Singular solutions for semi-linear parabolic equations on nonsmooth domains. (English) Zbl 1123.35023
The author studies the existence of positive singular solutions for the semi-linear parabolic equation
\[
\Delta_x u-{\partial \over \partial t} u+ \mu u^p=0\qquad \text{ on}\quad \Omega=D\times (0,\infty)
\]
where \(p>1,\) \(D\) is a bounded non-tangentially accessible domain in \({\mathbb R}^n,\) \(n\geq 2,\) and \(\mu \) is in a general class of signed Radon measures on \(D\) covering the elliptic Kato class of potentials adopted by Zhang and Zhao. A new proof of the result based on a simple fixed point theorem is also given.
Reviewer: Lubomira Softova (Bari)
References:
| [1] | Aikawa, H., Boundary Harnack principle and Martin boundary for a uniform domain, J. Math. Soc. Japan, 53, 1, 119-145 (2001) · Zbl 0976.31002 |
| [2] | Chung, K. L.; Zhao, Z., From Brownian Motion to Schrödinger’s Equation (1995), Springer: Springer New York · Zbl 0819.60068 |
| [3] | Hansen, W., Uniform boundary Harnack principle and generalized triangle property, J. Funct. Anal., 226, 452-484 (2005) · Zbl 1082.31003 |
| [4] | Jerison, D.; Kenig, C., Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. Math., 46, 80-147 (1982) · Zbl 0514.31003 |
| [5] | Mâatoug, L.; Riahi, L., Global existence of positive solutions for semilinear parabolic equations in a half-space, Differential Integral Equations, 17, 1273-1292 (2004) · Zbl 1150.35396 |
| [6] | Mâatoug, L.; Riahi, L., Global existence and asymptotic behavior of solutions for nonlinear parabolic equations on unbounded domains, J. Funct. Anal., 233, 583-618 (2006) · Zbl 1112.35097 |
| [7] | Möser, J., A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math., 17, 101-103 (1964) · Zbl 0149.06902 |
| [8] | Riahi, L., A \(3G\)-Theorem for Jordan domains in \(R^2\), Colloq. Math., 101, 1-7 (2004) · Zbl 1081.35017 |
| [9] | Riahi, L., The \(3G\)-inequality for general Schrödinger operators on Lipschitz domains, Manuscripta Math., 116, 211-227 (2005) · Zbl 1070.35053 |
| [10] | Zhang, Q. S., Global existence and local continuity of solution for semilinear parabolic equations, Comm. Partial Differential Equations, 22, 1529-1557 (1997) · Zbl 0883.35061 |
| [11] | Zhang, Q. S.; Zhao, Z., Global asymptotic behavior of solutions of semilinear parabolic equation, Proc. Amer. Math. Soc., 126, 1491-1500 (1998) · Zbl 0887.35074 |
| [12] | Zhang, Q. S.; Zhao, Z., Singular solutions of semilinear elliptic and parabolic equations, Math. Ann., 310, 777-794 (1998) · Zbl 0907.35047 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
