American Institute of Mathematical Sciences

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

 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 $q^{*} ≤ m+p < 3$.
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. [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. [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. [4] M. S. Baouendi, Sur une classe d'opérateurs elliptiques dégénérés,, Bull. Soc. Math. France, 95 (1967), 45. [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. [6] A. Bellaiche and J. J. Risler, "Sub-Riemannian Geometry,", Birkhauser, (1996). doi: 10.1007/978-3-0348-9210-0. [7] T. Bieske, Viscosity solutions on Grushin-planes,, Illinois J. Math, 46 (2002), 893. [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. [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. [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. [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. [12] C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems,, in, (1981), 590. [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. [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. [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. [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. [17] N. Garofalo and Z. Shen, Absence of positive eigenvalues for a class of subelliptic operators,, Math. Ann., 304 (1996), 701. [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. [19] J. A. Goldstein and I. Kombe, Instantaneous blow up,, Contemp. Math, 327 (2003), 141. 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. [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] 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. [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. [24] V. Grushin, A certain class of hypoelliptic operators,, Mat. Sb. (N.S), 83 (1970), 456. [25] V. Grushin, A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold,, Mat. Sb., 84 (1971), 163. [26] A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations,, Russian Math. Surveys, 42 (1987), 169. [27] J. L. Lions, "Quelque Methodes de Resolution des Problemes aux Limites Nonlineaire,", Springer, (). [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. [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. [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. [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. [32] I. Kombe, Nonlinear degenerate parabolic equations for Baouendi-Grushin operators,, Mathematische Nachrichten, 279 (2006), 756. doi: 10.1002/mana.200310391. [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. [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. [36] R. Monti and D. Morbidelli, Isoperimetric inequality in the Grushin plane,, J. Geom. Anal., 14 (2004), 355. 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. doi: 10.1007/BF02392539.

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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. [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. [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. [4] M. S. Baouendi, Sur une classe d'opérateurs elliptiques dégénérés,, Bull. Soc. Math. France, 95 (1967), 45. [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. [6] A. Bellaiche and J. J. Risler, "Sub-Riemannian Geometry,", Birkhauser, (1996). doi: 10.1007/978-3-0348-9210-0. [7] T. Bieske, Viscosity solutions on Grushin-planes,, Illinois J. Math, 46 (2002), 893. [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. [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. [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. [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. [12] C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems,, in, (1981), 590. [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. [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. [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. [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. [17] N. Garofalo and Z. Shen, Absence of positive eigenvalues for a class of subelliptic operators,, Math. Ann., 304 (1996), 701. [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. [19] J. A. Goldstein and I. Kombe, Instantaneous blow up,, Contemp. Math, 327 (2003), 141. 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. [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] 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. [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. [24] V. Grushin, A certain class of hypoelliptic operators,, Mat. Sb. (N.S), 83 (1970), 456. [25] V. Grushin, A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold,, Mat. Sb., 84 (1971), 163. [26] A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations,, Russian Math. Surveys, 42 (1987), 169. [27] J. L. Lions, "Quelque Methodes de Resolution des Problemes aux Limites Nonlineaire,", Springer, (). [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. [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. [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. [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. [32] I. Kombe, Nonlinear degenerate parabolic equations for Baouendi-Grushin operators,, Mathematische Nachrichten, 279 (2006), 756. doi: 10.1002/mana.200310391. [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. [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. [36] R. Monti and D. Morbidelli, Isoperimetric inequality in the Grushin plane,, J. Geom. Anal., 14 (2004), 355. 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. doi: 10.1007/BF02392539.
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