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Comparison results for unbounded solutions for a parabolic Cauchy-Dirichlet problem with superlinear gradient growth
1. | Dipartimento di Scienze di Base e Applicate per l'Ingegneria, ''Sapienza'' Università di Roma I, Via Antonio Scarpa 10, 00161, Rome, Italy |
2. | Dipartimento di Matematica, Università degli Studi di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy |
$\begin{equation*}\begin{cases}\begin{array}{ll} u_t- \Delta_p u = H (t,x,\nabla u) &\text{in}\quad \ Q_T,\\\displaystyle u (t,x) = 0 & \text{on}\quad(0,T)\times \partial \Omega,\\ \displaystyle u(0,x) = u_0(x) &\text{in }\quad \Omega,\end{array}\end{cases}\end{equation*}$ |
$Q_T = (0, T)\times \Omega$ |
$\Omega$ |
$\mathbb{R}^N$ |
$N\ge2$ |
$1 < p < N$ |
$\displaystyle H(t, x, \xi):(0, T)\times\Omega \times \mathbb{R}^N\to \mathbb{R}$ |
References:
[1] |
B. Abdellaoui, A. Dall'Aglio and I. Peral,
Some remarks on elliptic problems with critical growth in the gradient, J. Differential Equations, 222 (2006), 21-62.
doi: 10.1016/j.jde.2005.02.009. |
[2] |
B. Abdellaoui, A. Dall'Aglio and I. Peral,
Regularity and nonuniqueness results for parabolic problems arising in some physical models, having natural growth in the gradient, J. Math. Pures Appl., 90 (2008), 242-269.
doi: 10.1016/j.matpur.2008.04.004. |
[3] |
B. Alvino, M. F. Betta and A. Mercaldo,
Comparison principle for some classes of nonlinear elliptic equations, J. Diff. Eq., 249 (2010), 3279-3290.
doi: 10.1016/j.jde.2010.07.030. |
[4] |
G. Barles and F. Murat,
Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions, Arch. Rational Mech. Anal., 133 (1995), 77-101.
doi: 10.1007/BF00375351. |
[5] |
G. Barles and F. Da Lio,
On the generalized Dirichlet problem for viscous Hamilton-Jacobi equations, J. Math. Pures Appl., 83 (2004), 53-75.
doi: 10.1016/S0021-7824(03)00070-9. |
[6] |
G. Barles and A. Porretta,
Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations, Ann. Scuola Norm. Sup. di Pisa Cl. Sci., 5 (2006), 107-136.
|
[7] |
M. Ben-Artzi, P. Souplet and F. Weissler,
The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces, J. Math. Pures Appl., 81 (2002), 343-378.
doi: 10.1016/S0021-7824(01)01243-0. |
[8] |
M. Ben-Artzi, P. Souplet and F. Weissler,
Sur la non-existence et la non-unicité des solutions du problème de Cauchy pour une équation parabolique semi-linéaire, Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 329 (1999), 371-376.
doi: 10.1016/S0764-4442(00)88608-5. |
[9] |
P. Benilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez,
An L1 theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa, 22 (1995), 241-273.
|
[10] |
M. F. Betta, R. Di Nardo, T. Mercaldo, A. Gariepy and A. Perrotta,
Gradient estimates and comparison principle for some nonlinear elliptic equations, Commun. Pure Appl. Anal., 14 (2015), 897-922.
doi: 10.3934/cpaa.2015.14.897. |
[11] |
F. Betta, A. Mercaldo, F. Murat and M. Porzio,
Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in $L^1(\Omega)$
, a tribute to J. L. Lions, ESAIM Control Optim. Calc. Var., 8 (2002), 239-272.
doi: 10.1051/cocv:2002051. |
[12] |
F. Betta, A. Mercaldo, F. Murat and M. Porzio,
Uniqueness results for nonlinear elliptic equations with a lower order term, Nonlin. Anal., 63 (2005), 153-170.
doi: 10.1016/j.na.2005.03.097. |
[13] |
D. Blanchard and F. Murat,
Renormalised solutions of nonlinear parabolic problems with $L^1$
data: existence and uniqueness, Proceedings of the Royal Society of Edinburgh, 127 (1997), 1137-1152.
doi: 10.1017/S0308210500026986. |
[14] |
D. Blanchard and A. Porretta,
Stefan problems with nonlinear diffusion and convection, J. Diff. Eq., 210 (2005), 383-428.
doi: 10.1016/j.jde.2004.06.012. |
[15] |
L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina,
Nonlinear Parabolic Equations with Measure Data, J. Func. An., 147 (1997), 237-258.
doi: 10.1006/jfan.1996.3040. |
[16] |
M. Crandall, H. Ishii and P. L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992).
doi: 10.1090/S0273-0979-1992-00266-5. |
[17] |
R. Di Nardo, F. Feo and O. Guibé,
Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 1185-1208.
doi: 10.1017/S0308210511001831. |
[18] |
F. Feo,
A remark on uniqueness of weak solutions for some classes of parabolic problems, Ric. Mat., 63 (2014), S143-S155.
doi: 10.1007/s11587-014-0210-z. |
[19] |
N. Grenon, F. Murat and A. Porretta, Existence and a priori estimate for elliptic problems with subquadratic gradient dependent terms, C. R. Acad. Sci. Paris, Ser. I, 342 (2006), 23–28.,
doi: 10.1016/j.crma.2005.09.027. |
[20] |
N. Grenon, F. Murat and A. Porretta,
A priori estimates and existence for elliptic equations with gradient dependent terms, Ann. Sc. Norm. Super. Pisa Cl. Sci., 13 (2014), 137-205.
|
[21] |
T. Leonori and A. Porretta,
On the comparison principle for unbounded solutions of elliptic equations with first order terms, J. of Math. Anal. Appl., 457 (2018), 1492-1501.
doi: 10.1016/j.jmaa.2017.04.018. |
[22] |
T. Leonori, A. Porretta and G. Riey,
Comparison principles for p-Laplace equations with lower order terms, Ann. Mat. Pura Appl., 196 (2017), 877-903.
doi: 10.1007/s10231-016-0600-9. |
[23] |
M. Magliocca,
Existence results for a Cauchy-Dirichlet parabolic problem with a repulsive gradient term, Nonlin. Anal., 166 (2018), 102-143.
doi: 10.1016/j.na.2017.09.012. |
[24] |
A. Mercaldo, A priori estimates and comparison principle for some nonlinear elliptic equations, in Geometric Properties for Parabolic and Elliptic PDE's (R. Magnanini, S. Sakaguchi and A. Alvino eds). Springer INdAM Series, 2 (2013) Springer, Milano.,
doi: 10.1007/978-88-470-2841-8_14. |
[25] |
F. Petitta,
Renormalized solutions of nonlinear parabolic equations with general measure data, Ann. Mat. Pura Appl., 187 (2008), 563-604.
doi: 10.1007/s10231-007-0057-y. |
[26] |
F. Petitta, A. Ponce and A. Porretta,
Diffuse measures and nonlinear parabolic equations, J. Evol. Eq., 11 (2011), 861-905.
doi: 10.1007/s00028-011-0115-1. |
[27] |
A. Porretta,
Existence results for nonlinear parabolic equations via strong convergence of truncations, Ann. Mat. Pura e Appl., 177 (1999), 143-172.
doi: 10.1007/BF02505907. |
[28] |
A. Porretta, On the comparison principle for p-Laplace type operators with first order terms, in " On the notions of solution to nonlinear elliptic problems: results and developments", 459–497, Quad. Mat. 23, Dept. Math., Seconda Univ. Napoli, Caserta (2008). |
[29] |
G. Stampacchia, Le probléme de Dirichlet pour les équations elliptiques du second ordre á coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189–258. |
show all references
References:
[1] |
B. Abdellaoui, A. Dall'Aglio and I. Peral,
Some remarks on elliptic problems with critical growth in the gradient, J. Differential Equations, 222 (2006), 21-62.
doi: 10.1016/j.jde.2005.02.009. |
[2] |
B. Abdellaoui, A. Dall'Aglio and I. Peral,
Regularity and nonuniqueness results for parabolic problems arising in some physical models, having natural growth in the gradient, J. Math. Pures Appl., 90 (2008), 242-269.
doi: 10.1016/j.matpur.2008.04.004. |
[3] |
B. Alvino, M. F. Betta and A. Mercaldo,
Comparison principle for some classes of nonlinear elliptic equations, J. Diff. Eq., 249 (2010), 3279-3290.
doi: 10.1016/j.jde.2010.07.030. |
[4] |
G. Barles and F. Murat,
Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions, Arch. Rational Mech. Anal., 133 (1995), 77-101.
doi: 10.1007/BF00375351. |
[5] |
G. Barles and F. Da Lio,
On the generalized Dirichlet problem for viscous Hamilton-Jacobi equations, J. Math. Pures Appl., 83 (2004), 53-75.
doi: 10.1016/S0021-7824(03)00070-9. |
[6] |
G. Barles and A. Porretta,
Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations, Ann. Scuola Norm. Sup. di Pisa Cl. Sci., 5 (2006), 107-136.
|
[7] |
M. Ben-Artzi, P. Souplet and F. Weissler,
The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces, J. Math. Pures Appl., 81 (2002), 343-378.
doi: 10.1016/S0021-7824(01)01243-0. |
[8] |
M. Ben-Artzi, P. Souplet and F. Weissler,
Sur la non-existence et la non-unicité des solutions du problème de Cauchy pour une équation parabolique semi-linéaire, Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 329 (1999), 371-376.
doi: 10.1016/S0764-4442(00)88608-5. |
[9] |
P. Benilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez,
An L1 theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa, 22 (1995), 241-273.
|
[10] |
M. F. Betta, R. Di Nardo, T. Mercaldo, A. Gariepy and A. Perrotta,
Gradient estimates and comparison principle for some nonlinear elliptic equations, Commun. Pure Appl. Anal., 14 (2015), 897-922.
doi: 10.3934/cpaa.2015.14.897. |
[11] |
F. Betta, A. Mercaldo, F. Murat and M. Porzio,
Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in $L^1(\Omega)$
, a tribute to J. L. Lions, ESAIM Control Optim. Calc. Var., 8 (2002), 239-272.
doi: 10.1051/cocv:2002051. |
[12] |
F. Betta, A. Mercaldo, F. Murat and M. Porzio,
Uniqueness results for nonlinear elliptic equations with a lower order term, Nonlin. Anal., 63 (2005), 153-170.
doi: 10.1016/j.na.2005.03.097. |
[13] |
D. Blanchard and F. Murat,
Renormalised solutions of nonlinear parabolic problems with $L^1$
data: existence and uniqueness, Proceedings of the Royal Society of Edinburgh, 127 (1997), 1137-1152.
doi: 10.1017/S0308210500026986. |
[14] |
D. Blanchard and A. Porretta,
Stefan problems with nonlinear diffusion and convection, J. Diff. Eq., 210 (2005), 383-428.
doi: 10.1016/j.jde.2004.06.012. |
[15] |
L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina,
Nonlinear Parabolic Equations with Measure Data, J. Func. An., 147 (1997), 237-258.
doi: 10.1006/jfan.1996.3040. |
[16] |
M. Crandall, H. Ishii and P. L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992).
doi: 10.1090/S0273-0979-1992-00266-5. |
[17] |
R. Di Nardo, F. Feo and O. Guibé,
Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 1185-1208.
doi: 10.1017/S0308210511001831. |
[18] |
F. Feo,
A remark on uniqueness of weak solutions for some classes of parabolic problems, Ric. Mat., 63 (2014), S143-S155.
doi: 10.1007/s11587-014-0210-z. |
[19] |
N. Grenon, F. Murat and A. Porretta, Existence and a priori estimate for elliptic problems with subquadratic gradient dependent terms, C. R. Acad. Sci. Paris, Ser. I, 342 (2006), 23–28.,
doi: 10.1016/j.crma.2005.09.027. |
[20] |
N. Grenon, F. Murat and A. Porretta,
A priori estimates and existence for elliptic equations with gradient dependent terms, Ann. Sc. Norm. Super. Pisa Cl. Sci., 13 (2014), 137-205.
|
[21] |
T. Leonori and A. Porretta,
On the comparison principle for unbounded solutions of elliptic equations with first order terms, J. of Math. Anal. Appl., 457 (2018), 1492-1501.
doi: 10.1016/j.jmaa.2017.04.018. |
[22] |
T. Leonori, A. Porretta and G. Riey,
Comparison principles for p-Laplace equations with lower order terms, Ann. Mat. Pura Appl., 196 (2017), 877-903.
doi: 10.1007/s10231-016-0600-9. |
[23] |
M. Magliocca,
Existence results for a Cauchy-Dirichlet parabolic problem with a repulsive gradient term, Nonlin. Anal., 166 (2018), 102-143.
doi: 10.1016/j.na.2017.09.012. |
[24] |
A. Mercaldo, A priori estimates and comparison principle for some nonlinear elliptic equations, in Geometric Properties for Parabolic and Elliptic PDE's (R. Magnanini, S. Sakaguchi and A. Alvino eds). Springer INdAM Series, 2 (2013) Springer, Milano.,
doi: 10.1007/978-88-470-2841-8_14. |
[25] |
F. Petitta,
Renormalized solutions of nonlinear parabolic equations with general measure data, Ann. Mat. Pura Appl., 187 (2008), 563-604.
doi: 10.1007/s10231-007-0057-y. |
[26] |
F. Petitta, A. Ponce and A. Porretta,
Diffuse measures and nonlinear parabolic equations, J. Evol. Eq., 11 (2011), 861-905.
doi: 10.1007/s00028-011-0115-1. |
[27] |
A. Porretta,
Existence results for nonlinear parabolic equations via strong convergence of truncations, Ann. Mat. Pura e Appl., 177 (1999), 143-172.
doi: 10.1007/BF02505907. |
[28] |
A. Porretta, On the comparison principle for p-Laplace type operators with first order terms, in " On the notions of solution to nonlinear elliptic problems: results and developments", 459–497, Quad. Mat. 23, Dept. Math., Seconda Univ. Napoli, Caserta (2008). |
[29] |
G. Stampacchia, Le probléme de Dirichlet pour les équations elliptiques du second ordre á coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189–258. |
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