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Nonlinear degenerate parabolic equations for Baouendi-Grushin operators. (English) Zbl 1117.35039
In this paper, the author studies the following two nonlinear parabolic equations:
\[ \begin{cases} \frac{\partial u}{\partial t}=\Delta_\gamma(u^m)+V(z)u^{m} & \text{in } \Omega\times(0,T),\, 0<m<1,\\ u(z,0)=u_0(z)\geq 0 &\text{in } \Omega,\\ u(z,t)=0 &\text{on } \partial\Omega\times(0,T),\end{cases}\tag{1} \]
and
\[ \begin{cases} \frac {\partial u}{\partial t}=\nabla_\gamma \cdot (| \nabla_\gamma u| ^{p-2}\nabla_\gamma u)+V(z)u^{p-1} &\text{in }\;\Omega\times(0,T),\\ u(z,0)=u_0(z)\geq 0 &\text{in } \Omega,\\ u(z,t)=0 &\text{on } \partial\Omega\times(0,T), \end{cases}\tag{2} \]
where \(1<p<2\), \(\Omega\) is a Carnot-Carathéodory metric ball in \(\mathbb R^{d+k}\), with \(d,k \geq 1\) and \(d+k=N \), and where \(V\in L^1_{\text{ loc}}(\Omega)\). Moreover, \(\nabla_\gamma\) denotes the sub-elliptic gradient, which is the (\(d+k\))-dimensional vector field given by
\[ \nabla_\gamma := (\nabla_x,| x| ^\gamma\nabla_y),\quad \gamma>0. \]
Finally, \(\Delta_\gamma\) is the Baouendi-Grushin operator on \(\mathbb R^{d+k}\) defined by
\[ \Delta_\gamma: =\nabla_\gamma \cdot \nabla_\gamma=\Delta_x+| x| ^{2\gamma}\Delta_y. \]
These operators were introduced by M. S. Baouendi in [Bull. Soc. Math. Fr., 95, 45–87 (1967; Zbl 0179.19501)] and subsequently studied by V. V. Grushin in [Mat. Sb., 84, 163–195 (1971; Zbl 0215.49203)] and in [Mat. USSR-Sb., 12, No. 3, 458–476 (1970; Zbl 0252.35057)].
In accordance with the notation of the author, let \(\mathcal{K}\) be a closed Lebesgue null subset of \(\Omega\) and let \(\sigma_{\text{inf}} ( (1-\epsilon)V; \Omega)\) and \(\sigma^p_{\text{inf}} ( (1-\epsilon)V; \Omega) \) be defined, respectively, by
\[ \sigma_{\text{inf}} ( (1-\epsilon)V; \Omega) := \inf_{ 0 \not\equiv \phi \in C_c^{\infty}(\Omega \setminus \mathcal{K})} \frac{\int_{\Omega}| \nabla_{\gamma } \phi | ^2\,dz - \int_{\Omega} (1-\epsilon) V | \phi | ^2\, dz}{\int_{\Omega} | \phi| ^2 \,dz} \]
and by
\[ \sigma^p_{\text{inf}} ( (1-\epsilon)V; \Omega) : =\inf_{ 0 \not\equiv \phi \in C_c^{\infty}(\Omega \setminus \mathcal{K})} \frac{\int_{\Omega}| \nabla_{\gamma } \phi | ^p\, dz - \int_{\Omega} (1-\epsilon) V | \phi | ^p \,dz}{\int_{\Omega} | \phi| ^p \,dz}. \]
Now, the two main results can be restated as follows:
Theorem. Let \( \frac{Q-2}{Q} \leq m < 1\) (respectively, \(\frac{2Q}{Q+1} \leq p < 2 \)), where \(Q=d+(1+\gamma)k \), and let \(V \in L^1_{\text{Loc}}(\Omega \setminus \mathcal{K})\). If \(\sigma_{\text{inf}} ( (1-\epsilon)V; \Omega)= -\infty \) (respectively, \(\sigma_{\text{inf}} ( (1-\epsilon)V; \Omega)= -\infty \) ) for some \(\epsilon > 0\) then the problem (1) (respectively, the problem (2)) has no general positive local solution off \(\mathcal{K}\).

MSC:
35K65 Degenerate parabolic equations
35K55 Nonlinear parabolic equations
35R05 PDEs with low regular coefficients and/or low regular data
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