Existence of Nonnegative Solutions of Nonlinear Fractional Parabolic Inequalities

We study the existence of nontrivial nonlocal nonnegative solutions \(u(x,t)\) of the nonlinear initial value problems \[ (\partial_t -\Delta)^\alpha u\geq u^\lambda \quad \text{in } \mathbb{R}^n \times\mathbb{R},\,n\geq 1 \] \[ u=0 \quad\text{in } \mathbb{R}^n \times(-\infty,0) \] and \[ C_1 u^\lam...

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Bibliographic Details
Published in:arXiv.org 2020-05
Main Author: Taliaferro, Steven D
Format: Article
Language:English
Subjects:
Boundary value problems
Online Access:Get full text
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Summary:We study the existence of nontrivial nonlocal nonnegative solutions \(u(x,t)\) of the nonlinear initial value problems \[ (\partial_t -\Delta)^\alpha u\geq u^\lambda \quad \text{in } \mathbb{R}^n \times\mathbb{R},\,n\geq 1 \] \[ u=0 \quad\text{in } \mathbb{R}^n \times(-\infty,0) \] and \[ C_1 u^\lambda \leq(\partial_t -\Delta)^\alpha u\leq C_2 u^\lambda \quad\text{in } \mathbb{R}^n \times\mathbb{R},\,n\geq1 \] \[ u=0 \quad\text{in } \mathbb{R}^n \times(-\infty,0), \] where \(\lambda,\alpha,C_1\), and \(C_2\) are positive constants with \(C_1
ISSN:2331-8422