Global radial renormalized solution to a producer–scrounger model with singular sensitivities

2020 ◽  
Vol 30 (06) ◽  
pp. 1119-1165 ◽  
Author(s):  
Xinru Cao
Keyword(s):  
Initial Data ◽  
Boundary Problem ◽  
Parabolic System ◽  
Initial Boundary ◽  
Renormalized Solution ◽  
Initial Boundary Problem ◽  
Symmetric Initial Data

This paper is concerned with the parabolic system [Formula: see text] in a bounded ball [Formula: see text] ([Formula: see text]) with [Formula: see text]. Where [Formula: see text] and [Formula: see text]. It is shown that for arbitrarily radially symmetric initial data [Formula: see text], which are nonnegative and suitably regular, the corresponding Neumann initial-boundary problem admits a global renormalized solution, which is moreover smooth in [Formula: see text].

scholarly journals Fujita blow-up phenomena of solutions for a Dirichlet problem of parabolic equations with space-time coefficients

2021 ◽  
Author(s):  
Jiaqi Liu ◽  
Fengjie Li ◽  
Bingchen Liu
Keyword(s):  
Dirichlet Problem ◽  
Initial Data ◽  
Boundary Problem ◽  
Parabolic Equations ◽  
Blow Up ◽  
Initial Boundary ◽  
Space Time ◽  
Space Domain ◽  
Initial Boundary Problem

This paper deals with a homogeneous Dirichlet initial-boundary problem of parabolic equations with different space-time coefficients, $$u_t =\Delta u + t^{\sigma_1} u^{\alpha} + \langle x\rangle^{n} v^{p},\quad v_t =\Delta v + \langle x\rangle^{m} u^{q} + t^{\sigma_2} v^{\beta},$$ where the eight exponents are nonnegative constants and $\langle x\rangle$ is the Japanese brackets. We obtain the Fujita exponents of solutions, which are determined by the eight exponents and the dimension of the space domain. Moreover, simultaneous or non-simultaneous blow-up of the two components of blow-up solutions is discussed with or without conditions on the initial data.

scholarly journals Discrete-continuous three-element model of impact device

2021 ◽  
Vol 2131 (3) ◽  
pp. 032091
Author(s):  
A M Slidenko ◽  
V M Slidenko ◽  
S G Valyukhov
Keyword(s):  
Boundary Problem ◽  
Fourier Method ◽  
Mathematic Model ◽  
Initial Boundary ◽  
Element Model ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Initial Boundary Problem ◽  
Impact Device ◽  
The Impact

Abstract There have been examined the mathematic model of the impact device provided for geological materials destruction. Basic elements of the impact device are variable cross-section tool, striker and impact device body. The interaction of these elements is described as a movement of two discrete mass and the rod in the presence of rigid and dissipative connections. One equation in partial derivatives and two ordinary differential equations associated by initial and boundary conditions represent the initial-boundary problem. The numerical method parameters of which are determined at tests problems solution by Fourier method is used for looking for solutions of mixed initial-boundary problem. Researches are made, and parameters determining the damping efficiency of tool, striker and impact device body oscillations are evaluated.

scholarly journals Vanishing shear viscosity limit and boundary layer study for the planar MHD system

2019 ◽  
Vol 29 (06) ◽  
pp. 1139-1174 ◽  
Author(s):  
Xulong Qin ◽  
Tong Yang ◽  
Zheng-an Yao ◽  
Wenshu Zhou
Keyword(s):  
Boundary Layer ◽  
Initial Data ◽  
Global Existence ◽  
Shear Viscosity ◽  
Initial Boundary ◽  
Heat Conductivity Coefficient ◽  
Large Initial Data ◽  
Initial Boundary Problem ◽  
Viscosity Limit ◽  
Mhd System

We consider an initial boundary problem for the planar MHD system under the general condition on the heat conductivity coefficient that depends on both the temperature and the density. Firstly, the global existence of strong solution for large initial data is obtained, and then the limit of the vanishing shear viscosity is justified. In addition, the [Formula: see text] convergence rate is obtained together with the estimation on the thickness of the boundary layer.

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