(untitled) (Journal of Partial Differential Equations, 2020)
In this paper the existence and nonexistence of non-trivial solution for the Cauchy problem of the form ut = div(|∇u|^p-2∇u) - ∂∂x_ib_i(u) - u^q, (x, t) ∈ S_T = R^N × (0, T), u(x, 0) = 0, x ∈ R^N 0 are studied. We assume that |b^'_i(s)| ≤ Ms^m-1, and proved that if p > 2, 0 < q < p-1+ pN, 0 ≤ m < p-1+ pN, then the problem has a solution; if p > 2, q > p-1+ pN, 0 ≤ m ≤ q(p+Np-N-1)p+Np-N , then the problem has no solution; if p > 2,p-1 < q < p-1+ pN, 0 ≤ m < q, then the problem has a very singular solution; if p > 2, q > p-1 + pN, 0 < m < q - p2N, then the problem has no very singular solution. We use P.D.E. methods such as regularization, Moser iteration and Imbedding Theorem.