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

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.

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

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.