System Of Semilinear Equations Research Articles

Problem Without Initial Conditions for a Countable System of Semilinear Hyperbolic Equations of the First Order

We derive sufficient conditions for the solvability of the problem without initial conditions for a countable hyperbolic system of semilinear equations of the first order and establish conditions for the classical solvability of the initial-boundary value problem for countable hyperbolic systems of semilinear equations of the first order in a semistrip.

Estimation of Decay of the Solutions of Initial-Boundary-Value Problem for the System of Semilinear Equations of Magnetoelasticity in Exterior Domains

We study the behavior as t → ∞ of strong solutions of the initial-boundary-value problem for the system of semilinear equations of magnetoelasticity in exterior domains. For the vectors of magnetic induction and displacements, the estimates of decay are obtained in L ∞ and in the Sobolev spaces of fractional order, respectively.

Right Markov Processes and Systems of Semilinear Equations with Measure Data

In the paper we prove the existence of probabilistic solutions to systems of the form $-Au=F(x,u)+\mu$, where $F$ satisfies a generalized sign condition and $\mu$ is a smooth measure. As for $A$ we assume that it is a generator of a Markov semigroup determined by a right Markov process whose resolvent is order compact on $L^1$. This class includes local and nonlocal operators corresponding to Dirichlet forms as well as some operators which are not in the variational form. To study the problem we introduce new concept of compactness property relating the underlying Markov process to almost everywhere convergence. We prove some useful properties of the compactness property and provide its characterization in terms of Meyer's property (L) of Markov processes and in terms of order compactness of the associated resolvent.

Min-rank conjecture for log-depth circuits

A completion of an m-by- n matrix A with entries in { 0 , 1 , ⁎ } is obtained by setting all ⁎-entries to constants 0 and 1. A system of semi-linear equations over G F 2 has the form M x = f ( x ) , where M is a completion of A and f : { 0 , 1 } n → { 0 , 1 } m is an operator, the ith coordinate of which can only depend on variables corresponding to ⁎-entries in the ith row of A. We conjecture that no such system can have more than 2 n − ϵ ⋅ mr ( A ) solutions, where ϵ > 0 is an absolute constant and mr ( A ) is the smallest rank over G F 2 of a completion of A. The conjecture is related to an old problem of proving super-linear lower bounds on the size of log-depth boolean circuits computing linear operators x ↦ M x . The conjecture is also a generalization of a classical question about how much larger can non-linear codes be than linear ones. We prove some special cases of the conjecture and establish some structural properties of solution sets.

Limiting behavior of a controlled system at infinite time

The foundations of the dynamical theory of biological communities were laid down in the work of Volterra and Lotka [1, 2]. Both these studies were an outgrowth of the solution of problems arising in connection with growth of specific populations. All populations are characterized by a range of standard processes, which include birth and death of individuals, spreading of individuals, predation, competition, selection, etc. These processes are described by a large number of variables that show complex interactions and nonlinear dependencies. The Lotka–Volterra equations are based on a conjecture known as the “collision principle” or “paired interaction principle,” which assumes additive contribution of each population to the relative growth rate of each population—an assumption that has adequate biological justification [3]. The growth rate of a population in the absence of any other species is taken proportional to its size, and this assumption is much less adequate for processes observed in biological communities. For instance, higher organisms generally display sexual propagation. If the number of individuals of each sex is approximately half the total population and the number of progeny is a linear function of the number of encounters between individuals of opposite sexes, then the growth rate of the given population in the absence of other species is proportional to the square of its size. The evolution of the spatial distribution of individuals is commonly described by “reaction–diffusion” equations [4–7]. On the microscopic level, for a single individual, this assumes random, “accidental” motion. In this article we generalize the system of Lotka–Volterra equations allowing for the phenomenon of transport. A species dynamics determined by a model with transport arises, for instance, under conditions of strong water flows—fast rivers or sea currents. Let zi be the distribution density of species i among all individuals, i = 1, n; z0 i the initial population density of species i on the interval [0, l]; a the flow speed; bi the propagation coefficients; individuals propagate according to a quadratic law (this reflects the fact that the increase in the number of individuals is proportional to the number of encounters of individuals of opposite sex); the mortality coefficient determined by environmental constraints is proportional to the total birth rate. The model is described by the following system of semi-linear equations:

Local Bifurcation Results for Systems of Semilinear Equations

Local Bifurcation Results for Systems of Semilinear Equations

Regularity of solutions of an initial/boundary problem for a system of semilinear equations of magnetoelasticity

We obtain an existence and uniqueness theorem for a global strong solution of a system of semilinear equations of magnetoelasticity for a sufficiently small input data.

On the solvability of the initial- and boundary-value problem for the system of semilinear equations of magnetoelasticity

An existence theorem is obtained for the system of semilinear equations of magnetoelasticity. The asymptotic behavior of the solutions over time is established.

Two solution methods for hyperbolic systems of partial differential equations in chemical engineering

Hyperbolic systems of partial differential equations are found in modeling separation and reaction engineering problems. They can be divided into linear, semi-linear, quasi-linear and non-linear equations. In this paper two mathematical tools for the solution of PDEs: the numerical method of characteristics and Lax—Wendroff schemes are reviewed and applied to the solution of various problems: fixed-bed reactor for MTBE synthesis (system of semi-linear equations), separation of xylene isomers (reducible system of quasi-linear equations) and adsorptive reactors (non-reducible system of quasi-linear equations).