Some degenerate parabolic problems: Existence and decay properties

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August 2014, 7(4): 617-629. doi: 10.3934/dcdss.2014.7.617

Some degenerate parabolic problems: Existence and decay properties

Lucio Boccardo ^1, and Maria Michaela Porzio ^1,

1.	Dipartimento di Matematica, Sapienza Universitá di Roma, Piazzale A. Moro 5, 00185 Roma, Italy, Italy

Received October 2013 Revised December 2013 Published February 2014

Abstract
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We study the existence of solutions $u $ belonging to $L^1(0,T; W_0^{1,1}(\Omega)) \cap L^{\infty}(0,T;L^2(\Omega))$ of a class of nonlinear problems whose prototype is the following \begin{equation} \label{prob1} \left\{ \begin{array}{lll} \displaystyle u_t - {\rm div} \left( \frac{\nabla u}{(1+|u|)^2} \right) = 0, & \hbox{in} & \Omega_T; \\ u=0, & \hbox{on} & \partial\Omega \times (0,T); & & & & \hbox{(1)}\\ u(x,0)= u_0(x) \in L^2(\Omega), & \hbox{ in} & \Omega. \end{array} \right. \end{equation} We investigate also the asymptotic estimates satisfied by distributional solutions that we find and the uniqueness.

Keywords: unbounded solutions, 1}(\Omega))$-solutions, T;W_0^{1, $L^1(0, Nonlinear parabolic problems, decay estimates, degenerate parabolic equations, existence results..

Mathematics Subject Classification: Primary: 35K65, 35K55; Secondary: 35K10, 35K15, 35K2.

Citation: Lucio Boccardo, Maria Michaela Porzio. Some degenerate parabolic problems: Existence and decay properties. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 617-629. doi: 10.3934/dcdss.2014.7.617

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]	H. Brezis and T. Cazenave, Notes, unpublished.
[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]	M. M. Porzio, On uniform estimates, 2013.
[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.

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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]	H. Brezis and T. Cazenave, Notes, unpublished.
[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]	M. M. Porzio, On uniform estimates, 2013.
[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.

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