Boundedness of classical solutions for a chemotaxis system with general sensitivity function

Abstract The chemotaxis system u t = Δ u − ∇ ⋅ ( u ∇ v ) , v t = Δ v − u v , is considered under the boundary conditions ∂ u ∂ ν − u ∂ v ∂ ν = 0 and v = v ⋆ on ∂Ω, where Ω ⊂ R n is a ball and v ⋆ is a given positive constant. In the setting of radially symmetric and suitably regular initial data, a result on global existence of bounded classical solutions is derived in the case n = 2, while global weak solutions are constructed when n ∈ {3, 4, 5}. This is achieved by analyzing an energy-type inequality reminiscent of global structures previously observed in related homogeneous Neumann problems. Ill-signed boundary integrals newly appearing therein are controlled by means of spatially localized smoothing arguments revealing higher order regularity features outside the spatial origin. Additionally, unique classical solvability in the corresponding stationary problem is asserted, even in nonradial frameworks.

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Comments on “Bit synchronisation method based on sensitivity function”

IEE Proceedings F Communications Radar and Signal Processing ◽

10.1049/ip-f-1.1984.0066 ◽

1984 ◽

Vol 131 (4) ◽

pp. 424

Author(s):

V. Brajović

Keyword(s):

Sensitivity Function

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Remarks on nonlinear elastic waves with null conditions

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2020039 ◽

2020 ◽

Vol 26 ◽

pp. 121

Author(s):

Dongbing Zha ◽

Weimin Peng

Keyword(s):

Initial Data ◽

Global Existence ◽

Elastic Waves ◽

Wave Equations ◽

Alternative Proof ◽

Classical Solutions ◽

Small Initial Data ◽

Hyperelastic Materials ◽

Variational Structure ◽

Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

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Global classical solutions to the elastodynamic equations with damping

Journal of Inequalities and Applications ◽

10.1186/s13660-021-02608-9 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Mengmeng Liu ◽

Xueyun Lin

Keyword(s):

Initial Data ◽

Periodic Boundary ◽

Energy Estimate ◽

Periodic Boundary Conditions ◽

Deformation Tensor ◽

Classical Solutions ◽

Small Initial Data ◽

Weighted Energy ◽

Global Classical Solutions ◽

Weighted Energy Estimate

AbstractIn this paper, we show the global existence of classical solutions to the incompressible elastodynamics equations with a damping mechanism on the stress tensor in dimension three for sufficiently small initial data on periodic boxes, that is, with periodic boundary conditions. The approach is based on a time-weighted energy estimate, under the assumptions that the initial deformation tensor is a small perturbation around an equilibrium state and the initial data have some symmetry.

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