Advances in Differential Equations

Nonlinear degenerate prabolic equations with singular lower-order term

Jerome A. Goldstein and Ismail Kombe

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Abstract

We use variational methods to study the nonexistence of positive solutions for the following nonlinear parabolic partial differential equations: \[ \begin{cases} \frac{\partial u}{\partial t}=\Delta( u^m)+V(x)u^m & \text{in}\quad \Omega \times (0, T ) ,\\ u(x,0)=u_{0}(x)\geq 0 & \text{in} \quad\Omega, \\ u(x,t)=0 & \text{on}\quad \partial\Omega\times (0, T), \end{cases} \] and \[ \begin{cases} \frac{\partial u}{\partial t}=\Delta_p u+V(x)u^{p-1} & \text{in}\quad \Omega \times (0, T ) ,\\ u(x,0)=u_{0}(x)\geq 0 & \text{in} \quad\Omega ,\\ u(x,t)=0 & \text{on}\quad \partial\Omega\times (0, T ), \end{cases} \] where $0 < m < 1$, $1 < p < 2$, $V\in L_{loc}^1(\Omega)$ and $\Omega$ is a bounded domain with smooth boundary in $ \mathbb{R}^N$.

Article information

Source
Adv. Differential Equations Volume 8, Number 10 (2003), 1153-1192.

Dates
First available in Project Euclid: 19 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355926158

Mathematical Reviews number (MathSciNet)
MR2016679

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35A15: Variational methods 35K20: Initial-boundary value problems for second-order parabolic equations 35K65: Degenerate parabolic equations 47J30: Variational methods [See also 58Exx]

Citation

Goldstein, Jerome A.; Kombe, Ismail. Nonlinear degenerate prabolic equations with singular lower-order term. Adv. Differential Equations 8 (2003), no. 10, 1153--1192. https://projecteuclid.org/euclid.ade/1355926158.


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