TY  - JOUR
AB  - The 1D Burger's equation with Dirichlet boundary conditions exhibits a first transition from the trivial steady state to a sinusoidal patterned steady state as the parameter lambda which controls the linear term exceeds 1. The main goal of this paper is to present two different approaches regarding the transition of this patterned steady state. We believe that these approaches can be extended to study the dynamics of more interesting models. As a first approach, we consider an external forcing on the equation which supports a sinusoidal solution as a stable steady state which loses its stability at a critical threshold. We use the method of continued fractions to rigorously analyze the associated linear problem. In particular, we find that the system exhibits a mixed type transition with two distinct basins for initial conditions one of which leads to a local steady state and the other leaves a small neighborhood of the origin. As a second approach, we consider the dynamics on the center-unstable manifold of the first two modes of the unforced system. In this approach, the secondary transition produces two branches of steady state solutions. On one of these branches there is another transition which indicates a symmetry breaking phenomena.
AU  - Efendiyev, M.A.
AU  - Sengul, T.*
AU  - Tiryakioglu, B.*
C1  - 69167
C2  - 53852
CY  - Po Box 2604, Springfield, Mo 65801-2604, United States
SP  - 1621-1638
TI  - Two approaches to instability analysis of the viscous Burgers' equation.
JO  - Discret. Contin. Dyn. Syst.-Ser. S
VL  - 17
IS  - 4
PB  - Amer Inst Mathematical Sciences-aims
PY  - 2023
SN  - 1937-1632
ER  - 

TY  - JOUR
AB  - We demonstrate the existence in the sense of sequences of solutions for some integro-differential type problems in a square in two dimensions with periodic boundary conditions. They contain the normal diffusion in one direction and the superdiffusion in the other direction. We work in a constrained subspace of H2 using the fixed point technique. The elliptic equation involves a second order differential operator satisfying the Fredholm property. It is established that, under reasonable technical assumptions, the convergence in the appropriate function spaces of the integral kernels yields the existence and convergence in H02 of the solutions. We generalize the results obtained in our preceding work [11] for the analogous equation considered in the whole R2 which contained a non-Fredholm operator.
AU  - Efendiyev, M.A.
AU  - Vougalter, V.*
C1  - 69411
C2  - 53859
CY  - Po Box 2604, Springfield, Mo 65801-2604, United States
TI  - Solvability of some Fredholm integro-differential equations with mixed diffusion in a square.
JO  - Discret. Contin. Dyn. Syst.-Ser. S
PB  - Amer Inst Mathematical Sciences-aims
PY  - 2023
SN  - 1937-1632
ER  -