We introduce and analyze a new, nonlinear fourth-order regularization of forwardbackward parabolic equations. In one space dimension, under general assumptions on the potentials, which include those of Perona--Malik type, we prove existence of Radon measure-valued solutions under both natural and essential boundary conditions. If the decay at infinity of the nonlinearities is sufficiently fast, we also exhibit examples of local solutions whose atomic part arises and/or persists (in contrast to the linear fourth-order regularization) and even disappears within finite time (in contrast to pseudoparabolic regularizations).

Measure-Valued solutions to a nonlinear Fourth-Order regularization of forward-backward parabolic equations / Bertsch, Michiel; Giacomelli, Lorenzo; Tesei, Alberto. - In: SIAM JOURNAL ON MATHEMATICAL ANALYSIS. - ISSN 0036-1410. - 51:1(2019), pp. 374-402. [10.1137/18M1203821]

Measure-Valued solutions to a nonlinear Fourth-Order regularization of forward-backward parabolic equations

Giacomelli, Lorenzo;Tesei, Alberto
2019

Abstract

We introduce and analyze a new, nonlinear fourth-order regularization of forwardbackward parabolic equations. In one space dimension, under general assumptions on the potentials, which include those of Perona--Malik type, we prove existence of Radon measure-valued solutions under both natural and essential boundary conditions. If the decay at infinity of the nonlinearities is sufficiently fast, we also exhibit examples of local solutions whose atomic part arises and/or persists (in contrast to the linear fourth-order regularization) and even disappears within finite time (in contrast to pseudoparabolic regularizations).
2019
forward-backward parabolic equations; fourth-order parabolic equations; Radon measures; Perona--Malik equation
01 Pubblicazione su rivista::01a Articolo in rivista
Measure-Valued solutions to a nonlinear Fourth-Order regularization of forward-backward parabolic equations / Bertsch, Michiel; Giacomelli, Lorenzo; Tesei, Alberto. - In: SIAM JOURNAL ON MATHEMATICAL ANALYSIS. - ISSN 0036-1410. - 51:1(2019), pp. 374-402. [10.1137/18M1203821]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1278417
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