一类拟线性退化抛物方程的Cauchy问题

对方程α（xx）u＋uαyu-αtu=f（·,u）,（x,y,t）∈R^2×（0,T）的Cauchy问题,给出了一个整体BV-弱解的定义,证明了该弱解的连续性.     

奇异半线性热方程初值问题解的存在性与Blow-up问题

本文讨论如下奇异半线性热方程的初值问题其中γ＞1，σ＞O，f（x）连续有界非负但不恒为零．我们讨论了问题（1），（2）非负局部经典解的存在性和非负整体解的不存在问题，得到当0＜σ＜1且时，（1），（2）没有非负整体解．当σ＝1且时，（1），（2）没有非负整体解，当σ＞1时（1），（2）没有非负整体解．     

On the Cauchy Problem for the Third Order Benjamin-Ono Equation

The existence and uniqueness of global solutions to the Cauchyproblem for the third order Benjamin-Ono equation are established.In addition, we obtain an Iorio type result for the asymptoticbehaviour of its solutions.     

Two-Parameter Asymptotics in a Bisingular Cauchy Problem for a Parabolic Equation

The Cauchy problem for a quasilinear parabolic equation with a small parameter ε at the highest derivative is considered. The initial function, which has the form of a smoothed step, depends on a “stretched” variable x/ρ, where ρ is another small parameter. This problem statement is of interest for applications as a model of propagation of nonlinear waves in physical systems in the presence of small dissipation. In the case corresponding to a compression wave, asymptotic solutions of the problem are constructed in the parameters ε and ρ independently tending to zero. It is assumed that ε/ρ → 0. Far from the line of discontinuity of the limit solution, asymptotic solutions are constructed in the form of series in powers of ε and ρ. In a small domain of linear approximation, an asymptotic solution is constructed in the form of a series in powers of the ratio ρ/ε. The coefficients of the inner expansion are determined from a recursive chain of initial value problems. The asymptotics of these coefficients at infinity is studied. The time of reconstruction of the scale of the internal space variable is determined.     

Upper and Lower Bounds of a Solution of the Cauchy Problem for a Stochastic Differential Equation of Parabolic Type with Power Nonlinearities (Weak Source)

We study the time evolution of a solution of the Cauchy problem for a stochastic differential equation of the parabolic type with power nonlinearities. We construct upper and lower bounds for this solution.     

The Cauchy Problem for a Higher Order Telegraph Equation

Using the integral operator that defines a solution of the Cauchy problem for the equation

Asymptotics of the Solution to the Cauchy Problem for Linear Parabolic Equations of Second Order with Small Diffusion

This paper is devoted to constructing an asymptotics of the solution to the Cauchy problem for a linear parabolic equation of second order with variable coefficients containing a small parameter at the highest derivative. Sufficient conditions for the existence and uniqueness of the “multiplicative” asymptotic expansion of the global solution of the problem are given.     

带吸收项的非线性双重退缩抛物方程的Cauchy问题

讨论带吸收项的非线性双重退缩抛物方程ut=div(│↓△u^m│^p-2↓△u^m)-u^q u(x,0)=u0(x)的Cauchy问题。     

Coefficient Stability of the Solution of a Difference Scheme Approximating a Mixed Problem for a Semilinear Parabolic Equation

We study the coefficient stability of a difference scheme approximating a mixed problem for a one-dimensional semilinear parabolic equation. We obtain sufficient conditions on the input data under which the solutions of the differential and difference problems are bounded. We also obtain estimates of perturbations of the solution of a linearized difference scheme with respect to perturbations of the coefficients; these estimates agree with the estimates for the differential problem.     

拟线性退化抛物方程Cauchy问题的解

对来自金融数学领域的方程_xxu+u_yu-_tu=c（x,y,t,u）,（x,y,t）∈Q_T=R²×[0,T)的Cauchy问题,给出了一种新的熵解的定义,得到了其适定性结果.可以证明所得到的解还是强解,即方程中所出现的各阶偏导数几乎处处连续.最后讨论了解的爆破性质以及与解的间断点相关的几何性质.     

On the Restoration of Initial Data of a Boundary-Value Problem for a Second-Order Parabolic Equation

On the basis of the Fourier expansion of a generalized function in a complete orthonormal system of fundamental functions of an elliptic operator, we propose an approximate method for determination of the initial data of a boundary-value problem for a second-order parabolic equation.     

Numerical Solution of the Cauchy Problem for a Second-Order Integro-Differential Equation

Differential Equations - In a finite-dimensional Hilbert space, we consider the Cauchy problem for a second-order integro-differential evolution equation with memory where the integrand is the...     

Regularization of the Solution of a Cauchy Problem for a Hyperbolic Equation

Journal of Applied and Industrial Mathematics - Given a hyperbolic equation with variable coefficients, we construct a regularizing algorithm to solve the problem of continuation of the wave field...