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November  2019, 18(6): 2923-2960. doi: 10.3934/cpaa.2019131

Comparison results for unbounded solutions for a parabolic Cauchy-Dirichlet problem with superlinear gradient growth

Tommaso Leonori 1, and  Martina Magliocca 2,

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

Received  March 2018 Revised  March 2018 Published  May 2019

In this paper we deal with uniqueness of unbounded solutions to the following problem
$\begin{equation*}\begin{cases}\begin{array}{ll} u_t- \Delta_p u = H (t,x,\nabla u) &\text{in}\quad \ Q_T,\\\displaystyle u (t,x) = 0 & \text{on}\quad(0,T)\times \partial \Omega,\\ \displaystyle u(0,x) = u_0(x) &\text{in }\quad \Omega,\end{array}\end{cases}\end{equation*}$
where
$Q_T = (0, T)\times \Omega$
 is the parabolic cylinder,
$\Omega$
 is an open subset of
$\mathbb{R}^N$
 ,
$N\ge2$
 ,
$1 < p < N$
 , and the right hand side
$\displaystyle H(t, x, \xi):(0, T)\times\Omega \times \mathbb{R}^N\to \mathbb{R}$
 exhibits a superlinear growth with respect to the gradient term.
Keywords: Uniqueness, nonlinear parabolic equations, unbounded solutions, nonlinear lower order terms.
Mathematics Subject Classification: 35B51, 35K55, 35K61.
Citation: Tommaso Leonori, Martina Magliocca. Comparison results for unbounded solutions for a parabolic Cauchy-Dirichlet problem with superlinear gradient growth. Communications on Pure & Applied Analysis, 2019, 18 (6) : 2923-2960. doi: 10.3934/cpaa.2019131
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

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