TY - JOUR
T1 - Singular Solution of a Quasilinear Convection Diffusion Degenerate Parabolic Equation with Absorption
AU - Peixin Zhang 
JO - Journal of Partial Differential Equations
VL - 4
SP - 349
EP - 364
PY - 2007
DA - 2007/11
SN - 20
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jpde/5314.html
KW - Convect diffusion equation
KW - Cauchy problem
KW - non-trivial solution
AB -  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^&#39;_i(s)| ≤ Ms^{m-1}, and proved that if p &gt; 2, 0 &lt; q &lt; p-1+ \frac{p}{N}, 0 ≤ m &lt; p-1+ \frac{p}{N}, then the problem has a solution; if p &gt; 2, q &gt; p-1+ \frac{p}{N}, 0 ≤ m ≤ \frac{q(p+Np-N-1)}{p+Np-N} , then the problem has no solution; if p &gt; 2,p-1 &lt; q &lt; p-1+ \frac{p}{N}, 0 ≤ m &lt; q, then the problem has a very singular solution; if p &gt; 2, q &gt; p-1 + \frac{p}{N}, 0 &lt; m &lt; 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.