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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

Bonforte and G. Grillo, Super and ultracontractive bounds for doubly nonlinear evolution equations, Rev. Mat. Iberoamericana, 22 (2006), 111-129.  Google Scholar

[7]

H. Brezis and T. Cazenave, Notes,, unpublished., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. L. Lions, Quelques Méthodes de Resolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[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.  Google Scholar

[17]

M. M. Porzio, On decay estimates, Journal of Evolution Equations, 9 (2009), 561-591. doi: 10.1007/s00028-009-0024-8.  Google Scholar

[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.  Google Scholar

[19]

M. M. Porzio, On uniform estimates,, 2013., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

Bonforte and G. Grillo, Super and ultracontractive bounds for doubly nonlinear evolution equations, Rev. Mat. Iberoamericana, 22 (2006), 111-129.  Google Scholar

[7]

H. Brezis and T. Cazenave, Notes,, unpublished., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. L. Lions, Quelques Méthodes de Resolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[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.  Google Scholar

[17]

M. M. Porzio, On decay estimates, Journal of Evolution Equations, 9 (2009), 561-591. doi: 10.1007/s00028-009-0024-8.  Google Scholar

[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.  Google Scholar

[19]

M. M. Porzio, On uniform estimates,, 2013., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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