Strong solutions to stochastic Volterra equations

In this paper stochastic Volterra equations admitting exponentially bounded resolvents are studied. After obtaining convergence of resolvents, some properties for stochastic convolutions are studied. Our main results provide sufficient conditions for strong solutions to stochastic Volterra equations.     

Finite-dimensional attractors for the Kirchhoff equation with a strong dissipation

The paper studies the existence of the finite-dimensional global attractors and exponential attractors for the dynamical system associated with the Kirchhoff type equation with a strong dissipation utt−M(‖∇u^‖2)Δu−Δut+h(ut)+g(u)=f(x). It proves that the above mentioned dynamical system possesses a global attractor which has finite fractal dimension and an exponential attractor. For application, the fact shows that for the concerned viscoelastic flow the permanent regime (global attractor) can be observed when the excitation starts from any bounded set in phase space, and the dimension of the attractor, that is, the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow, is finite.     

Trajectory attractors for nonclassical diffusion equations with fading memory

In this article, we consider the existence of trajectory and global attractors for nonclassical diffusion equations with linear fading memory. For this purpose, we will apply the method presented by Chepyzhov and Miranville [7,8], in which the authors provide some new ideas in describing the trajectory attractors for evolution equations with memory.     

On the structure of positive solutions to an elliptic problem arising in thin film equations

We consider the structure of positive radial solutions of

     

Strong solutions to a parabolic equation with linear growth with respect to the gradient variable

In this paper we prove existence and uniqueness of strong solutions to the homogeneous Neumann problem associated to a parabolic equation with linear growth with respect to the gradient variable. This equation is a generalization of the time-dependent minimal surface equation. Existence and regularity in time of the solution is proved by means of a suitable pseudoparabolic relaxed approximation of the equation and a passage to the limit.     

Stability of attractors for the Kirchhoff wave equation with strong damping and critical nonlinearities

The paper investigates longtime dynamics of the Kirchhoff wave equation with strong damping and critical nonlinearities: $u_{tt}?(1+?{\Vert \mathrm{?}u\Vert }^2)\mathrm{\Delta }u?\mathrm{\Delta }{u}_t+h(u_t)+g(u)=f(x)$, with $?\in [0,1]$. The well-posedness and the existence of global and exponential attractors are established, and the stability of the attractors on the perturbation parameter ? is proved for the IBVP of the equation provided that both nonlinearities $h(s)$ and $g(s)$ are of critical growth.     

Pullback attractors for the non-autonomous Benjamin-Bona-Mahony equation in unbounded domains

The existence of a pullback attractor is proven for the non-autonomous Benjamin-Bona-Mahony equation in unbounded domains.The asymptotic compactness of the solution operator is obtained by the uniform estimates on the tails of solutions.     

Strong solutions to the Richards equation in the unsaturated zone

We study the unsaturated case of the Richards equation in three space dimensions with Dirichlet boundary data. We first establish an a priori L^∞-estimate. With its help, by means of a fixed point argument we prove global in time existence of a unique weak solution in Sobolev spaces. Finally, we are able to improve the regularity of this weak solution in order to gain a strong one.     

Strong solutions to the equations of electrically conductive magnetic fluids

We study the equations of flow of an electrically conductive magnetic fluid, when the fluid is subjected to the action of an external applied magnetic field. The system is formed by the incompressible Navier–Stokes equations, the magnetization relaxation equation of Bloch type and the magnetic induction equation. The system takes into account the Kelvin and Lorentz force densities. We prove the local-in-time existence of the unique strong solution to the system equipped with initial and boundary conditions. We also establish a blow-up criterion for the local strong solution.     

Rainbow solutions to the Sidon equation

We prove that for every 4-coloring of {1,2,…,n}, with each color class having cardinality more than (n+1)/6, there exists a solution of the equation x+y=z+w with x, y, z and w belonging to different color classes. The lower bound on a color class cardinality is tight.     

Pullback attractors for a nonautonomous damped wave equation with infinite delays in weighted space

In this paper, we investigate the existence of solutions for a damped wave equation with infinite delays in the weighted space and , respectively. In addition, we study the existence of pullback attractor for the process associated to the problem by a direct and simple compactness method.     

Invariant sets and solutions to the generalized thin film equation

The invariant sets and the solutions of the 1 2-dimensional generalized thin film equation are discussed. It is shown that there exists a class of solutions to the equations, which are invariant with respect to the set E0 = {u : ux = vxF(u), uy = vyF(u)}, where v is a smooth function of variables x, y and F is a smooth function of u. This extends the results of Galaktionov (2001) and for the l l-dimensional nonlinear evolution equations.     

Global solutions of the fast diffusion equation with gradient absorption terms

We investigate the existence of nonnegative weak solutions to the problem ut=Δ(um)−p|∇u| in Rn×(0,∞) with ⁺(1−2/n)<m<1. It will be proved that: (i) When 1<p<2, if the initial datum u₀∈D(Rn) then there exists a solution; (ii) When 1<p<(2+mn)/(n+1), if the initial datum u₀(x) is a bounded and nonnegative measure then the solution exists; (iii) When (2+mn)/(n+1)?p<2, if the initial datum is a Dirac mass then the solution does not exist. We also study the large time behavior of the L¹-norm of solutions for 1<p?(2+mn)/(n+1), and the large time behavior of t1/β‖u(⋅,t)−Ec(⋅,t)L^∞_‖ for (2+mn)/(n+1)<p<2.     

Strong stability and uniform decay of solutions to a wave equation with semilinear porous acoustic boundary conditions

We consider a wave equation with semilinear porous acoustic boundary conditions. This is a coupled system of second and first order in time partial differential equations, with possibly semilinear boundary conditions on the interface. The results obtained are (i) strong stability for the linear model, (ii) exponential decay rates for the energy of the linear model, and (iii) local exponential decay rates for the energy of the semilinear model. This work builds on a previous result showing generation of a well-posed dynamical system. The main tools used in the proofs are (i) the Stability Theorem of Arendt-Batty, (ii) energy methods used in the study of a wave equation with boundary damping, and (iii) an abstract result of I. Lasiecka applicable to hyperbolic-like systems with nonlinearly perturbed boundary conditions.     

Finite dimensionality of global attractors for a non-classical reaction-diffusion equation with memory

In this paper, we consider a periodic boundary value problem for a non-classical reaction-diffusion equation with memory. In other paper, we use the ω-limit compactness of the solution semigroup {S(t)}_t≥0 to get the existence of a global attractor. The main goal here is to give an estimate of the fractal dimension of the global attractor. By the fractal dimension theorem given by A.O. Celebi et al., we obtain that the fractal dimension of the global attractor for the problem is finite; this makes the results for the non-classical reaction-diffusion equations more substantial and perfect.     

一类带有非牛顿位势的可压缩Navier-Stokes方程整体强解的存在性

主要研究了一类带有非牛顿位势的可压缩Navier-Stokes方程:其中粘性系数μ依赖于密度ρ,Φ是非牛顿位势.证明了上述问题的强解的存在性.在相容性条件下,得到了强解的唯一性.     

The existence of global attractors for a class nonlinear evolution equation

In this paper, using a new method (or framework), we establish the existence of global attractors for a class nonlinear evolution equation in , where the nonlinear term f satisfies a critical exponential growth condition.