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

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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

Abstract
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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 and Continuous Dynamical Systems, 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), 1270-1294. doi: 10.1137/S0036141098341137.
[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), 119-148.
[3]	P. Baras and J. A. Goldstein, The heat equation with a singular potential, Trans. AMS, 284 (1984), 121-139. doi: 10.1090/S0002-9947-1984-0742415-3.
[4]	M. S. Baouendi, Sur une classe d'opérateurs elliptiques dégénérés, Bull. Soc. Math. France, 95 (1967), 45-87.
[5]	W. Beckner, On the Grushin operator and hyperbolic symmetry, Proc. Amer. Math. Soc., 129 (2001), 1233-1246. doi: 10.1090/S0002-9939-00-05630-6.
[6]	A. Bellaiche and J. J. Risler, "Sub-Riemannian Geometry," Birkhauser, Basel, 1996. doi: 10.1007/978-3-0348-9210-0.
[7]	T. Bieske, Viscosity solutions on Grushin-planes, Illinois J. Math, 46 (2002), 893-911.
[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), 973-978. doi: 10.1016/S0764-4442(00)88588-2.
[9]	Lorenzo D'Ambrosio, Hardy inequalities related to Grushin type operators, Proc. Amer. Math. Soc., 132 (2004), 725-734. doi: 10.1090/S0002-9939-03-07232-0.
[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), 691-716. doi: 10.1137/S0036141003432353.
[11]	J. Garcia Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Diff. Equations, 144 (1998), 441-476. doi: 10.1006/jdeq.1997.3375.
[12]	C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, in "Conference on Harmonic Analysis" ( eds. W. Beckner et al.), Wadsworth, (1981), 590-606.
[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), 281-300.
[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), 523-541.
[15]	B. Franchi, C. E. Gutiérrez and R. L. Wheeden, Weighted Sobolev-Poincare inequalities for Grushin type operators, Comm. Partial Differential Equations, 19 (1994), 523-604. doi: 10.1080/03605309408821025.
[16]	N. Garofalo, Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension, J. Diff. Equations, 104 (1993), 117-146. doi: 10.1006/jdeq.1993.1065.
[17]	N. Garofalo and Z. Shen, Absence of positive eigenvalues for a class of subelliptic operators, Math. Ann., 304 (1996), 701-715.
[18]	J. A. Goldstein, G. Ruiz Goldstein and I. Kombe, Nonlinear parabolic equations with singular coefficient and critical exponent, Applicable Analysis, 84 (2005), 571-583. doi: 10.1080/00036810500047709.
[19]	J. A. Goldstein and I. Kombe, Instantaneous blow up, Contemp. Math, 327 (2003), 141-149. doi: 10.1090/conm/327/05810.
[20]	J. A. Goldstein and I. Kombe, Nonlinear parabolic differentail equations with singular lower order term, Adv. Differential Equations, 10 (2003), 1153-1192.
[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), 1-22.
[22]	J. A. Goldstein and Q. S. Zhang, On a degenerate heat equation with a singular potential, J. Functional Analysis, 186 (2001), 342-359. doi: 10.1006/jfan.2001.3792.
[23]	J. A. Goldstein and Q. S. Zhang, Linear parabolic equations with strong singular potentials, Trans. AMS, 355 (2003), 197-211. doi: 10.1090/S0002-9947-02-03057-X.
[24]	V. Grushin, A certain class of hypoelliptic operators, Mat. Sb. (N.S), 83 (1970), 456-473.
[25]	V. Grushin, A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold, Mat. Sb., 84 (1971), 163-195.
[26]	A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations, Russian Math. Surveys, 42 (1987), 169-222.
[27]	J. L. Lions, "Quelque Methodes de Resolution des Problemes aux Limites Nonlineaire," Springer, Berlin.
[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), 1-17.
[29]	G. Savar and V. Vespri, The asymptotic profile of solutions of a class of doubly nonlinear equations, Nonlinear Anal., 22 (1994), 1553-1565. doi: 10.1016/0362-546X(94)90188-0.
[30]	I. Kombe, The linear heat equation with a highly oscillating potential, Proc. Amer. Math. Soc., 132 (2004), 2683-2691. doi: 10.1090/S0002-9939-04-07392-7.
[31]	I. Kombe, Doubly nonlinear parabolic equations with singular lower order term, Nonlinear Analysis, 56 (2004), 185-199. doi: 10.1016/j.na.2003.09.006.
[32]	I. Kombe, Nonlinear degenerate parabolic equations for Baouendi-Grushin operators, Mathematische Nachrichten, 279 (2006), 756-773. doi: 10.1002/mana.200310391.
[33]	I. Kombe, Cauchy-Dirichlet problem for the nonlinear degenerate parabolic equations, Abstract and Applied Analysis, 6 (2005), 607-617. doi: 10.1155/AAA.2005.607.
[34]	I. Kombe, Hardy and Rellich type inequalities with remainders for Baouendi-Grushin vector fields, to appear in Houston Journal of Mathematics.
[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), 139-152.
[36]	R. Monti and D. Morbidelli, Isoperimetric inequality in the Grushin plane, J. Geom. Anal., 14 (2004), 355-368. doi: 10.1007/BF02922077.
[37]	S. Nagel, E. M. Stein and S. Wainger, Balls and metrics defined by vector fields, I: Basic properties, Acta Math., 155 (1985), 103-147. doi: 10.1007/BF02392539.

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), 1270-1294. doi: 10.1137/S0036141098341137.
[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), 119-148.
[3]	P. Baras and J. A. Goldstein, The heat equation with a singular potential, Trans. AMS, 284 (1984), 121-139. doi: 10.1090/S0002-9947-1984-0742415-3.
[4]	M. S. Baouendi, Sur une classe d'opérateurs elliptiques dégénérés, Bull. Soc. Math. France, 95 (1967), 45-87.
[5]	W. Beckner, On the Grushin operator and hyperbolic symmetry, Proc. Amer. Math. Soc., 129 (2001), 1233-1246. doi: 10.1090/S0002-9939-00-05630-6.
[6]	A. Bellaiche and J. J. Risler, "Sub-Riemannian Geometry," Birkhauser, Basel, 1996. doi: 10.1007/978-3-0348-9210-0.
[7]	T. Bieske, Viscosity solutions on Grushin-planes, Illinois J. Math, 46 (2002), 893-911.
[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), 973-978. doi: 10.1016/S0764-4442(00)88588-2.
[9]	Lorenzo D'Ambrosio, Hardy inequalities related to Grushin type operators, Proc. Amer. Math. Soc., 132 (2004), 725-734. doi: 10.1090/S0002-9939-03-07232-0.
[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), 691-716. doi: 10.1137/S0036141003432353.
[11]	J. Garcia Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Diff. Equations, 144 (1998), 441-476. doi: 10.1006/jdeq.1997.3375.
[12]	C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, in "Conference on Harmonic Analysis" ( eds. W. Beckner et al.), Wadsworth, (1981), 590-606.
[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), 281-300.
[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), 523-541.
[15]	B. Franchi, C. E. Gutiérrez and R. L. Wheeden, Weighted Sobolev-Poincare inequalities for Grushin type operators, Comm. Partial Differential Equations, 19 (1994), 523-604. doi: 10.1080/03605309408821025.
[16]	N. Garofalo, Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension, J. Diff. Equations, 104 (1993), 117-146. doi: 10.1006/jdeq.1993.1065.
[17]	N. Garofalo and Z. Shen, Absence of positive eigenvalues for a class of subelliptic operators, Math. Ann., 304 (1996), 701-715.
[18]	J. A. Goldstein, G. Ruiz Goldstein and I. Kombe, Nonlinear parabolic equations with singular coefficient and critical exponent, Applicable Analysis, 84 (2005), 571-583. doi: 10.1080/00036810500047709.
[19]	J. A. Goldstein and I. Kombe, Instantaneous blow up, Contemp. Math, 327 (2003), 141-149. doi: 10.1090/conm/327/05810.
[20]	J. A. Goldstein and I. Kombe, Nonlinear parabolic differentail equations with singular lower order term, Adv. Differential Equations, 10 (2003), 1153-1192.
[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), 1-22.
[22]	J. A. Goldstein and Q. S. Zhang, On a degenerate heat equation with a singular potential, J. Functional Analysis, 186 (2001), 342-359. doi: 10.1006/jfan.2001.3792.
[23]	J. A. Goldstein and Q. S. Zhang, Linear parabolic equations with strong singular potentials, Trans. AMS, 355 (2003), 197-211. doi: 10.1090/S0002-9947-02-03057-X.
[24]	V. Grushin, A certain class of hypoelliptic operators, Mat. Sb. (N.S), 83 (1970), 456-473.
[25]	V. Grushin, A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold, Mat. Sb., 84 (1971), 163-195.
[26]	A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations, Russian Math. Surveys, 42 (1987), 169-222.
[27]	J. L. Lions, "Quelque Methodes de Resolution des Problemes aux Limites Nonlineaire," Springer, Berlin.
[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), 1-17.
[29]	G. Savar and V. Vespri, The asymptotic profile of solutions of a class of doubly nonlinear equations, Nonlinear Anal., 22 (1994), 1553-1565. doi: 10.1016/0362-546X(94)90188-0.
[30]	I. Kombe, The linear heat equation with a highly oscillating potential, Proc. Amer. Math. Soc., 132 (2004), 2683-2691. doi: 10.1090/S0002-9939-04-07392-7.
[31]	I. Kombe, Doubly nonlinear parabolic equations with singular lower order term, Nonlinear Analysis, 56 (2004), 185-199. doi: 10.1016/j.na.2003.09.006.
[32]	I. Kombe, Nonlinear degenerate parabolic equations for Baouendi-Grushin operators, Mathematische Nachrichten, 279 (2006), 756-773. doi: 10.1002/mana.200310391.
[33]	I. Kombe, Cauchy-Dirichlet problem for the nonlinear degenerate parabolic equations, Abstract and Applied Analysis, 6 (2005), 607-617. doi: 10.1155/AAA.2005.607.
[34]	I. Kombe, Hardy and Rellich type inequalities with remainders for Baouendi-Grushin vector fields, to appear in Houston Journal of Mathematics.
[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), 139-152.
[36]	R. Monti and D. Morbidelli, Isoperimetric inequality in the Grushin plane, J. Geom. Anal., 14 (2004), 355-368. doi: 10.1007/BF02922077.
[37]	S. Nagel, E. M. Stein and S. Wainger, Balls and metrics defined by vector fields, I: Basic properties, Acta Math., 155 (1985), 103-147. doi: 10.1007/BF02392539.

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