# Pseudo-Parabolic Regularization of Forward-Backward Parabolic Equations with Bounded Nonlinearities

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## Abstract

We study the initial-boundary value problem

$$ \Big\{{\displaystyle \begin{array}{l} ut={\left[\varphi (u)\right]}_{xx}+\varepsilon {\left[\psi (u)\right]}_{txx}\kern1em \mathrm{in}\;\varOmega \times \left(0,T\right]\\ {}\varphi (u)+\varepsilon {\left[\psi (u)\right]}_t=0\kern3em \mathrm{in}\;\partial \varOmega \times \left(0,T\right]\\ {}u={u}_0\ge 0\kern7em \mathrm{in}\;\varOmega \times \left\{0\right\},\end{array}} $$

with Radon measure-valued initial data, by assuming that the regularizing term *ψ* is bounded and increasing (the cases of power-type or logarithmic *ψ* were examined in [2, 3] for spaces on any dimension). The function 𝜑 is nonmonotone and bounded, and either (i) decreases and vanishes at infinity, or (ii) increases at infinity. The existence of solutions in a space of positive Radon measures is proved in both cases. Moreover, a general result on the *spontaneous appearance of singularities* in he case (i) is presented. The case of a cubic-like 𝜑 is also discussed to point out the influence of the behavior at infinity of 𝜑 on the regularity of solutions.

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## References

- 1.G. I. Barenblatt, M. Bertsch, R. Dal Passo, and M. Ughi, “A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow,”
*SIAM J. Math. Anal.*,**24**, 1414–1439 (1993).MathSciNetCrossRefGoogle Scholar - 2.M. Bertsch, F. Smarrazzo, and A. Tesei, “Pseudo-parabolic regularization of forward-backward parabolic equations: A logarithmic nonlinearity,”
*Anal. PDE*,**6**, 1719–1754 (2013).MathSciNetCrossRefGoogle Scholar - 3.M. Bertsch, F. Smarrazzo, and A. Tesei, “Pseudo-parabolic regularization of forward-backward parabolic equations: Power-type nonlinearities,”
*J. Reine Angew. Math.*,**712**, 51–80 (2014).MathSciNetzbMATHGoogle Scholar - 4.M. Bertsch, F. Smarrazzo, and A. Tesei, “Forward-backward parabolic equations with pseudoparabolic regularization and bounded nonlinearities decreasing at infinity: Existence of solutions,” preprint (2015).Google Scholar
- 5.M. Bertsch, F. Smarrazzo, and A. Tesei, “Pseudo-parabolic regularization of forward-backward parabolic equations: Bounded nonlinearities increasing at infinity,” preprint (2015).Google Scholar
- 6.M. Bertsch, F. Smarrazzo, and A. Tesei, “Forward-backward parabolic equations with pseudoparabolic regularization and bounded nonlinearities decreasing at infinity: Qualitative properties of solutions,” in preparation.Google Scholar
- 7.M. Brokate and J. Sprekels,
*Hysteresis and Phase Transitions*, Springer-Verlag, Berlin (1996).CrossRefGoogle Scholar - 8.L. C. Evans,
*Weak Convergence Methods for Nonlinear Partial Differential Equations*, Am. Math. Soc., Providence, Rhode Island (1990).Google Scholar - 9.M. Giaquinta, G. Modica, and J. Souček,
*Cartesian Currents in the Calculus of Variations*, Springer-Verlag, Berlin (1998).CrossRefGoogle Scholar - 10.C. Mascia, A. Terracina, and A. Tesei, “Two-phase entropy solutions of a forward-backward parabolic equation,”
*Arch. Ration. Mech. Anal.*,**194**, 887–925 (2009).MathSciNetCrossRefGoogle Scholar - 11.A. Novick-Cohen and R. L. Pego, “Stable patterns in a viscous diffusion equation,”
*Trans. Am. Math. Soc.*,**324**, 331–351 (1991).MathSciNetCrossRefGoogle Scholar - 12.V. Padròn, “Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations,”
*Commun. Part. Differ. Equ.*,**23**, 457–486 (1998).MathSciNetCrossRefGoogle Scholar - 13.P. Perona and J. Malik, “Scale space and edge detection using anisotropic diffusion,”
*IEEE Trans. Pattern Anal. Mach. Intell.*,**12**, 629–639 (1990).CrossRefGoogle Scholar - 14.P. I. Plotnikov, “Passing to the limit with respect to viscosity in an equation with variable parabolicity direction,”
*Differ. Equ.*,**30**, 614–622 (1994).zbMATHGoogle Scholar - 15.D. Serre,
*Systems of Conservation Laws, Vol. 1: Hyperbolicity, Entropies, Shock Waves*, Cambridge Univ. Press, Cambridge (1999).CrossRefGoogle Scholar - 16.F. Smarrazzo, “On a class of equations with variable parabolicity direction,”
*Discr. Contin. Dyn. Syst.*,**22**, 729–758 (2008).MathSciNetCrossRefGoogle Scholar - 17.F. Smarrazzo and A. Tesei, “Degenerate regularization of forward-backward parabolic equations: The regularized problem,”
*Arch. Ration. Mech. Anal.*,**204**, 85–139 (2012).MathSciNetCrossRefGoogle Scholar - 18.F. Smarrazzo and A. Tesei, “Degenerate regularization of forward-backward parabolic equations: The vanishing viscosity limit,”
*Math. Ann.*,**355**, 551–584 (2013).MathSciNetCrossRefGoogle Scholar

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