The Role of Data on the Regularity of Solutions to Some Evolution Equations
Abstract
:1. Introduction
2. Statement of Results
3. Proofs of the Results
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
References
- Nash, J. Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 1958, 80, 931–954. [Google Scholar] [CrossRef]
- Moser, J. A Harnack inequality for parabolic differential equations. Commun. Pure Appl. Math. 1964, 17, 101–134. [Google Scholar] [CrossRef]
- Ladyženskaja, O.; Solonnikov, V.A.; Ural’ceva, N.N. Linear and Quasilinear Equations of Parabolic Type; Translations of the American Mathematical Society; American Mathematical Society: Providence, RI, USA, 1968. [Google Scholar]
- Lions, J.L. Quelques Methodes de Resolution des Problemes Aux Limites Non Lineaires; Dunod et Gauthiers-Villars: Paris, France, 1969. [Google Scholar]
- Aronson, D.G.; Serrin, J. Local behavior of solutions of quasilinear parabolic equations. Arch. Ration. Mech. Anal. 1967, 25, 81–122. [Google Scholar] [CrossRef]
- Bénilan, P.; Crandall, M.G. Regularizing Effects of Homogeneous Evolution Equations; MRC Technical Report N. 2076; MRC: Madison, WI, USA, 1980. [Google Scholar]
- Boccardo, L.; Gallouet, T. Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 1989, 87, 149–169. [Google Scholar] [CrossRef]
- Boccardo, L.; Dall’Aglio, A.; Gallouet, T.; Orsina, L. Existence and regularity results for nonlinear parabolic equations. Adv. Math. Sci. Appl. 1999, 9, 1017–1031. [Google Scholar]
- Di Benedetto, E.; Herrero, M.A. On the Cauchy problem and initial traces for a degenerate parabolic equation. Trans. AMS 1989, 314, 187–224. [Google Scholar] [CrossRef]
- Herrero, M.A.; Vazquez, J.L. Asymptotic behaviour of the solutions of a strongly nonlinear parabolic problem. Ann. Fac. Sci. Toulose Math. 1981, 3, 113–127. [Google Scholar] [CrossRef]
- Porzio, M.M. Existence of solutions for some “noncoercive” parabolic equations. Discret. Contin. Dyn. Syst. 1999, 5, 553–568. [Google Scholar] [CrossRef]
- Dall’Aglio, A.; Orsina, L. Existence results for some nonlinear parabolic equations with nonregular data. Differ. Integral Equ. 1992, 5, 1335–1354. [Google Scholar] [CrossRef]
- Baras, P.; Pierre, M. Problémes paraboliques semilinéaires avec donnés mesures. Appl. Anal. 1984, 18, 11–149. [Google Scholar] [CrossRef]
- Andreu, F.; Mazon, J.M.; Segura De Leon, S.; Toledo, J. Existence and uniqueness for a degenerate parabolic equation with L1(Ω) data. Trans. Am. Math. Soc. 1999, 351, 285–306. [Google Scholar] [CrossRef]
- Prignet, A. Existence and uniqueness of “entropy” solutions of parabolic problems with L1 data. Nonlinear Anal. TMA 1997, 28, 1943–1954. [Google Scholar] [CrossRef]
- Dall’Aglio, A. Approximated solutions of equations with L1 data. Application to the H-convergence of quasi-linear parabolic equations. Ann. Mat. Pura Appl. 1996, 170, 207–240. [Google Scholar] [CrossRef]
- Porretta, A. Regularity for entropy solutions of a class of parabolic equations with nonregular initial datum. Dyn. Syst. Appl. 1998, 7, 53–71. [Google Scholar]
- Kuusi, T.; Mingione, G. Riesz Potentials and Nonlinear Parabolic Equations. Arch. Ration. Mech. Anal. 2014, 212, 727–780. [Google Scholar] [CrossRef]
- Blanchard, D.; Murat, F.; Redwane, H. Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems. J. Differ. Equ. 2001, 177, 331–374. [Google Scholar] [CrossRef]
- Cipriani, F.; Grillo, G. Uniform bounds for solutions to quasilinear parabolic equations. J. Differ. Equ. 2001, 177, 209–234. [Google Scholar] [CrossRef]
- Di Benedetto, E. Degenerate Parabolic Equations; Springer: New York, NY, USA, 1993. [Google Scholar]
- Vazquez, J.L. Smoothing and Decay Estimates for Nonlinear Diffusion Equations; Oxford University Press: Oxford, UK, 2006. [Google Scholar]
- Porzio, M.M. On decay estimates. J. Evol. Equ. 2009, 9, 561–591. [Google Scholar] [CrossRef]
- Veron, L. Effects regularisants des semi-groupes non linéaires dans des espaces de Banach. Ann. Fac. Sci. Toulose Math. 1979, 1, 171–200. [Google Scholar] [CrossRef]
- Bonforte, M.; Grillo, G. Super and ultracontractive bounds for doubly nonlinear evolution equations. Rev. Mat. Iberoam. 2006, 22, 11–129. [Google Scholar]
- Kalashnikov, A.S. Cauchy’s problem in classes of increasing functions for certain quasi-linear degenerate parabolic equations of the second order. Differ. Uravn. 1973, 9, 682–691. [Google Scholar]
- Porzio, M.M. Existence, uniqueness and behavior of solutions for a class of nonlinear parabolic problems. Nonlinear Anal. TMA 2011, 74, 5359–5382. [Google Scholar] [CrossRef]
- Porzio, M.M. Asymptotic behavior and regularity properties of strongly nonlinear parabolic equations. Ann. Mat. Pura Appl. 2019, 198, 1803–1833. [Google Scholar] [CrossRef]
- Porzio, M.M. Regularity and time behavior of the solutions of linear and quasilinear parabolic equations. Adv. Differ. Equ. 2018, 23, 329–372. [Google Scholar] [CrossRef]
- Porzio, M.M. Regularity and time behavior of the solutions to weak monotone parabolic equations. J. Evol. Equ. 2021, 21, 3849–3889. [Google Scholar] [CrossRef]
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Porzio, M.M. The Role of Data on the Regularity of Solutions to Some Evolution Equations. Mathematics 2024, 12, 761. https://doi.org/10.3390/math12050761
Porzio MM. The Role of Data on the Regularity of Solutions to Some Evolution Equations. Mathematics. 2024; 12(5):761. https://doi.org/10.3390/math12050761
Chicago/Turabian StylePorzio, Maria Michaela. 2024. "The Role of Data on the Regularity of Solutions to Some Evolution Equations" Mathematics 12, no. 5: 761. https://doi.org/10.3390/math12050761
APA StylePorzio, M. M. (2024). The Role of Data on the Regularity of Solutions to Some Evolution Equations. Mathematics, 12(5), 761. https://doi.org/10.3390/math12050761