Existence of a renormalized solution of nonlinear parabolic equations with general measure data

In this paper we prove the existence of a renormalized solution for nonlinear parabolic equations of the type:$$\displaystyle{\partial b(x,u)\over\partial t} - {\rm div}\Big(a(x,t,\nabla u)\Big)=\mu\qquad \text{in}\ \Omega\times (0,T),$$ where the right handside is a general measure, $b(x,u)$ is anunbounded function of $u$ and $- {\rm div}(a(x,t,\nabla u))$is a Leray--Lions type operator with growth $|\nabla u|^{p-1}$ in$\nabla u$.

An Unconditional Positivity-Preserving Difference Scheme for Models of Cancer Migration and Invasion

In this paper, we consider models of cancer migration and invasion, which consist of two nonlinear parabolic equations (one of the convection–diffusion reaction type and the other of the diffusion–reaction type) and an additional nonlinear ordinary differential equation. The unknowns represent concentrations or densities that cannot be negative. Widely used approximations, such as difference schemes, can produce negative solutions because of truncation errors and can become unstable. We propose a new difference scheme that guarantees the positivity of the numerical solution for arbitrary mesh step sizes. It has explicit and fast performance even for nonlinear reaction terms that consist of sums of positive and negative functions. The numerical examples illustrate the simplicity and efficiency of the method. A numerical simulation of a model of cancer migration is also discussed.

Existence and asymptotic behavior of Radon measure-valued solutions for a class of nonlinear parabolic equations

In this paper we address the weak Radon measure-valued solutions associated with the Young measure for a class of nonlinear parabolic equations with initial data as a bounded Radon measure. This problem is described as follows: $$ \textstyle\begin{cases} u_{t}=\alpha u_{xx}+\beta [\varphi (u) ]_{xx}+f(u) &\text{in} \ Q:=\Omega \times (0,T), \\ u=0 &\text{on} \ \partial \Omega \times (0,T), \\ u(x,0)=u_{0}(x) &\text{in} \ \Omega , \end{cases} $$ { u t = α u x x + β [ φ ( u ) ] x x + f ( u ) in Q : = Ω × ( 0 , T ) , u = 0 on ∂ Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) in Ω , where $T>0$ T > 0 , $\Omega \subset \mathbb{R}$ Ω ⊂ R is a bounded interval, $u_{0}$ u 0 is nonnegative bounded Radon measure on Ω, and $\alpha , \beta \geq 0$ α , β ≥ 0 , under suitable assumptions on φ and f. In this work we prove the existence and the decay estimate of suitably defined Radon measure-valued solutions for the problem mentioned above. In particular, we study the asymptotic behavior of these Radon measure-valued solutions.

An inverse problem for a system of nonlinear parabolic equations

The inverse problem for a system of nonlinear parabolic equations is considered in the present paper. Namely, it is required to restore the initial condition by a given time-average value of the solution to the system of the nonlinear parabolic equations. An exact in the order error estimate of the optimal method for solving the inverse problem through the error estimate for the corresponding linear problem is obtained. A stable approximate solution to the unstable nonlinear problem under study is constructed by means of the projection regularization method which consists of using the representation of the approximate solution as a partial sum of the Fourier series. An exact in the order estimate for the error of the projection regularization method is obtained on one of the standard correctness classes. As a consequence, it is proved the optimality of the projection regularization method. As an example of a nonlinear system of parabolic equations, which has important practical applications, a spatially distributed model of blood coagulation is considered.

An Empirical Study for Real Options of Water Management in the Three Gorges Reservoir Area

This paper investigates an empirical evaluation of water management on agricultural irrigation. To address this problem, a real options model was proposed. This model analysis the choice of investment in water savings. Also, the model discusses a linear complementary problem that can be transformed into the inequalities of parabolic variational. By using a power penalty method, we solved the parabolic variational inequalities. The results depicted that the nonlinear parabolic equations’ solution is converges to the rate of order O(h−k2). A numerical example is given at the end of the paper to demonstrate the theoretical analysis follows from the Three Gorges Reservoir Area.