A doubly degenerate diffusion equation not in divergence form with gradient term

In this paper, we investigate positive solutions to the doubly degenerate parabolic equation not in divergence form with gradient term $u_{t}=u^{m}\operatorname{div}(|\nabla u|^{p-2}\nabla u)+ \lambda u^{q}+ \gamma u^{r}|\nabla u|^{p}$ , subject to the null Dirichlet boundary condition. We first establish the local existence of weak solutions to the problem, and then determine in what way the gradient term affects the behavior of solutions. The conditions for global and non-global solutions are obtained with the critical exponent $r_{c}= \frac{pm-q}{p-1}$ . Here we introduce some precise technique for the ‘concavity method’ to deal with the difficult non-divergence form of the model.

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