Anton Savostianov

• Doctor of Philosophy
• Researcher at Uppsala University

The paper gives a comprehensive study of infinite-energy solutions and their long-time behavior for semi-linear weakly damped wave equations in R 3 \mathbb {R}^3 with quintic nonlinearities. This study includes global well-posedness of the so-called Shatah-Struwe solutions, their dissipativity, the existence of a locally compact global attractors (...

The paper gives a comprehensive study of infinite-energy solutions and their long-time behavior for semi-linear weakly damped wave equations in $\mathbb{R}^3$ with quintic nonlinearities. This study includes global well-posedness of the so-called Shatah-Struwe solutions, their dissipativity, the existence of a locally compact global attractors (in...

This is a detailed study of damped quintic wave equations with non-regular and non-autonomous external forces which are measures in time. In the 3D case with periodic boundary conditions, uniform energy-to- Strichartz estimates are established for the solutions, the existence of uniform attractors in a weak or strong topology in the energy phase sp...

В работе исследованы диссипативные волновые уравнения с нелинейностью пятой степени и нерегулярными неавтономными внешними силами, которые являются мерами по времени. В случае трeхмерной области и периодических граничных условий получены оценки норм Штрихарца решений через соответствующие энергетические нормы, доказано существование равномерных атт...

Homogenisation of global 𝓐ε and exponential 𝓜ε attractors for the damped semi-linear anisotropic wave equation ∂t2uε+y∂tuε−divaxε∇uε+f(uε)=g,$\begin{array}{} \displaystyle \partial_t ^2u^\varepsilon + y \partial_t u^\varepsilon-\operatorname{div} \left(a\left( \tfrac{x}{\varepsilon} \right)\nabla u^\varepsilon \right)+f(u^\varepsilon)=g, \end{array...

We give a detailed study of attractors for measure driven quintic damped wave equations with periodic boundary conditions. This includes uniform energy-to-Strichartz estimates, the existence of uniform attractors in a weak or strong topology in the energy phase space, the possibility to present them as a union of all complete trajectories, further...

Homogenisation of global A^ε and exponential M^ε attractors for the damped semi-linear anisotropic wave equation ∂t^2 u_ε +γ∂t u_ε −div a (x/ε) u_ε +f (u_ε) = g, on a bounded domain Ω ⊂ R^3 , is performed. Order-sharp estimates between trajectories u_ε(t) and their homogenised trajectories u_0(t) are established. These estimates are given in terms...

Homogenisation of global $\mathcal{A}^\epsilon$ and exponential $\mathcal{M}^\epsilon$ attractors for the damped semi-linear anisotropic wave equation $\partial_t^2 u^\epsilon +\gamma\partial_t u^\epsilon-{\rm div} \left(a\left( \tfrac{x}{\epsilon} \right)\nabla u^\epsilon \right)+f(u^\epsilon)=g$, on a bounded domain $\Omega \subset \mathbb{R}^3$,...

This work is devoted to infinite-energy solutions of semi-linear wave equations in unbounded smooth domains of $\mathbb{R}^3$ with fractional damping of the form $(-\Delta_x+1)^\frac{1}{2}\partial_t u$. The work extends previously known results for bounded domains in finite energy case. Furthermore, well-posedness and existence of locally-compact s...

In this paper, we continue the study of the hyperbolic relaxation of the Cahn-Hilliard-Oono equation with the sub-quintic non-linearity in the whole space R 3 started in our previous paper and verify that under the natural assumptions on the non-linearity and the external force, the fractal dimension of the associated global attractor in the natura...

The work is devoted to Dirichlet problem for sub-quintic semi-linear wave equation with damping damping term of the form $(-\Delta)^\alpha\partial_t u$, $\alpha\in(0,\frac{1}{2})$, in bounded smooth domains of $\Bbb R^3$. It appears that to prove well-posedness and develop smooth attractor theory for the problem we need additional regularity of the...

In this paper, we continue the study of the hyperbolic relaxation of the Cahn-Hilliard-Oono equation with the sub-quintic non-linearity in the whole space $\R^3$ started in our previous paper and verify that under the natural assumptions on the non-linearity and the external force, the fractal dimension of the associated global attractor in the nat...

We prove the global well-posedness of the so-called hyperbolic relaxation of the Cahn–Hilliard–Oono equation in the whole space ℝ3 with the nonlinearity of the sub-quintic growth rate. Moreover, the dissipativity and the existence of a smooth global attractor in the naturally defined energy space is also verified. The result is crucially based on t...

Dissipative wave equations with critical quintic nonlinearity and damping term involving the fractional Laplacian are considered. The additional regularity of energy solutions is established by constructing the new Lyapunov-type functional and based on this, the global well-posedness and dissipativity of the energy solutions as well as the existenc...

We report on new results concerning the global well-posedness, dissipativity and attractors of the damped quintic wave equations in bounded domains of R^3.

The dissipative wave equation with a critical quintic nonlinearity in smooth bounded three dimensional domain is considered. Based on the recent extension of the Strichartz estimates to the case of bounded domains, the existence of a compact global attractor for the solution semigroup of this equation is established. Moreover, the smoothness of the...