Positive solutions with a singular set for semilinear parabolic equations
DOI:
https://doi.org/10.12775/TMNA.2024.039Keywords
Partial differential equations, semilinear parabolic equation, semilinear elliptic equation, Dirichlet boundary condition, positive solution, singular solution, Kato class, asymptotic behaviorAbstract
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 $E$.References
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Copyright (c) 2025 Sidi Hamidou Jah, Lotfi Riahi
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