Nonuniqueness of solutions for a singular diffusion problem (Q854040)

The authors prove nonuniqueness for the singular parabolic equation \(u_t=u\Delta_pu-\gamma| \nabla u| ^p\) in \(\Omega\times(0,T)\) complemented by homogeneous Dirichlet boundary conditions and initial condition \(u(\cdot,0)=u_0\) in \(\Omega\). Here \(\Omega\) is a smooth bounded domain in \(\mathbb R^N\), \(\Delta_p\) denotes the \(p\)-Laplacian, \(p\in(1,2)\), \(\gamma\in(0,(p-1)/p)\) and \(u_0\) is nonnegative and satisfies suitable regularity and compatibility conditions. The authors first prove the existence of a maximal weak solution whose support is nondecreasing in time. Then they provide an explicit example of a weak solution with shrinking support.