Pseudo-Parabolic Regularization of Forward-Backward Parabolic Equations with Bounded Nonlinearities
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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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- 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
- 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
- 8.L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, Am. Math. Soc., Providence, Rhode Island (1990).Google Scholar