The Cauchy problem of the generalized Kuramoto-Sivashinsky equation in multidimensions(n ≥ 3) is considered. Based on Green's function method, some ingenious energy estimates are given. Then the global existence and pointwise convergence rates of the classical solutions are established. Furthermore, the L^p convergence rate of the solution is obtained.
In this paper,we consider a semilinear parabolic equation with a general nonlinearity.We establish a new finite time blow-up criterion and also derive the upper bound for the blow-up time.The results partially generalize some recent ones obtained by He Ma et al.
This paper studies the existence and long time behavior of the solutions to the coupled Burgers-complex Ginzburg-Landau （Burgers-CGL） equations, which are derived from the nonlinear evolution of the coupled long-scale oscillatory and monotonic instabilities of a uniformly propagating combustion wave governed by a sequential chem- ical reaction, having two flame fronts corresponding to two reaction zones with a finite separation distance between them. This paper firstly shows the existence of the global solutions to these coupled equations via subtle transforms, delicate a priori estimates and a so-called continuity method, then prove the existence of the global attractor and establish the estimates of the upper bounds of Hausdorff and fractal dimensions for the attractor.
In this paper, we consider the stochastic version of the 3D Bardina model arising from the turbulent flows of fluids. We obtain the existence of probabilistie weak solution for the model with the non-Lipschitz condition.
In this paper,a Riesz space fractional advection-dispersion equation with fractional Robin boundary condition is considered.By applying the fractional central di erence formula and the weighted and shifted Grunwald-Letnikov formula,we derive a weighted implicit nite difference scheme with accuracy O(△t^2+h^2).The solvability,stability,and convergence of the proposed numerical scheme are proved.A numerical example is presented to confirm the accuracy and efficiency of the scheme.
