We derive an upper bound on the waiting time for a variational weak solution to Inverse Mean Curvature Flow in $\mathbb{R}^{n+1}$ become star-shaped. As consequence, we demonstrate that any connected surface moving by flow which is not initially topological sphere develops singularity or self-intersection within prescribed interval depending only initial data. Finally, establish existence of ei...

In this paper we study a convection–diffusion equation on star-shaped graph composed by n incoming edges and m outgoing with nonlinearity $$f\in C^1({\mathbb {R}})$$ satisfying some additional general conditions. First, prove the global well-posedness of solutions system under consideration. Next, in particular case that nonlinear convection is given $$\partial _x(f(u(t, x))$$ $$f(s)=-a|s|^{q-1...