The stochastic parabolic partial differential equation with non-Lipschitz coefficients on the unbounded domain

We consider the existence and uniqueness of the stochastic heat equation driven by the space-time white noise under the weaker conditions of the coefficients than the Lipschitz conditions. In order to show the existence of the solution, we investigate the sequence of the successive approximations under two kinds of conditions.     

Comparison theorems for anticipated BSDEs with non-Lipschitz coefficients

We study comparison theorems for one dimensional anticipated backward stochastic differential equations under one kind of non-Lipschitz assumption. In the results, the generator functions are allowed to contain the anticipated term of z, neither generator function needs to be necessarily monotone in the anticipated term of y  , and the anticipated times of the anticipated terms of (y,z)$(y,z)$ in one generator function can differ from those in the other.     

Regularity of solutions to differential equations with non-Lipschitz coefficients

We study the ordinary and stochastic differential equations whose coefficients satisfy certain non-Lipschitz conditions, namely, we study the behaviors of small subsets under the flows generated by these equations.     

Existence, uniqueness, and quenching properties of solutions for degenerate semilinear parabolic problems with second boundary conditions

Let q?0, p?0, T?∞, D=(0,a), , Ω=D×(0,T), and Lu=xqut−uxx. This article considers the following degenerate semilinear parabolic initial-boundary value problem,

     

Existence of solutions for a higher-order semilinear parabolic equation with singular initial data

We establish the existence of solutions of the Cauchy problem for a higher-order semilinear parabolic equation by introducing a new majorizing kernel. We also study necessary conditions on the initial data for the existence of local-in-time solutions and identify the strongest singularity of the initial data for the solvability of the Cauchy problem.     

Existence and uniqueness of weak solutions for a non-uniformly parabolic equation

In this paper we develop a unifying method to prove the existence and uniqueness of weak solutions for the initial-boundary value problem of a non-uniformly parabolic equation. Some well-known parabolic equations are the special cases of this equation.     

Existence and uniqueness of local solutions to a class of quasilinear degenerate parabolic systems

In this paper, the authors study an initial and boundary value problem to a system of evolution p-Laplacian equations coupled with general nonlinear terms. The authors use the method of regularization to construct a sequence of approximation solutions with the help of monotone iteration technique and hence obtain the existence of solutions to a regularized system of equations. Then the existence of solutions to the system of evolution p-Laplacian equations is obtained with the use of a standard limiting process. The uniqueness of the solution is also proven.     

Existence of positive solutions for some nonlinear parabolic systems in exteriors domains

We prove some existence results of positive continuous solutions to the semilinear parabolic system , in an unbounded domain D with compact boundary subject to some Dirichlet conditions, where λ and μ are nonnegative parameters. The functions f, g are nonnegative continuous monotone on (0,∞) and the potentials p, q are nonnegative and satisfy some hypotheses related to the parabolic Kato class J^∞(D).     

Existence and uniqueness of weak solutions for a fourth-order nonlinear parabolic equation

In this paper we establish the existence and uniqueness of weak solutions for the initial-boundary value problem of a fourth-order nonlinear parabolic equation.     

Homogenization of periodic semilinear parabolic degenerate PDEs

In this paper a second order semilinear parabolic PDE with rapidly oscillating coefficients is homogenized. The novelty of our result lies in the fact that we allow the second order part of the differential operator to be degenerate in some part of Rd${\mathbf{R}}^d$.     

Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order

In this paper, we prove some existence and uniqueness of mild solutions for a semilinear integrodifferential equation of fractional order with nonlocal initial conditions in α-norm. We assume that the linear part generates a noncompact analytic semigroup. Our results cover the cases that the nonlinearity F takes values in different spaces such as X,X_α and X_β, where α≤β(0,1). Finally, some practical consequences are also obtained.     

On uniqueness for a semilinear parabolic system coupled in an equation and a boundary condition

We consider the system

     

Euler-Maruyama approximations for SDEs with non-Lipschitz coefficients and applications

In this paper Euler-Maruyama approximation for SDE with non-Lipschitz coefficients is proved to converge uniformly to the solution in Lp-space with respect to the time and starting points. As an application, we also study the existence of solution and large deviation principle for anticipative SDE with random initial condition.     

Existence and uniqueness of positive solutions for a class of semilinear elliptic systems

The authors prove the uniqueness and existence of positive solutions for the semilinear elliptic system which involves nonlinearities with sublinear growth conditions.     

Strong solutions for jump-type stochastic differential equations with non-Lipschitz coefficients

In this paper, the existence and pathwise uniqueness of strong solutions for jump-type stochastic differential equations are investigated under non-Lipschitz conditions. A sufficient condition is obtained for ensuring the non-confluent property of strong solutions of jump-type stochastic differential equations. Moreover, some examples are given to illustrate our results.     

Oscillation of solutions for certain impulsive vector parabolic differential equations with delays

In this paper, sufficient conditions are established for H-oscillation of solutions of the following impulsive vector parabolic differential equations with delays