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Consider weakly nonlinear complex Ginzburg--Landau (CGL) equation of the form: $$ u_t+i(-\Delta u+V(x)u)=\epsilon\mu\Delta u+\epsilon \mathcal{P}( u),\quad x\in {R^d}\,, \quad(*) $$ under the periodic boundary conditions, where…

We consider asymptotic stability of a small solitary wave to supercritical 2-dimensional nonlinear Schr\"{o}dinger equations $$ iu_t+\Delta u=Vu\pm |u|^{p-1}u \quad\text{for $(x,t)\in\mathbb{R}^2\times\mathbb{R}$,}$$ in the energy class.

We study stability of solutions of the Cauchy problem on the line for the Camassa-Holm equation $u_t-u_{xxt}+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ with initial data $u_0$. In particular, we derive a new Lipschitz metric $d_\D$ with the property that…

We generalize the Beurling--Deny--Ouhabaz criterion for parabolic evolution equations governed by forms to the non-autonomous, non-homogeneous and semilinear case. Let $V, H$ are Hilbert spaces such that $V$ is continuously and densely…

This paper is concerned with the asymptotic behavior of bounded solutions of the Cauchy problem \begin{equation*} \left\{ \begin{array}{ll} u_t=u_{xx} +f(t,u), & x\in\mathbb{R},\,t>0,\\ u(x,0)= u_0, & x\in\mathbb{R}, \end{array}\right.…

In this paper we investigate a discrete approximation in time and in space of a Hilbert space valued stochastic process $\{u(t)\}_{t\in [0,T]}$ satisfying a stochastic linear evolution equation with a positive-type memory term driven by an…

We examine the fourth order problem $\Delta^2 u = \lambda f(u) $ in $ \Omega$ with $ \Delta u = u =0 $ on $ \partial \Omega$, where $ \lambda > 0$ is a parameter, $ \Omega$ is a bounded domain in $ R^N$ and where $f$ is one of the following…

We investigate the following fractional order in time Cauchy problem \begin{equation*} \begin{cases} \mathbb{D}_{t}^{\alpha }u(t)+Au(t)=f(u(t)), & 1<\alpha <2, \\ u(0)=u_{0},\,\,\,u^{\prime }(0)=u_{1}. & \end{cases}% \end{equation*}% where…

In a separable Hilbert space $X$, we study the controlled evolution equation \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0, \end{equation*} where $A\geq-\sigma I$ ($\sigma\geq0$) is a self-adjoint linear operator, $B$ is a bounded linear…

We consider the second order Cauchy problem $$u''+\m{u}Au=0, u(0)=u_{0}, u'(0)=u_{1},$$ where $m:[0,+\infty)\to[0,+\infty)$ is a continuous function, and $A$ is a self-adjoint nonnegative operator with dense domain on a Hilbert space. It is…

In this paper we study a convection-reaction-diffusion equation of the form \begin{equation*} u_t=\varepsilon(h(u)u_x)_x-f(u)_x+f'(u), \quad t>0, \end{equation*} with a nonlinear diffusion in a bounded interval of the real line. In…

The asymptotic stable region and long-time decay rate of solutions to linear homogeneous Caputo time fractional ordinary differential equations (F-ODEs) are known to be completely determined by the eigenvalues of the coefficient matrix.…

We consider periodic homogenization of boundary value problems for quasilinear second-order ODE systems in divergence form of the type $a(x,x/\varepsilon,u(x),u'(x))'= f(x,x/\varepsilon,u(x),u'(x))$ for $x \in [0,1]$. For small…

New explicit conditions of asymptotic and exponential stability are obtained for the scalar nonautonomous linear delay differential equation $$ \dot{x}(t)+\sum_{k=1}^m a_k(t)x(h_k(t))=0 $$ with measurable delays and coefficients. These…

Fractional difference equations provide a flexible mathematical framework for modeling complex systems with memory, hereditary, and non-local effects. In this work, we study the stability of higher-order two-term fractional linear…

In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form $D^{\alpha}_Cu(t)=Au(t)+f(t)$ on the half line, where $D^{\alpha}_Cu(t)$ is the derivative of the function $u$ in Caputo's sense,…

We consider systems of stochastic evolutionary equations of the type $$du=\mathrm{div}\,S(\nabla u)\,dt+\Phi(u)dW_t$$ where $S$ is a non-linear operator, for instance the $p$-Laplacian $$S(\xi)=(1+|\xi|)^{p-2}\xi,\quad \xi\in\mathbb…

Some uniform decay estimates are established for solutions of the following type of retarded integral inequalities: $$y(t)\leq E(t,\tau)||y_\tau||+\int_\tau^t K_1(t,s)||y_s||ds+\int_t^\infty K_2(t,s)||y_s||ds+\rho, \hspace{0.5cm}…

In this paper we derive new criterion for uniform stability assessment of the linear periodic time-varying systems $\dot x=A(t)x,$ $A(t+T)=A(t).$ As a corollary, the lower and upper bounds for the Floquet characteristic exponents are…

We examine the weighted elliptic system \begin{equation*} \begin{cases} -\Delta u=(1+|x|^2)^{\frac{\alpha}{2}} v,\\ -\Delta v=(1+|x|^2)^{\frac{\alpha}{2}} u^p, \end{cases} \quad \mbox{in}\;\ \mathbb{R}^N, \end{equation*}where $N \ge 5$,…