Fading absorption in non-linear elliptic equations
Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 2, p. 315-336
We study the equation $-\Delta u+h\left(x\right){|u|}^{q-1}u=0$, $q>1$, in ${ℝ}_{+}^{N}={ℝ}^{N-1}×{ℝ}_{+}$ where $h\in C\left(\overline{{ℝ}_{+}^{N}}\right)$, $h⩾0$. Let $\left({x}_{1},\cdots ,{x}_{N}\right)$ be a coordinate system such that ${ℝ}_{+}^{N}=\left[{x}_{N}>0\right]$ and denote a point $x\in {ℝ}^{N}$ by $\left({x}^{\text{'}},{x}_{N}\right)$. Assume that $h\left({x}^{\text{'}},{x}_{N}\right)>0$ when ${x}^{\text{'}}\ne 0$ but $h\left({x}^{\text{'}},{x}_{N}\right)\to 0$ as $|{x}^{\text{'}}|\to 0$. For this class of equations we obtain sharp necessary and sufficient conditions in order that singularities on the boundary do not propagate in the interior.
@article{AIHPC_2013__30_2_315_0,
author = {Marcus, Moshe and Shishkov, Andrey},
title = {Fading absorption in non-linear elliptic equations},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {30},
number = {2},
year = {2013},
pages = {315-336},
doi = {10.1016/j.anihpc.2012.08.002},
zbl = {1295.35208},
mrnumber = {3035979},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2013__30_2_315_0}
}

Marcus, Moshe; Shishkov, Andrey. Fading absorption in non-linear elliptic equations. Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 2, pp. 315-336. doi : 10.1016/j.anihpc.2012.08.002. http://www.numdam.org/item/AIHPC_2013__30_2_315_0/

[1] C. Bandle, M. Marcus, Asymptotic behavior of solutions and their derivatives, for semilinear, elliptic problems with blowup at the boundary, Ann. Inst. H. Poincare 12 (1995), 155-171 | Numdam | MR 1326666 | Zbl 0840.35033

[2] A. Brada, Comportement asymptotique de solutions dʼéquations elliptiques semi-linéares dans un cylindre, Asymptot. Anal. 10 (1995), 335-366 | MR 1338253 | Zbl 0860.35012

[3] J.B. Keller, On solutions of $\Delta u=f\left(u\right)$, Comm. Pure Appl. Math. 10 (1957), 503-510 | MR 91407 | Zbl 0090.31801

[4] M. Marcus, L. Véron, The boundary trace of positive solutions of semilinear elliptic equations: The subcritical case, Arch. Ration. Mech. Anal. 144 (1998), 201-231 | MR 1658392 | Zbl 0924.35050

[5] M. Marcus, L. Véron, Initial trace of positive solutions to semilinear parabolic inequalities, Adv. Nonlinear Stud. 2 (2002), 395-436 | MR 1936045 | Zbl 1021.35051

[6] M. Marcus, L. Véron, Boundary trace of positive solutions of nonlinear elliptic inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) III (2004), 481-533 | Numdam | MR 2099247 | Zbl 1121.35057

[7] R. Osserman, On the inequality $\Delta u⩾f\left(u\right)$, Pacific J. Math. 7 (1957), 1641-1647 | MR 98239 | Zbl 0083.09402

[8] A. Shishkov, L. Véron, The balance between diffusion and absorption in semilinear parabolic equations, Rend. Lincei Mat. Appl. 18 (2007), 59-96 | MR 2314465 | Zbl 1139.35366

[9] A. Shishkov, L. Véron, Diffusion versus absorption in semilinear elliptic equations, J. Math. Anal. Appl. 352 (2009), 206-217 | MR 2499898 | Zbl 1168.35015

[10] A. Shishkov, L. Véron, Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption, Calc. Var. Partial Differential Equations 33 (2008), 343-375 | MR 2429535 | Zbl 1158.35058

[11] A. Shishkov, L. Véron, Propagation of singularities of nonlinear heat flow in fissured media, arXiv:1103.5893v1 | MR 2997541 | Zbl 1267.35006