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November  2013, 33(11&12): 5167-5176. doi: 10.3934/dcds.2013.33.5167

On the nonexistence of positive solutions to doubly nonlinear equations for Baouendi-Grushin operators

Ismail Kombe 1,

1. 

Department of Mathematics, Faculty of Science and Letters, Istanbul Commerce University, Uskudar, Istanbul, Turkey

Received  January 2012 Published  May 2013

The purpose of this paper is to study the nonexistence of positive solutions of the doubly nonlinear equation \[\begin{cases} \frac{\partial u}{\partial t}=\nabla_{\gamma}\cdot (u^{m-1}|\nabla_{\gamma} u|^{p-2}\nabla_{\gamma} u) +Vu^{m+p-2} & \text{in}\quad \Omega \times (0, T ) ,\\ u(x,0)=u_{0}(x)\geq 0 & \text{in} \quad\Omega, \\ u(x,t)=0 & \text{on}\quad \partial\Omega\times (0, T),\end{cases}\] where $\nabla_{\gamma}=(\nabla_x, |x|^{2\gamma}\nabla_y)$, $x\in \mathbb{R}^d, y\in \mathbb{R}^k$, $\gamma>0$, $\Omega$ is a metric ball in $\mathbb{R}^{N}$, $V\in L_{\text{loc}}^1(\Omega)$, $m\in \mathbb{R}$, $1 < p < d+k$ and $m + p - 2 > 0$. The exponents $q^{*}$ are found and the nonexistence results are proved for $q^{*} ≤ m+p < 3$.
Keywords: positive solutions, Baouendi-Grushin vector fields, nonlinear parabolic equations, Hardy inequality..
Mathematics Subject Classification: Primary: 35K55, 35K65; Secondary: 26D1.
Citation: Ismail Kombe. On the nonexistence of positive solutions to doubly nonlinear equations for Baouendi-Grushin operators. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5167-5176. doi: 10.3934/dcds.2013.33.5167
References:
[1]

J. A. Aguilar Crespo and I. Peral Alonso, Global behaviour of the Cauchy problem for some critical nonlinear parabolic equations,, SIAM J. Math. Anal., 31 (2000), 1270.  doi: 10.1137/S0036141098341137.  Google Scholar

[2]

B. Abdellaoui, Eduardo Colorado and I. Peral, Existence and nonexistence results for a class of linear and semilinear parabolic equations related to some Caffarelli-Kohn-Nirenberg inequalities,, Journal of the European Mathematical Society, 6 (2004), 119.   Google Scholar

[3]

P. Baras and J. A. Goldstein, The heat equation with a singular potential,, Trans. AMS, 284 (1984), 121.  doi: 10.1090/S0002-9947-1984-0742415-3.  Google Scholar

[4]

M. S. Baouendi, Sur une classe d'opérateurs elliptiques dégénérés,, Bull. Soc. Math. France, 95 (1967), 45.   Google Scholar

[5]

W. Beckner, On the Grushin operator and hyperbolic symmetry,, Proc. Amer. Math. Soc., 129 (2001), 1233.  doi: 10.1090/S0002-9939-00-05630-6.  Google Scholar

[6]

A. Bellaiche and J. J. Risler, "Sub-Riemannian Geometry,", Birkhauser, (1996).  doi: 10.1007/978-3-0348-9210-0.  Google Scholar

[7]

T. Bieske, Viscosity solutions on Grushin-planes,, Illinois J. Math, 46 (2002), 893.   Google Scholar

[8]

X. Cabré and Y. Martel, Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier,, C. R. Acad. Sci. Paris, 329 (1999), 973.  doi: 10.1016/S0764-4442(00)88588-2.  Google Scholar

[9]

Lorenzo D'Ambrosio, Hardy inequalities related to Grushin type operators,, Proc. Amer. Math. Soc., 132 (2004), 725.  doi: 10.1090/S0002-9939-03-07232-0.  Google Scholar

[10]

A. Dall' Aglio, D. Giachetti and I. Peral, Results on Parabolic Equations Related to some Caffarelli-Kohn-Nirenberg inequalities,, SIAM J. Math. Anal., 36 (2004), 691.  doi: 10.1137/S0036141003432353.  Google Scholar

[11]

J. Garcia Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems,, J. Diff. Equations, 144 (1998), 441.  doi: 10.1006/jdeq.1997.3375.  Google Scholar

[12]

C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems,, in, (1981), 590.   Google Scholar

[13]

F. Ferrari and B. Franchi, Geometry of the boundary and doubling property of the harmonic measure for Grushin type operators,, Rend. Sem. Univ. e Politec. Torino, 58 (2000), 281.   Google Scholar

[14]

B. Franchi and E. Lanconelli, Hölder regularity theorem for a class of linear nonlinear uniformly elliptic operators with measurable coefficients,, Ann. Scuola Norm. Sup. Pisa (4), 10 (1983), 523.   Google Scholar

[15]

B. Franchi, C. E. Gutiérrez and R. L. Wheeden, Weighted Sobolev-Poincare inequalities for Grushin type operators,, Comm. Partial Differential Equations, 19 (1994), 523.  doi: 10.1080/03605309408821025.  Google Scholar

[16]

N. Garofalo, Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension,, J. Diff. Equations, 104 (1993), 117.  doi: 10.1006/jdeq.1993.1065.  Google Scholar

[17]

N. Garofalo and Z. Shen, Absence of positive eigenvalues for a class of subelliptic operators,, Math. Ann., 304 (1996), 701.   Google Scholar

[18]

J. A. Goldstein, G. Ruiz Goldstein and I. Kombe, Nonlinear parabolic equations with singular coefficient and critical exponent,, Applicable Analysis, 84 (2005), 571.  doi: 10.1080/00036810500047709.  Google Scholar

[19]

J. A. Goldstein and I. Kombe, Instantaneous blow up,, Contemp. Math, 327 (2003), 141.  doi: 10.1090/conm/327/05810.  Google Scholar

[20]

J. A. Goldstein and I. Kombe, Nonlinear parabolic differentail equations with singular lower order term,, Adv. Differential Equations, 10 (2003), 1153.   Google Scholar

[21]

J. A. Goldstein and I. Kombe, Nonlinear degenerate differential equations with singular lower order term on the Heisenberg group,, International Journal of Evolution Equations, 1 (2005), 1.   Google Scholar

[22]

J. A. Goldstein and Q. S. Zhang, On a degenerate heat equation with a singular potential,, J. Functional Analysis, 186 (2001), 342.  doi: 10.1006/jfan.2001.3792.  Google Scholar

[23]

J. A. Goldstein and Q. S. Zhang, Linear parabolic equations with strong singular potentials,, Trans. AMS, 355 (2003), 197.  doi: 10.1090/S0002-9947-02-03057-X.  Google Scholar

[24]

V. Grushin, A certain class of hypoelliptic operators,, Mat. Sb. (N.S), 83 (1970), 456.   Google Scholar

[25]

V. Grushin, A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold,, Mat. Sb., 84 (1971), 163.   Google Scholar

[26]

A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations,, Russian Math. Surveys, 42 (1987), 169.   Google Scholar

[27]

J. L. Lions, "Quelque Methodes de Resolution des Problemes aux Limites Nonlineaire,", Springer, ().   Google Scholar

[28]

J. Juan Manfredi and V. Vespri, Large time behavior of solutions to a class of doubly nonlinear parabolic equations,, Electron. J. Differential Equations, (1994), 1.   Google Scholar

[29]

G. Savar and V. Vespri, The asymptotic profile of solutions of a class of doubly nonlinear equations,, Nonlinear Anal., 22 (1994), 1553.  doi: 10.1016/0362-546X(94)90188-0.  Google Scholar

[30]

I. Kombe, The linear heat equation with a highly oscillating potential,, Proc. Amer. Math. Soc., 132 (2004), 2683.  doi: 10.1090/S0002-9939-04-07392-7.  Google Scholar

[31]

I. Kombe, Doubly nonlinear parabolic equations with singular lower order term,, Nonlinear Analysis, 56 (2004), 185.  doi: 10.1016/j.na.2003.09.006.  Google Scholar

[32]

I. Kombe, Nonlinear degenerate parabolic equations for Baouendi-Grushin operators,, Mathematische Nachrichten, 279 (2006), 756.  doi: 10.1002/mana.200310391.  Google Scholar

[33]

I. Kombe, Cauchy-Dirichlet problem for the nonlinear degenerate parabolic equations,, Abstract and Applied Analysis, 6 (2005), 607.  doi: 10.1155/AAA.2005.607.  Google Scholar

[34]

I. Kombe, Hardy and Rellich type inequalities with remainders for Baouendi-Grushin vector fields,, to appear in Houston Journal of Mathematics., ().   Google Scholar

[35]

F. Lascialfari and D. Pardo, Compact embedding of a degenerate Sobolev space and existence of entire solutions to semilinear equation for a Grushin-type operator,, Rend. Sem. Mat. Univ. Padova, 107 (2002), 139.   Google Scholar

[36]

R. Monti and D. Morbidelli, Isoperimetric inequality in the Grushin plane,, J. Geom. Anal., 14 (2004), 355.  doi: 10.1007/BF02922077.  Google Scholar

[37]

S. Nagel, E. M. Stein and S. Wainger, Balls and metrics defined by vector fields, I: Basic properties,, Acta Math., 155 (1985), 103.  doi: 10.1007/BF02392539.  Google Scholar

show all references

References:
[1]

J. A. Aguilar Crespo and I. Peral Alonso, Global behaviour of the Cauchy problem for some critical nonlinear parabolic equations,, SIAM J. Math. Anal., 31 (2000), 1270.  doi: 10.1137/S0036141098341137.  Google Scholar

[2]

B. Abdellaoui, Eduardo Colorado and I. Peral, Existence and nonexistence results for a class of linear and semilinear parabolic equations related to some Caffarelli-Kohn-Nirenberg inequalities,, Journal of the European Mathematical Society, 6 (2004), 119.   Google Scholar

[3]

P. Baras and J. A. Goldstein, The heat equation with a singular potential,, Trans. AMS, 284 (1984), 121.  doi: 10.1090/S0002-9947-1984-0742415-3.  Google Scholar

[4]

M. S. Baouendi, Sur une classe d'opérateurs elliptiques dégénérés,, Bull. Soc. Math. France, 95 (1967), 45.   Google Scholar

[5]

W. Beckner, On the Grushin operator and hyperbolic symmetry,, Proc. Amer. Math. Soc., 129 (2001), 1233.  doi: 10.1090/S0002-9939-00-05630-6.  Google Scholar

[6]

A. Bellaiche and J. J. Risler, "Sub-Riemannian Geometry,", Birkhauser, (1996).  doi: 10.1007/978-3-0348-9210-0.  Google Scholar

[7]

T. Bieske, Viscosity solutions on Grushin-planes,, Illinois J. Math, 46 (2002), 893.   Google Scholar

[8]

X. Cabré and Y. Martel, Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier,, C. R. Acad. Sci. Paris, 329 (1999), 973.  doi: 10.1016/S0764-4442(00)88588-2.  Google Scholar

[9]

Lorenzo D'Ambrosio, Hardy inequalities related to Grushin type operators,, Proc. Amer. Math. Soc., 132 (2004), 725.  doi: 10.1090/S0002-9939-03-07232-0.  Google Scholar

[10]

A. Dall' Aglio, D. Giachetti and I. Peral, Results on Parabolic Equations Related to some Caffarelli-Kohn-Nirenberg inequalities,, SIAM J. Math. Anal., 36 (2004), 691.  doi: 10.1137/S0036141003432353.  Google Scholar

[11]

J. Garcia Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems,, J. Diff. Equations, 144 (1998), 441.  doi: 10.1006/jdeq.1997.3375.  Google Scholar

[12]

C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems,, in, (1981), 590.   Google Scholar

[13]

F. Ferrari and B. Franchi, Geometry of the boundary and doubling property of the harmonic measure for Grushin type operators,, Rend. Sem. Univ. e Politec. Torino, 58 (2000), 281.   Google Scholar

[14]

B. Franchi and E. Lanconelli, Hölder regularity theorem for a class of linear nonlinear uniformly elliptic operators with measurable coefficients,, Ann. Scuola Norm. Sup. Pisa (4), 10 (1983), 523.   Google Scholar

[15]

B. Franchi, C. E. Gutiérrez and R. L. Wheeden, Weighted Sobolev-Poincare inequalities for Grushin type operators,, Comm. Partial Differential Equations, 19 (1994), 523.  doi: 10.1080/03605309408821025.  Google Scholar

[16]

N. Garofalo, Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension,, J. Diff. Equations, 104 (1993), 117.  doi: 10.1006/jdeq.1993.1065.  Google Scholar

[17]

N. Garofalo and Z. Shen, Absence of positive eigenvalues for a class of subelliptic operators,, Math. Ann., 304 (1996), 701.   Google Scholar

[18]

J. A. Goldstein, G. Ruiz Goldstein and I. Kombe, Nonlinear parabolic equations with singular coefficient and critical exponent,, Applicable Analysis, 84 (2005), 571.  doi: 10.1080/00036810500047709.  Google Scholar

[19]

J. A. Goldstein and I. Kombe, Instantaneous blow up,, Contemp. Math, 327 (2003), 141.  doi: 10.1090/conm/327/05810.  Google Scholar

[20]

J. A. Goldstein and I. Kombe, Nonlinear parabolic differentail equations with singular lower order term,, Adv. Differential Equations, 10 (2003), 1153.   Google Scholar

[21]

J. A. Goldstein and I. Kombe, Nonlinear degenerate differential equations with singular lower order term on the Heisenberg group,, International Journal of Evolution Equations, 1 (2005), 1.   Google Scholar

[22]

J. A. Goldstein and Q. S. Zhang, On a degenerate heat equation with a singular potential,, J. Functional Analysis, 186 (2001), 342.  doi: 10.1006/jfan.2001.3792.  Google Scholar

[23]

J. A. Goldstein and Q. S. Zhang, Linear parabolic equations with strong singular potentials,, Trans. AMS, 355 (2003), 197.  doi: 10.1090/S0002-9947-02-03057-X.  Google Scholar

[24]

V. Grushin, A certain class of hypoelliptic operators,, Mat. Sb. (N.S), 83 (1970), 456.   Google Scholar

[25]

V. Grushin, A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold,, Mat. Sb., 84 (1971), 163.   Google Scholar

[26]

A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations,, Russian Math. Surveys, 42 (1987), 169.   Google Scholar

[27]

J. L. Lions, "Quelque Methodes de Resolution des Problemes aux Limites Nonlineaire,", Springer, ().   Google Scholar

[28]

J. Juan Manfredi and V. Vespri, Large time behavior of solutions to a class of doubly nonlinear parabolic equations,, Electron. J. Differential Equations, (1994), 1.   Google Scholar

[29]

G. Savar and V. Vespri, The asymptotic profile of solutions of a class of doubly nonlinear equations,, Nonlinear Anal., 22 (1994), 1553.  doi: 10.1016/0362-546X(94)90188-0.  Google Scholar

[30]

I. Kombe, The linear heat equation with a highly oscillating potential,, Proc. Amer. Math. Soc., 132 (2004), 2683.  doi: 10.1090/S0002-9939-04-07392-7.  Google Scholar

[31]

I. Kombe, Doubly nonlinear parabolic equations with singular lower order term,, Nonlinear Analysis, 56 (2004), 185.  doi: 10.1016/j.na.2003.09.006.  Google Scholar

[32]

I. Kombe, Nonlinear degenerate parabolic equations for Baouendi-Grushin operators,, Mathematische Nachrichten, 279 (2006), 756.  doi: 10.1002/mana.200310391.  Google Scholar

[33]

I. Kombe, Cauchy-Dirichlet problem for the nonlinear degenerate parabolic equations,, Abstract and Applied Analysis, 6 (2005), 607.  doi: 10.1155/AAA.2005.607.  Google Scholar

[34]

I. Kombe, Hardy and Rellich type inequalities with remainders for Baouendi-Grushin vector fields,, to appear in Houston Journal of Mathematics., ().   Google Scholar

[35]

F. Lascialfari and D. Pardo, Compact embedding of a degenerate Sobolev space and existence of entire solutions to semilinear equation for a Grushin-type operator,, Rend. Sem. Mat. Univ. Padova, 107 (2002), 139.   Google Scholar

[36]

R. Monti and D. Morbidelli, Isoperimetric inequality in the Grushin plane,, J. Geom. Anal., 14 (2004), 355.  doi: 10.1007/BF02922077.  Google Scholar

[37]

S. Nagel, E. M. Stein and S. Wainger, Balls and metrics defined by vector fields, I: Basic properties,, Acta Math., 155 (1985), 103.  doi: 10.1007/BF02392539.  Google Scholar

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