Positive solutions with a singular set for semilinear parabolic equations
DOI:
https://doi.org/10.12775/TMNA.2024.039Słowa kluczowe
Partial differential equations, semilinear parabolic equation, semilinear elliptic equation, Dirichlet boundary condition, positive solution, singular solution, Kato class, asymptotic behaviorAbstrakt
We study the existence and large time behavior of positive solutions for the semilinear parabolic equation $\frac{\partial u}{\partial t}(x,t) =\Delta u(x,t)+ V(x)u(x,t) +f(x,u(x,t))$ with initial Dirichlet boundary conditions on $(D\setminus E)\times (0,\infty)$, where $D$ is a bounded Lipschitz domain in $\mathbb{R}^n$, $ n\geq 3$, $E$ is a prescribed compact set of $D$, and $V$ and $f$ are real-valued functions satisfying some general conditions. Our results cover various types of nonlinearities and extend known results proved for the power nonlinearity $f(x,u)=W(x)u^p$ and a singular one-point set $E$.Bibliografia
H. Aikawa, Boundary Harnack principle and Martin boundary for a uniform domain, J. Math. Soc. Japan 53 (2001), no. 1, 119–145.
H. Brézis, L.A. Peletier and D. Terman, A very singular solution of the heat equation with absorption, Arch. Ration. Mech. Anal. 95 (1986), 185–209.
X. Chen, Y. Qi and M. Wang, Classification of singular solutions of porous medium equations with absorption, Proc. Roy. Soc. Edinburgh Ser. A 135 (2005), 563–584.
X. Chen, Y. Qi and M. Wang, Singular solutions of parabolic p-Laplacian with absorption, Trans. Amer. Math. Soc. 359 (2007), 5653–5668.
E.B. Fabes and M.V. Safonov, Behavior near the boundary of positive solutions of second order parabolic equations, J. Fourier Anal. Appl. 3 (1997), 871–882.
K. Hirata, Positive solutions with a time-independent boundary singularity of semilinear heat equations in bounded Lipschitz domains, Nonlinear Anal. 134 (2016), 144–163.
K. Hirata and T. Ono, Removable singularities and singular solutions of semilinear elliptic equations, Nonlinear Anal. 105 (2014), 10–23.
S.H. Jah and L. Riahi, Singular solutions for nonlinear elliptic equations on bounded domains, J. Fixed Point Theory Appl. 24 (2022), article no. 6.
S. Kamin, L.A. Peletier and J.L. Vasquez, Classification of singular solutions of a nonlinear heat equation, Duke Math. J. 58 (1989), 601–615.
S.C. Port and C.J. Stone, Brownian Motion and Classical Potential Theory, Academic, New York, 1978.
L. Riahi, Singular solutions for semilinear parabolic equations on nonsmooth domains, J. Math. Anal. Appl. 333 (2007), 604–613.
L. Riahi, Estimates for Dirichlet heat kernels, intrinsic ultracontractivity and expected exit time on Lipschitz domains, Comm. Math. Anal. 15 (2013), no. 1, 115–130.
S. Sato, A singular solution with smooth initial data for a semilinear parabolic equation, Nonlinear Anal. 74 (2011), 1383–1392.
S. Sato and E. Yanagida:, Solutions with moving singularities for a semilinear parabolic equation, J. Differential Equations 246 (2009), 724–748.
S. Sato and E. Yanagida, Forward self-similar solution with a moving singularity for a semilinear parabolic equation, Discrete Contin. Dyn. Syst. 26 (2010), 313–331.
S. Shishkov and L. Véron, Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption, Calc. Var. 33 (2008), 343–375.
Q.S. Zhang and Z. Zhao, Singular solutions of semilinear elliptic and parabolic equations, Math. Ann. 310 (1998), 777–794.
Pobrania
Opublikowane
Jak cytować
Numer
Dział
Licencja
Prawa autorskie (c) 2025 Sidi Hamidou Jah, Lotfi Riahi
Utwór dostępny jest na licencji Creative Commons Uznanie autorstwa – Bez utworów zależnych 4.0 Międzynarodowe.
Statystyki
Liczba wyświetleń i pobrań: 0
Liczba cytowań: 0
