-
Previous Article
An excess-decay result for a class of degenerate elliptic equations
- DCDS-S Home
- This Issue
-
Next Article
Ultrafunctions and applications
Some degenerate parabolic problems: Existence and decay properties
1. | Dipartimento di Matematica, Sapienza Universitá di Roma, Piazzale A. Moro 5, 00185 Roma, Italy, Italy |
References:
[1] |
L. Boccardo and H. Brezis, Some remarks on a class of elliptic equations with degenerate coercivity, Boll. Unione Mat. Ital., 6 (2003), 521-530. |
[2] |
L. Boccardo, G. Croce and L. Orsina, Nonlinear degenerate elliptic problems with $W_0^{1,1}$ solutions, Manuscripta Mathematica, 137 (2012), 419-439.
doi: 10.1007/s00229-011-0473-6. |
[3] |
L. Boccardo, A. Dall'Aglio and L. Orsina, Existence and regularity results for some elliptic equations with degenerate coercivity, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 51-81. |
[4] |
L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with right hand side measures, J. Funct. Anal., 147 (1997), 237-258.
doi: 10.1006/jfan.1996.3040. |
[5] |
L. Boccardo, T. Gallouet and F. Murat, Unicité de la solution pour des equations elliptiques non linéaires, C. R. Acad. Sc. Paris, 315 (1992), 1159-1164. |
[6] |
Bonforte and G. Grillo, Super and ultracontractive bounds for doubly nonlinear evolution equations, Rev. Mat. Iberoamericana, 22 (2006), 111-129. |
[7] | |
[8] |
F. Cipriani and G. Grillo, Uniform bounds for solutions to quasilinear parabolic equations, J. Differential Equations, 177 (2001), 209-234.
doi: 10.1006/jdeq.2000.3985. |
[9] |
A. Dall'Aglio, Approximated solutions of equations with $L^1$ data. Application to the $H$-convergence of quasi-linear parabolic equations, Ann. Mat. Pura Appl., 170 (1996), 207-240.
doi: 10.1007/BF01758989. |
[10] |
D. Giachetti and M. M. Porzio, Existence results for some non uniformly elliptic equations with irregular data, J. Math. Anal. Appl., 257, (2001), 100-130.
doi: 10.1006/jmaa.2000.7324. |
[11] |
D. Giachetti and M. M. Porzio, Elliptic equations with degenerate coercivity: Gradient regularity, Acta. Mathematica Sinica, 19 (2003), 1-11.
doi: 10.1007/s10114-002-0235-1. |
[12] |
G. Grillo, On the equivalence between p-Poincaré inequalities and $L^r-L^q$ regularization and decay estimates of certain nonlinear evolution, J. Differential Equations, 249 (2010), 2561-2576.
doi: 10.1016/j.jde.2010.05.022. |
[13] |
G. Grillo, M. Muratori and M. M. Porzio, Porous media equations with two weights: Existence, uniqueness, smoothing and decay properties of energy solutions via Poincaré inequalities, Discrete and Continuous Dynamical Systems, 33 (2013), 3599-3640.
doi: 10.3934/dcds.2013.33.3599. |
[14] |
O. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of the American Mathematical Society, American Mathematical Society, Providence, 1968. |
[15] |
J. L. Lions, Quelques Méthodes de Resolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. |
[16] |
A. Porretta, Uniqueness and homogenization for a class of noncoercive operators in divergence form, dedicated to Prof. C. Vinti (Perugia, 1996), Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 915-936. |
[17] |
M. M. Porzio, On decay estimates, Journal of Evolution Equations, 9 (2009), 561-591.
doi: 10.1007/s00028-009-0024-8. |
[18] |
M. M. Porzio, Existence, uniqueness and behavior of solutions for a class of nonlinear parabolic problems, Nonlinear Analysis TMA, 74 (2011), 5359-5382.
doi: 10.1016/j.na.2011.05.020. |
[19] | |
[20] |
M. M. Porzio and M. A. Pozio, Parabolic equations with non-linear, degenerate and space-time dependent operators, Journal of Evolution Equations, 8 (2008), 31-70.
doi: 10.1007/s00028-007-0317-8. |
[21] |
M. M. Porzio, F. Smarrazzo and A. Tesei, Radon measure-valued solutions for a class of quasilinear parabolic equations, Arch. Ration. Mech. Anal., 210 (2013), 713-772.
doi: 10.1007/s00205-013-0666-0. |
[22] |
M. M. Porzio, F. Smarrazzo and A. Tesei, Radon measure-valued solutions of nonlinear strongly degenerate parabolic equations, Calculus of Variations and PDEs, (2014), 37 pp.
doi: 10.1007/s00526-013-0680-y. |
[23] |
M. M. Porzio and F. Smarrazzo, Radon Measure-Valued Solutions for some quasilinear degenerate elliptic equations, Annali di Matematica Pura ed Applicata, (2014), 38 pp.
doi: 10.1007/s10231-013-0386-y. |
[24] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[25] |
L. Veron, Effects regularisants des semi-groupes non linéaires dans des espaces de Banach, Ann. Fac. Sci., Toulose Math., 1 (1979), 171-200.
doi: 10.5802/afst.535. |
show all references
References:
[1] |
L. Boccardo and H. Brezis, Some remarks on a class of elliptic equations with degenerate coercivity, Boll. Unione Mat. Ital., 6 (2003), 521-530. |
[2] |
L. Boccardo, G. Croce and L. Orsina, Nonlinear degenerate elliptic problems with $W_0^{1,1}$ solutions, Manuscripta Mathematica, 137 (2012), 419-439.
doi: 10.1007/s00229-011-0473-6. |
[3] |
L. Boccardo, A. Dall'Aglio and L. Orsina, Existence and regularity results for some elliptic equations with degenerate coercivity, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 51-81. |
[4] |
L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with right hand side measures, J. Funct. Anal., 147 (1997), 237-258.
doi: 10.1006/jfan.1996.3040. |
[5] |
L. Boccardo, T. Gallouet and F. Murat, Unicité de la solution pour des equations elliptiques non linéaires, C. R. Acad. Sc. Paris, 315 (1992), 1159-1164. |
[6] |
Bonforte and G. Grillo, Super and ultracontractive bounds for doubly nonlinear evolution equations, Rev. Mat. Iberoamericana, 22 (2006), 111-129. |
[7] | |
[8] |
F. Cipriani and G. Grillo, Uniform bounds for solutions to quasilinear parabolic equations, J. Differential Equations, 177 (2001), 209-234.
doi: 10.1006/jdeq.2000.3985. |
[9] |
A. Dall'Aglio, Approximated solutions of equations with $L^1$ data. Application to the $H$-convergence of quasi-linear parabolic equations, Ann. Mat. Pura Appl., 170 (1996), 207-240.
doi: 10.1007/BF01758989. |
[10] |
D. Giachetti and M. M. Porzio, Existence results for some non uniformly elliptic equations with irregular data, J. Math. Anal. Appl., 257, (2001), 100-130.
doi: 10.1006/jmaa.2000.7324. |
[11] |
D. Giachetti and M. M. Porzio, Elliptic equations with degenerate coercivity: Gradient regularity, Acta. Mathematica Sinica, 19 (2003), 1-11.
doi: 10.1007/s10114-002-0235-1. |
[12] |
G. Grillo, On the equivalence between p-Poincaré inequalities and $L^r-L^q$ regularization and decay estimates of certain nonlinear evolution, J. Differential Equations, 249 (2010), 2561-2576.
doi: 10.1016/j.jde.2010.05.022. |
[13] |
G. Grillo, M. Muratori and M. M. Porzio, Porous media equations with two weights: Existence, uniqueness, smoothing and decay properties of energy solutions via Poincaré inequalities, Discrete and Continuous Dynamical Systems, 33 (2013), 3599-3640.
doi: 10.3934/dcds.2013.33.3599. |
[14] |
O. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of the American Mathematical Society, American Mathematical Society, Providence, 1968. |
[15] |
J. L. Lions, Quelques Méthodes de Resolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. |
[16] |
A. Porretta, Uniqueness and homogenization for a class of noncoercive operators in divergence form, dedicated to Prof. C. Vinti (Perugia, 1996), Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 915-936. |
[17] |
M. M. Porzio, On decay estimates, Journal of Evolution Equations, 9 (2009), 561-591.
doi: 10.1007/s00028-009-0024-8. |
[18] |
M. M. Porzio, Existence, uniqueness and behavior of solutions for a class of nonlinear parabolic problems, Nonlinear Analysis TMA, 74 (2011), 5359-5382.
doi: 10.1016/j.na.2011.05.020. |
[19] | |
[20] |
M. M. Porzio and M. A. Pozio, Parabolic equations with non-linear, degenerate and space-time dependent operators, Journal of Evolution Equations, 8 (2008), 31-70.
doi: 10.1007/s00028-007-0317-8. |
[21] |
M. M. Porzio, F. Smarrazzo and A. Tesei, Radon measure-valued solutions for a class of quasilinear parabolic equations, Arch. Ration. Mech. Anal., 210 (2013), 713-772.
doi: 10.1007/s00205-013-0666-0. |
[22] |
M. M. Porzio, F. Smarrazzo and A. Tesei, Radon measure-valued solutions of nonlinear strongly degenerate parabolic equations, Calculus of Variations and PDEs, (2014), 37 pp.
doi: 10.1007/s00526-013-0680-y. |
[23] |
M. M. Porzio and F. Smarrazzo, Radon Measure-Valued Solutions for some quasilinear degenerate elliptic equations, Annali di Matematica Pura ed Applicata, (2014), 38 pp.
doi: 10.1007/s10231-013-0386-y. |
[24] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[25] |
L. Veron, Effects regularisants des semi-groupes non linéaires dans des espaces de Banach, Ann. Fac. Sci., Toulose Math., 1 (1979), 171-200.
doi: 10.5802/afst.535. |
[1] |
Alexandre N. Carvalho, Jan W. Cholewa. Strongly damped wave equations in $W^(1,p)_0 (\Omega) \times L^p(\Omega)$. Conference Publications, 2007, 2007 (Special) : 230-239. doi: 10.3934/proc.2007.2007.230 |
[2] |
Xin Li, Chunyou Sun, Na Zhang. Dynamics for a non-autonomous degenerate parabolic equation in $\mathfrak{D}_{0}^{1}(\Omega, \sigma)$. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7063-7079. doi: 10.3934/dcds.2016108 |
[3] |
P. R. Zingano. Asymptotic behavior of the $L^1$ norm of solutions to nonlinear parabolic equations. Communications on Pure and Applied Analysis, 2004, 3 (1) : 151-159. doi: 10.3934/cpaa.2004.3.151 |
[4] |
Renato Manfrin. On the boundedness of solutions of the equation $u''+(1+f(t))u=0$. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 991-1008. doi: 10.3934/dcds.2009.23.991 |
[5] |
Denis R. Akhmetov, Renato Spigler. $L^1$-estimates for the higher-order derivatives of solutions to parabolic equations subject to initial values of bounded total variation. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1051-1074. doi: 10.3934/cpaa.2007.6.1051 |
[6] |
Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 763-800. doi: 10.3934/dcds.2008.21.763 |
[7] |
Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1047-1069. doi: 10.3934/dcds.2008.21.1047 |
[8] |
C. García Vázquez, Francisco Ortegón Gallego. On certain nonlinear parabolic equations with singular diffusion and data in $L^1$. Communications on Pure and Applied Analysis, 2005, 4 (3) : 589-612. doi: 10.3934/cpaa.2005.4.589 |
[9] |
Rosaria Di Nardo. Nonlinear parabolic equations with a lower order term and $L^1$ data. Communications on Pure and Applied Analysis, 2010, 9 (4) : 929-942. doi: 10.3934/cpaa.2010.9.929 |
[10] |
Mostafa Bendahmane, Kenneth Hvistendahl Karlsen, Mazen Saad. Nonlinear anisotropic elliptic and parabolic equations with variable exponents and $L^1$ data. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1201-1220. doi: 10.3934/cpaa.2013.12.1201 |
[11] |
Yu Ichida. Classification of nonnegative traveling wave solutions for the 1D degenerate parabolic equations. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022114 |
[12] |
H. Merdan, G. Caginalp. Decay of solutions to nonlinear parabolic equations: renormalization and rigorous results. Discrete and Continuous Dynamical Systems - B, 2003, 3 (4) : 565-588. doi: 10.3934/dcdsb.2003.3.565 |
[13] |
Huijiang Zhao. Large time decay estimates of solutions of nonlinear parabolic equations. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 69-114. doi: 10.3934/dcds.2002.8.69 |
[14] |
Shu-Cherng Fang, David Y. Gao, Ruey-Lin Sheu, Soon-Yi Wu. Canonical dual approach to solving 0-1 quadratic programming problems. Journal of Industrial and Management Optimization, 2008, 4 (1) : 125-142. doi: 10.3934/jimo.2008.4.125 |
[15] |
Xiaoling Sun, Hongbo Sheng, Duan Li. An exact algorithm for 0-1 polynomial knapsack problems. Journal of Industrial and Management Optimization, 2007, 3 (2) : 223-232. doi: 10.3934/jimo.2007.3.223 |
[16] |
Bojing Shi. $ W^{1, p} $ estimates for elliptic problems with drift terms in Lipschitz domains. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 537-553. doi: 10.3934/dcds.2021127 |
[17] |
Francesco Mainardi. On some properties of the Mittag-Leffler function $\mathbf{E_\alpha(-t^\alpha)}$, completely monotone for $\mathbf{t> 0}$ with $\mathbf{0<\alpha<1}$. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2267-2278. doi: 10.3934/dcdsb.2014.19.2267 |
[18] |
Song Li, Junhong Lin. Compressed sensing with coherent tight frames via $l_q$-minimization for $0 < q \leq 1$. Inverse Problems and Imaging, 2014, 8 (3) : 761-777. doi: 10.3934/ipi.2014.8.761 |
[19] |
Hongyong Cui, Peter E. Kloeden, Wenqiang Zhao. Strong $ (L^2,L^\gamma\cap H_0^1) $-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension. Electronic Research Archive, 2020, 28 (3) : 1357-1374. doi: 10.3934/era.2020072 |
[20] |
Emmanuele DiBenedetto, Ugo Gianazza and Vincenzo Vespri. Intrinsic Harnack estimates for nonnegative local solutions of degenerate parabolic equations. Electronic Research Announcements, 2006, 12: 95-99. |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]