The authors study qualitative properties of the solutions to forward-backward parabolic equations of the form \[ u_t = \varphi''(u_x)u_{xx} - G(u), \quad (x,t) \in (-1,1)\times [0,T] \] supplemented by the conditions \[ \begin{aligned} u_x(-1,t) &= u_x(1,t) = 0, \quad \forall t \in [0,T),\\ u(x,0) & = u_0(x), \quad \forall x \in (-1,1) \end{aligned} \] and their generalization to an \(n\)-dimensional open domain \(\Omega\), \[ \begin{aligned} &u_t = \text{div}\left[\varphi'(|\nabla u|)\frac{\nabla u}{|\nabla u|}\right] - G(u), \quad \text{in } \Omega\times [0,T),\\ &\frac{\partial u}{\partial n} = 0, \quad \text{on } \partial\Omega\times [0,T),\\ &u(x,0) = u_0(x), \quad \forall x \in \Omega. \end{aligned} \] Here, \(\varphi\) is a nonlinear function of class \(C^2\) such that \(\varphi'(0) = 0\) without any sign constraint. The above-mentioned problems are well-posed if \(\varphi'' > 0\) and ill-posed if \(\varphi'' < 0\).
First, the author proves a series of results devoted to a priori estimates of solutions to the initial-boundary value problems of the form \( \|u(x,t)\|_{L^p(\Omega)} \leq \|u(x,0)\|_{L^p(\Omega)}\) for every \(p \in [1,\infty]\) and for every \(t \in [0,T)\) and \(\|\nabla u(x,t)\|_{L^1(\Omega)} \leq \|\nabla u(x,0)\|_{L^1(\Omega)}\) \(\forall t \in [0,T)\) under additional assumptions on \(\varphi\).
Finally, the global nonexistence of solutions for the initial-boundary value problem related to the equation \[ u_t = \varphi''(u_x)u_{xx} \pm u^2, \quad \text{in } (-1,1)\times (0,\infty) \] is presented, with a suitable hypothesis on the initial data.