Cauchy-Dirichlet problem for the nonlinear degenerate parabolic equations.(English)Zbl 1103.35053

The article deals the nonexistence of positive solutions to the following nonlinear parabolic equation: \begin{alignedat}{2} & u_t = Lu + V(w)u^{p-1}&&\quad \text{in } \Omega\times (0,T),\;1 < p < 2, \\ & u(w,0) = u_0(w) \geq 0,&&\quad \text{in } \Omega,\\ & u(w,t) = 0&&\quad \text{on } \partial\Omega \times (0,T), \end{alignedat} where $$\Omega$$ is a Carnot-Carathéodory metric ball in $$\mathbb R^{2n+1}$$ and $$v \in L_{\text{loc}}^1(\Omega)$$. The nonlinear operator $$L$$ is the subelliptic $$p$$-Laplacian $Lu = \sum_{j=1}^{2n}X_j(| Xu| ^{p-2}X_ju),$ where $X_j = \frac{\partial}{\partial x_j} + 2ky_j| z| ^{2k-2}\frac{\partial}{\partial l},\quad X_{n+j} = \frac{\partial}{\partial y_j} - 2kx_j| z| ^{2k-2}\frac{\partial}{\partial l},\quad j = 1,\dots,n.$ are the smooth vector fields satisfying Hörmander condition for any $$k\in N$$. Here $$Xu = (X_1u,\dots,X_{2n}u)$$. The main result reads: Let $$(4n + 4k)/(2n + 2k + 1) \leq p <2$$ and $$V \in L^1_{\text{loc}}(\Omega\setminus K)$$, where $$K$$ is a closed Lebesgue null subset of $$\Omega$$. If $\inf_{0\neq \phi \in C_{c}^{\infty}}\frac{\int_{\Omega}| X\phi| ^{p}\,dw - \int_{\Omega}(1-\epsilon)V| \phi| ^p\,dw}{\int_{\Omega}| \phi| ^p\,dw} = -\infty$ for some $$\epsilon > 0$$, then the above mentioned problem has no general positive solution exterior to $$K$$.

MSC:

 35K65 Degenerate parabolic equations 35K55 Nonlinear parabolic equations 35H20 Subelliptic equations
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