scholarly journals Weak Solution to a Parabolic Nonlinear System Arising in Biological Dynamic in the Soil

2011 ◽  
Vol 2011 ◽  
pp. 1-24
Author(s):  
Côme Goudjo ◽  
Babacar Lèye ◽  
Mamadou Sy
Keyword(s):  
Weak Solution ◽  
Nonlinear System ◽  
Global Existence ◽  
Boundary Conditions ◽  
Parabolic System ◽  
Initial Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Nonlinear Parabolic System ◽  
Nonlinear Parabolic

We study a nonlinear parabolic system governing the biological dynamic in the soil. We prove global existence (in time) and uniqueness of weak and positive solution for this reaction-diffusion semilinear system in a bounded domain, completed with homogeneous Neumann boundary conditions and positive initial conditions.

scholarly journals Global weak solution to a generic reaction-diffusion nonlinear parabolic system

2021 ◽  
Author(s):  
Matallah Hana ◽  
MESSAOUD MAOUNI ◽  
Hakim Lekhal
Keyword(s):  
Fixed Point ◽  
Weak Solution ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Point Theorem ◽  
Parabolic System ◽  
Reaction Diffusion ◽  
Global Weak Solution ◽  
Nonlinear Parabolic System ◽  
Nonlinear Parabolic

We consider a new generic reaction-diffusion system, given as the following form: ∂u/∂t - div(g(│(∇u_σ)│)∇u)=f(t,x,u,v,∇v), in Q_T ∂v/∂t - d_v Δv=p(t,x,u,v,∇u), in Q_T u(0,.)=u_0, v(0,.)=v_0, in Ω (1) ∂u/∂η=0, ∂v/∂η=0, in ∑_T. Where Ω=]0,1[?×]0,1[, Q_T =]0,T [? and T =]0,T [?, (T > 0), η is an outward normal to domain Ω and u_0, v_0 is the image to be processed, x ∈Ω, σ >0, ∇u_σ= u∗ ∇G_σ and G_σ= 1/√2πσ exp(-│x│^2/4σ). In this study we are going to proof that there is a global weak solution to the ptoblem (1), we truncate the system and show that it can be solved by using Schauder fixed point theorem in Banach spaces. Finally by making some estimations, we prove that the solution of the truncated system converge to the solution of the problem.

Global Existence and Blow-Up in Finite Time of Solutions to One Kind of Reaction-Diffusion Educations with Neumann Boundary Conditions

2014 ◽  
Vol 971-973 ◽  
pp. 1017-1020
Author(s):  
Jun Zhou Shao ◽  
Ji Jun Xu
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Finite Time ◽  
Blow Up ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Blowing Up ◽  
Processing Techniques

This paper deals with the properties of one kind of reaction-diffusion equations with Neumann boundary conditions based on the comparison principles. The relations of parameter and the situation of the coupled about equations are used to construct the global existent super-solutions and the blowing-up sub-solutions, and then we obtain the conditions of the global existence and blow-up in finite time solutions with the processing techniques of inequality.

scholarly journals Global existence and boundedness of solutions in a Lotka-Volterra reaction-diffusion system of predator-prey model with nonlinear prey-taxis

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 5023-5035
Author(s):  
Demou Luo
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Classical Solutions ◽  
Boundedness Of Solutions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Global Classical Solutions ◽  
Prey Model

In this paper, we investigate a diffusive Lotka-Volterra predator-prey model with nonlinear prey-taxis under Neumann boundary conditions. This system describes a prey-taxis mechanism that is an immediate movement of the predator u in response to a change of the prey v (which lead to the collection of u). We apply some methods to overcome the substantial difficulty of the existence of nonlinear prey-taxis term and prove that the unique global classical solutions of Lotka-Volterra predator-prey model are globally bounded.

scholarly journals Analytical Diffusion Wave-type Solutions to a Nonlinear Parabolic System with Cylindrical and Spherical Symmetry

2021 ◽  
Vol 37 ◽  
pp. 31-46
Author(s):  
A. L. Kazakov ◽  
◽  
P. A. Kuznetsov ◽  
◽  
Keyword(s):  
Population Dynamics ◽  
Parabolic System ◽  
Spherical Symmetry ◽  
Parabolic Type ◽  
Reaction Diffusion ◽  
Nonlinear Parabolic System ◽  
Wave Type ◽  
Diffusion Wave ◽  
Plane Symmetric ◽  
Nonlinear Parabolic

The paper deals with a second-order nonlinear parabolic system that describes heat and mass transfer in a binary liquid mixture. The nature of nonlinearity is such that the system has a trivial solution where its parabolic type degenerates. This circumstance allows us to consider a class of solutions having the form of diffusion waves propagating over a zero background with a finite velocity. We focus on two spatially symmetric cases when one of the two independent variables is time, and the second is the distance to a certain point or line. The existence and uniqueness theorem of the diffusion wave-type solution with analytical components is proved. The solution is constructed as a power series with recursively determined coefficients, which convergence is proved by the majorant method. In one particular case, we reduce the considered problem to the Cauchy problem for a system of ordinary differential equations that inherits all the specific features of the original one. We present the form of exact solutions for exponential and power fronts. Thus, we extend the results previously obtained for a nonlinear parabolic reaction-diffusion system in the plane-symmetric form to more general cylindrical and spherical symmetry cases. Parabolic equations and systems often underlie population dynamics models. Such modeling allows one to determine the properties of populations and predict changes in population size. The results obtained, in particular, may be useful for mathematical modeling of the population dynamics of Baikal microorganisms.

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

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