x (\text{sign}(x) u) + \partial_x (u^2\partial_x u)$ . The one I have above it’s the steady state solution, and I want to compute it directly, instead of integrating in time.