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

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@Article{JPDE-20-349,
author = {},
title = {Singular Solution of a Quasilinear Convection Diffusion Degenerate Parabolic Equation with Absorption},
journal = {Journal of Partial Differential Equations},
year = {2007},
volume = {20},
number = {4},
pages = {349--364},
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.},
issn = {2079-732X},
doi = {https://doi.org/},
url = {http://global-sci.org/intro/article_detail/jpde/5314.html}
}

TY - JOUR
T1 - Singular Solution of a Quasilinear Convection Diffusion Degenerate Parabolic Equation with Absorption
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^'_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.

Peixin Zhang . (2019). Singular Solution of a Quasilinear Convection Diffusion Degenerate Parabolic Equation with Absorption.

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*Journal of Partial Differential Equations*.*20*(4). 349-364. doi: