Singular Solution of a Quasilinear Convection Diffusion Degenerate Parabolic Equation with Absorption

Abstract

In this paper the existence and nonexistence of non-trivial solution for the Cauchy problem of the form {ut = div(|∇u|^{p-2}∇u) - \frac{∂}{∂x_i}b_i(u) - u^q, \qquad(x, t) ∈ S_T = R^N × (0, T), u(x, 0) = 0, \qquad \qquad 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+ \frac{p}{N}, 0 ≤ m < p-1+ \frac{p}{N}, then the problem has a solution; if p > 2, q > p-1+ \frac{p}{N}, 0 ≤ m ≤ \frac{q(p+Np-N-1)}{p+Np-N} , then the problem has no solution; if p > 2,p-1 < q < p-1+ \frac{p}{N}, 0 ≤ m < q, then the problem has a very singular solution; if p > 2, q > p-1 + \frac{p}{N}, 0 < m < q - \frac{p}{2N}, then the problem has no very singular solution. We use P.D.E. methods such as regularization, Moser iteration and Imbedding Theorem.