On the Cauchy problem for the evolution p-Laplacian equations with gradient term and source

In this paper, the authors study the equation ut=div(|Du|p−2Du)+|u|q−1u−λl|Du| in RN with p>2. We first prove that for 1⩽l⩽p−1, the solution exists at least for a short time; then for , the existence and nonexistence of global (in time) solutions are studied in various situations.