The authors consider an initial boundary value problem for the degenerate pseudoparabolic equation (P) \[ \begin{aligned} u_t = [\phi(u)]_{xx} + \epsilon [\psi(u)]_{txx}, & \text{ in } Q: = \Omega \times (0, T) \\ \phi(u) + \epsilon [\psi(u)]_t = \gamma^*, & \text{on \;} \partial \Omega \times (0, T) \\ u(\cdot, 0) = u_0\end{aligned} \] where \(\epsilon, T, \gamma^*\) are positive constants, \(\Omega = (a,b) \subseteq R\), and \(u_0\) is a nonnegative Radon measure on \(\Omega\). The function \(\phi\) is nonmonotonic; there exists \(0 < \alpha < \beta < \infty\) such that \(\phi^{\prime} >0\) on \([0, \alpha), \phi^{\prime} < 0\) on \((\alpha, \beta)\), and \(\phi^{\prime} >0\) on \((\beta, \infty)\). Moreover, \(\phi(0) = 0, 0 \leq \phi(u) \leq \gamma^*, \phi(u) \to \gamma^*\), as \(u \to \infty\).
This is a quasilinear forward-backward-forward parabolic equation with regularizing term \(\epsilon \left[ \psi(u) \right]_{txx}\). It is closely related to the forward-backward problem (FB) \[ \begin{aligned} z_t = [\chi(z)]_{xx} + \epsilon [\psi(z)]_{txx}, & \text{\;in \;} Q \\ \chi(z) + \epsilon [\psi(z)]_t = 0, & \text{\;\;on \;} \partial \Omega \times (0, T) \\ z(\cdot, 0) = z_0 , \end{aligned} \] where the function \(\chi\) is nonmonotonic; there exists \(0 < \omega < \infty\) such that \(\chi^{\prime} >0\) on \([0, \omega), \chi^{\prime} < 0\) on \((\omega, \infty)\). Moreover, \(\chi(0) = 0, 0 \leq \chi(z) \leq \chi^* = \chi(\omega), \chi(z) \to 0\), as \( z \to \infty\).
The solution of problem (FB) can have spontaneous appearance of singularities; there are solutions with smooth initial data which become singular Radon measure valued solutions in time. If you change variables, \( s = T - t\) (reverse time) the related equation has solutions with spontaneous disappearance of singularities.
The authors prove the existence of suitably defined Radon measure valued solutions of problem (P) and several monotonicity properties of these solutions in time. The singular part of the measure valued solution is nonincreasing in time. They discuss the extinction in finite time of the singular part, examining how a Dirac mass can have its singular part extinguished in finite time.
They remark that the uniqueness of solutions of (P) remains an open problem, but recently examples of nonuniqueness have been given for problem (FB). The nonuniqueness is related to the forward-backward nature of the problem.
They finally observe that the forward-backward-forward problem (P) is considerably simpler than the forward-backward problem (FB), but that they will in future papers use the techniques in this paper to give results for problem (FB). These results will necessarily be more complicated.