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

Some degenerate parabolic problems: Existence and decay properties

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.
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.

[2]

L. Boccardo, G. Croce and L. Orsina, Nonlinear degenerate elliptic problems with $W_0^{1,1}$ solutions,, Manuscripta Mathematica, 137 (2012), 419. 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.

[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. 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.

[6]

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

[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H. Brezis and T. Cazenave, Notes,, unpublished., ().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. 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. 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. 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. 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. 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. 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, (1968).

[15]

J. L. Lions, Quelques Méthodes de Resolution des Problèmes aux Limites non Linéaires,, Dunod, (1969).

[16]

A. Porretta, Uniqueness and homogenization for a class of noncoercive operators in divergence form,, dedicated to Prof. C. Vinti (Perugia, 46 (1998), 915.

[17]

M. M. Porzio, On decay estimates,, Journal of Evolution Equations, 9 (2009), 561. 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. 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. 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. 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). 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). 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. doi: 10.1007/BF01762360.

[25]

L. Veron, Effects regularisants des semi-groupes non linéaires dans des espaces de Banach,, Ann. Fac. Sci., 1 (1979), 171. 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.

[2]

L. Boccardo, G. Croce and L. Orsina, Nonlinear degenerate elliptic problems with $W_0^{1,1}$ solutions,, Manuscripta Mathematica, 137 (2012), 419. 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.

[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. 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.

[6]

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

[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. 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. 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. 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. 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. 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. 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, (1968).

[15]

J. L. Lions, Quelques Méthodes de Resolution des Problèmes aux Limites non Linéaires,, Dunod, (1969).

[16]

A. Porretta, Uniqueness and homogenization for a class of noncoercive operators in divergence form,, dedicated to Prof. C. Vinti (Perugia, 46 (1998), 915.

[17]

M. M. Porzio, On decay estimates,, Journal of Evolution Equations, 9 (2009), 561. 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. 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. 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. 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). 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). 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. doi: 10.1007/BF01762360.

[25]

L. Veron, Effects regularisants des semi-groupes non linéaires dans des espaces de Banach,, Ann. Fac. Sci., 1 (1979), 171. doi: 10.5802/afst.535.

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