Long-time asymptotics for the damped Boussinesq equation in a disk.
Time estimates for the Cauchy problem for a third-order hyperbolic equation.
Riesz potentials for Korteweg-de Vries solitons and Sturm-Liouville problems.
Asymptotic behavior of solutions of the damped Boussinesq equation in two space dimensions.
Long-time asymptotics of solutions for the second initial-boundary value problem for the damped Boussinesq equation.
Nonlinear Heat Equation with a Fractional Laplacian in a Disk
For the nonlinear heat equation with a fractional Laplacian , 1 < α ≤ 2, the first initial-boundary value problem in a disk is considered. Small initial conditions, homogeneous boundary conditions, and periodicity conditions in the angular coordinate are imposed. Existence and uniqueness of a global-in-time solution is proved, and the solution is constructed in the form of a series of eigenfunctions of the Laplace operator in the disk. First-order long-time asymptotics of the solution is obtained....
On the Kuramoto-Sivashinsky equation in a disk
We consider the first initial-boundary value problem for the 2-D Kuramoto-Sivashinsky equation in a unit disk with homogeneous boundary conditions, periodicity conditions in the angle, and small initial data. Apart from proving the existence and uniqueness of a global in time solution, we construct it in the form of a series in a small parameter present in the initial conditions. In the stable case we also obtain the uniform in space long-time asymptotic expansion of the constructed solution and...
Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball
The nonlinear heat equation with a fractional Laplacian , is considered in a unit ball . Homogeneous boundary conditions and small initial conditions are examined. For 3/2 + ε₁ ≤ α ≤ 2, where ε₁ > 0 is small, the global-in-time mild solution from the space with κ < α - 1/2 is constructed in the form of an eigenfunction expansion series. The uniqueness is proved for 0 < κ < α - 1/2, and the higher-order long-time asymptotics is calculated.
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