Related Experiment Video
Updated: Feb 23, 2026
06:44
Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
Published on: September 23, 2025
616
Remark on the Cauchy problem for the evolution p-Laplacian equation.
Liangwei Wang1, Jngxue Yin2, Jinde Cao3
1School of Mathematics and Statistics, Chongqing Three Gorges University, No. 666, Tian Xing Road, Wanzhou District, Chongqing, 404100 China.
Journal of Inequalities and Applications
|September 1, 2017
Summary
This study demonstrates that the semigroup for the evolution p-Laplacian equation is continuous. This continuity reveals the equation generates chaotic dynamical systems on compact sets.
Area of Science:
- Nonlinear Partial Differential Equations
- Dynamical Systems Theory
- Mathematical Physics
Background:
- The evolution p-Laplacian equation is a key model in nonlinear analysis.
- Understanding the behavior of solutions is crucial for applications.
- Semigroup theory provides a framework for analyzing time-dependent problems.
Purpose of the Study:
- To establish the continuity of the semigroup generated by the evolution p-Laplacian equation.
- To demonstrate that this equation generates a chaotic dynamical system.
- To analyze the properties of solutions, including propagation and decay estimates.
Main Methods:
- Utilizing semigroup theory for the Cauchy problem.
- Establishing continuity from weighted L2 spaces to continuous spaces.
- Deriving propagation and space-time decay estimates for solutions.
- Analyzing chaotic dynamics on compact subsets.
Main Results:
- The semigroup generated by the evolution p-Laplacian equation is proven to be continuous.
- The evolution p-Laplacian equation generates a chaotic dynamical system.
- Propagation and space-time decay estimates for solutions were successfully established.
Conclusions:
- The continuity of the semigroup is a fundamental property of the evolution p-Laplacian equation.
- The identified chaotic dynamics have implications for understanding complex behaviors in nonlinear systems.
- The established estimates are essential for the analysis of the dynamical system.
More Related Videos
Related Concept Videos
01:25Poisson's And Laplace's Equation
4.4K
The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.
4.4K
01:22Second Derivatives and Laplace Operator
2.7K
The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.
Consider a scalar function. The curl of its...
Consider a scalar function. The curl of its...
2.7K
01:17Differential Form of Maxwell's Equations
1.3K
James Clerk Maxwell (1831–1879) was one of the significant contributors to physics in the nineteenth century. He is probably best known for having combined existing knowledge of the laws of electricity and the laws of magnetism with his insights to form a complete overarching electromagnetic theory, represented by Maxwell's equations. The four basic laws of electricity and magnetism were discovered experimentally through the work of physicists such as Oersted, Coulomb, Gauss, and...
1.3K
01:20Region of Convergence of Laplace Tarnsform
1.3K
The Region of Convergence (ROC) is a fundamental concept in signal processing and system analysis, particularly associated with the Laplace transform. The ROC represents an area in the complex plane where the Laplace transform of a given signal converges, determining the transform's applicability and utility.
Consider a decaying exponential signal that begins at a specific time. When deriving its Laplace transform, the time-domain variable is replaced with a complex variable. This...
Consider a decaying exponential signal that begins at a specific time. When deriving its Laplace transform, the time-domain variable is replaced with a complex variable. This...
1.3K
01:28Navier–Stokes Equations
2.4K
For incompressible Newtonian fluids, where density remains constant, stresses show a linear relationship with the deformation rate, defined by normal and shear stresses. Normal stresses depend on the pressure exerted on the fluid and the rate of deformation in specific directions, which determines how fluid flows under varying pressures. Shear stresses, on the other hand, act tangentially across fluid layers. They explain how adjacent fluid layers slide relative to one another, connecting...
2.4K
01:20Separable Differential Equations
150
A separable differential equation is a type of first-order differential equation where the derivative dy/dx can be expressed as a product of two functions: one that depends only on x and another that depends only on y. This allows for the rearrangement of the equation so that all terms involving y are on one side, and all terms involving x are on the other. This process, known as the separation of variables, simplifies the process of solving the equation by enabling the integration of both...
150