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Two-phase forward solutions for one-dimensional forward-backward parabolic equations with linear convection and reaction

We study the existence and properties of Lipschitz continuous weak solutions to the Neumann boundary value problem for a class of one-dimensional quasilinear forward-backward diffusion equations with linear convection and reaction. The diffusion flux function is assumed to be of a forward-backward type that contains two forward-diffusion phases. We prove that, for all smooth initial data, there exists at least one weak solution whose spatial derivative stays in the two forward phases. Also, for all smooth initial data that have a derivative value lying in a certain phase transition range, we show that there exist infinitely many such solutions that exhibit instantaneous phase transitions between the two forward phases. Moreover, we introduce the notion of transition gauge for such forward solutions and prove that the gauge of all constructed two-phase forward solutions can be arbitrarily close to a certain fixed constant.