Liouville type theorems for degenerate parabolic systems with advection terms

Abstract

We study a parabolic equation of the form

$$\begin{aligned} u_t - \Delta _\lambda u+a\cdot \nabla _\lambda u=u^p \hbox { in}\,\,{\mathbb {R}}^{{N}}\times {\mathbb {R}}, \end{aligned}$$

and a parabolic system

$$\left\{ {\begin{array}{*{20}l} {u_{t} - \Delta _{\lambda } u + a \cdot \nabla _{\lambda } u = v^{p} } \hfill \\ {u_{t} - \Delta _{\lambda } v + a \cdot \nabla _{\lambda } v = u^{q} } \hfill \\ \end{array} } \right.{\text{in}}\;\mathbb{R}^{N} \times \mathbb{R},$$

where p is a real number, a(x) is a smooth vector field, \(\Delta _\lambda\) is a strongly degenerate operator given by

$$\begin{aligned} \Delta _\lambda =\sum _{i=1}^N\partial _{x_i}\left( \lambda _i^2\partial _{x_i}\right) \end{aligned}$$

and \(\nabla _\lambda\) is the gradient operator associated to \(\Delta _\lambda\). Under some general condition of \(\lambda _i\) introduced in Kogoj and Lanconelli [Nonlinear Anal 75: 4637–4649, 2012], we establish a Liouville type theorem for positive supersolutions to the problems above. In particular, we compute explicitly the critical exponent depending on both the homogeneous dimension of \({\mathbb {R^N}}\) associated to \(\Delta _\lambda\) and the behavior of the advection term a at infinity. This critical exponent is sharp in the case of Laplace operator without advection term according to [15].

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Acknowledgements

This work is supported by Vietnam Ministry of Education and Training under grant number B2019-SPH-02.

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Correspondence to Vu Trong Luong.

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Luong, V.T., Pham, D.H. & Thi, H.A.V. Liouville type theorems for degenerate parabolic systems with advection terms. J Elliptic Parabol Equ 6, 871–882 (2020). https://doi.org/10.1007/s41808-020-00086-6

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Keywords

  • Liouville-type theorems
  • Positive supersolutions
  • Degenerate parabolic systems
  • Advection terms
  • \(\Delta _\lambda\)-Laplacian

Mathematics Subject Classification

  • Primary: 35K65
  • Secondary: 35B53
  • 35B33