Doubly nonlinear parabolic equations with singular lower order term.
The positive solutions of the doubly nonlinear parabolic equation \[ \begin{cases} u_t= \text{div}(u^{m-1}| \nabla u| ^{p-2} \nabla u) +V u^{m+p-2} \quad &\text{ in }\quad \Omega \times (0,T), \\ u(\cdot,0)=u_0(\cdot) \geq 0 \qquad &\text{ in }\quad \Omega, \\ u(x,t)=0 \qquad &\text{ on }\quad \partial \Omega \times (0,T), \end{cases}\tag{1} \] here \(\Omega \subset \mathbb R^N\) is a bounded domain with smooth boundary, \(V \in L_{\text{loc}}^1 (\Omega),\) \(m \in \mathbb R,\) \(1<p<N\) and \(m+p-2>0\) is studied. Equation (1) is good combination form of the porous media equation \((p=2),\) fast diffusion equation (\(p=2\), \(0<m<1\)), very fast diffusion equation \((p=2,\) \(m \leq 0).\) The main point of the considered problem is to perturb equation \[ \frac{\partial u}{\partial \tau}=\text{div} \left( u^{m-1} | \nabla u| ^{p-2} \nabla u \right) \] by a suitable term involving the potential function \(V\) and to search for a positive solution. The critical exponents \(q^*\) are found and the nonexistence results are also proved for \(q^* \leq m+p<3 \). The problem (1) is investigated on the Heisenberg group, too.
scientific article; zbMATH DE number 2056119