A nonlinear inequality and evolution problems

Assume that $g(t)\geq 0$, and $$\dot{g}(t)\leq -\gamma(t)g(t)+\alpha(t,g(t))+\beta(t),\ t\geq 0;\quad g(0)=g_0;\quad \dot{g}:=\frac{dg}{dt},$$ on any interval $[0,T)$ on which $g$ exists and has bounded derivative from the right, $\dot{g}(t):=\lim_{s\to +0}\frac{g(t+s)-g(t)}{s}$. It is assumed that $\gamma(t)$, and $\beta(t)$ are nonnegative continuous functions of $t$ defined on $\R_+:=[0,\infty)$, the function $\alpha(t,g)$ is defined for all $t\in \R_+$, locally Lipschitz with respect to $g$ uniformly with respect to $t$ on any compact subsets$[0,T]$, $T<\infty$, and non-decreasing with respect to $g$, $\alpha(t,g_1)\geq \alpha(t,g_2)$ if $g_1\ge g_2$. If there exists a function $\mu(t)>0$, $\mu(t)\in C^1(\R_+)$, such that $$\alpha\left(t,\frac{1}{\mu(t)}\right)+\beta(t)\leq \frac{1}{\mu(t)}\left(\gamma(t)-\frac{\dot{\mu}(t)}{\mu(t)}\right),\quad \forall t\ge 0;\quad \mu(0)g(0)\leq 1,$$ then $g(t)$ exists on all of $\R_+$, that is $T=\infty$, and the following estimate holds: $$0\leq g(t)\le \frac 1{\mu(t)},\quad \forall t\geq 0.$$ If $\mu(0)g(0)< 1$, then $0\leq g(t)< \frac 1{\mu(t)},\quad \forall t\geq 0.$ A discrete version of this result is obtained. The nonlinear inequality, obtained in this paper, is used in a study of the Lyapunov stability and asymptotic stability of solutions to differential equations in finite and infinite-dimensional spaces

Solving real time evolution problems by constructing excitation operators

In this paper we study the time evolution of an observable in the interacting fermion systems driven out of equilibrium. We present a method for solving the Heisenberg equations of motion by constructing excitation operators which are defined as the operators A satisfying [H,A]=\lambda A. It is demonstrated how an excitation operator and its excitation energy \lambda can be calculated. By an appropriate supposition of the form of A we turn the problem into the one of diagonalizing a series of matrices whose dimension depends linearly on the size of the system. We perform this method to calculate the evolution of the creation operator in a toy model Hamiltonian which is inspired by the Hubbard model and the nonequilibrium current through the single impurity Anderson model. This method is beyond the traditional perturbation theory in Keldysh-Green's function formalism, because the excitation energy \lambda is modified by the interaction and it will appear in the exponent in the function of time.Comment: 8 page

Nonlinear Evolution Problems

[no abstract available

Nonlinear Evolution Problems

In this workshop geometric evolution equations of parabolic type, nonlinear hyperbolic equations, and dispersive equations and their interrelations were the subject of 21 talks and several shorter special presentations

Nonlinear Evolution Problems

In this workshop three types of nonlinear evolution problems— geometric evolution equations (essentially of parabolic type), nonlinear hyperbolic equations, and dispersive equations— were the subject of 22 talks

Nonlinear Evolution Problems

[no abstract available