A generalization of Bitsadze–Samarskii problem for mixed type equation

We study a boundary-value problem with Bitsadze–Samarskii conditions on boundary characteristic on a special inner curve and on a segment of degeneration of mixed type equation. Its solvability is proved by method of integral equations, and uniqueness of solution is established by means of the maximum principle.     

Bitsadze–Samarskii problem for a parabolic system on the plane

The solvability (in classical sense) of the Bitsadze–Samarskii nonlocal initial–boundary value problem for a one-dimensional (in x) second-order parabolic system in a semibounded domain with a nonsmooth lateral boundary is proved by applying the method of boundary integral equations. The only condition imposed on the right-hand side of the nonlocal boundary condition is that it has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.     

On Periodic Solutions of Quasilinear Parabolic Equations with Boundary Conditions of Bitsadze–Samarskii Type

Doklady Mathematics - We consider a quasilinear parabolic boundary value problem with a nonlocal boundary condition of Bitsadze–Samarskii type. A theorem on the existence and uniqueness of a...     

Problem with a Bitsadze–Samarskii Condition on Parallel Characteristics for a Mixed Type Equation of the Second Kind

Differential Equations - For a mixed type equation of the second kind, we prove the uniqueness and existence of a solution of the boundary value problem with the Tricomi condition on part of the...     

Solution of the Bitsadze–Samarskii problem for a parabolic system in a semibounded domain

We consider a nonlocal initial–boundary value Bitsadze–Samarskii problem for a spatially one-dimensional parabolic second-order system in a semibounded domain with nonsmooth lateral boundary. The boundary integral equation method is used to construct a classical solution of this problem under the condition that the vector function on the right-hand side in the nonlocal boundary condition only has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.     

An Artificial Boundary Condition for the Vortex Movements in Two Dimensions

1 Motion of vortices and cloud in cell method The motion of incompressible inviscid ?ow in two dimensions can be described by the equations ?u ?t (u ?) ρ1 ?P = f (1) ?u = 0 , (2) where u = (u, v), ρ, P , and f = (f1, f2) denote ?uid velocity, densit…     

Sharp Estimates for the Global Attractor of Scalar Reaction–Diffusion Equations with a Wentzell Boundary Condition

In this paper, we derive optimal upper and lower bounds on the dimension of the attractor A_W\mathcal{A}_{\mathrm{W}} for scalar reaction–diffusion equations with a Wentzell (dynamic) boundary condition. We are also interested in obtaining explicit bounds on the constants involved in our asymptotic estimates, and to compare these bounds to previously known estimates for the dimension of the global attractor A_K\mathcal{A}_{K}, K∈{D,N,P}, of reaction–diffusion equations subject to Dirichlet, Neumann and periodic boundary conditions. The explicit estimates we obtain show that the dimension of the global attractor A_W\mathcal {A}_{\mathrm{W}} is of different order than the dimension of A_K\mathcal{A}_{K}, for each K∈{D,N,P}, in all space dimensions that are greater than or equal to three.     

Fujita Exponent for Porous Medium Equation with Convection and Nonlinear Boundary Condition

This paper is concerned with the critical exponent of the porous medium equation with convection and nonlinear boundary condition. It is shown that the coefficient of the lower order term is an important factor that determines the critical exponent.     

Nonlinear Transmission Problem for Wave Equation with?Boundary Condition of Memory Type

In this paper we consider a transmission problem with a boundary damping condition of memory type, that is, the wave propagation over bodies consisting of two physically different types of materials. One component is clamped, while the other is in a viscoelastic fluid producing a dissipative mechanism on the boundary. We will study the global existence of solutions for the transmission problem, and moreover we show that if the relaxation function decays exponentially or polynomially, then the solutions for the problem have the same decay rates.     

Existence for a Class of Non-Newtonian Fluids with a Nonlocal Friction Boundary Condition

We deal with a variational inequality describing the motion of incompressible fluids, whose viscous stress tensors belong to the subdifferential of a functional at the point given by the symmetric part of the velocity gradient, with a nonlocal friction condition on a part of the boundary obtained by a generalized mollification of the stresses. We establish an existence result of a solution to the nonlocal friction problem for this class of non-Newtonian flows. The result is based on the Faedo-Galerkin and Moreau Yosida methods, the duality theory of convex analysis and the Tychonov-Kakutani-Glicksberg fixed point theorem for multi-valued mappings in an appropriate functional space framework.     

P_1-Compact Mappings and Leary-Schauder Boundary Condition

Let X be a Banach space and let Γ_a={X_n,P_n} be a projectionally complete scheme. Let D be a bounded open subset of X and T:D→X an P_lcompact mapping undel the condition weaker than Leary Schauder boundarycondition, we show several fixed point theorems of T and consider he relation between the fixed points and eigenvectors of T.     

Nonlinear Degenerate Parabolic Equation with Nonlinear Boundary Condition

In this paper, we obtain the necessary and sufficient conditions on the global existence of all positive （weak） solutions to a nonlinear degenerate parabolic equation with nonlinear boundary condition.     

Boundary interface for the Allen–Cahn equation

We consider the Allen–Cahn equation

A Smallness Condition Ensuring Boundedness in a Two-dimensional Chemotaxis-Navier–Stokes System involving Dirichlet Boundary Conditions for the Signal

The chemotaxis-Navier-Stokes system ■is considered in a smoothly bounded planar domain Ω under the boundary conditions■ with a given nonnegative constant c_*.It is shown that if(n₀,c₀,u₀) is sufficiently regular and such that the product ■is suitably small,an associated initial value problem possesses a bounded classical solution with(n,c,u)|_t=0=(n₀,c₀,u₀).     

Tricomi problem for a functional-differential advanced-retarded Lavrent’ev–Bitsadze equation

We study a boundary value problem for the Lavrent’ev–Bitsadze equation with functional delay and advance. The general solution is constructed. The problem is uniquely solvable.     

Dirichlet Problem for Lavrent’ev–Bitsadze Equation With Loaded Summands

We study the first boundary-value problem for loaded equation of elliptic-hyperbolic type in rectangular domain. We establish a criterion of uniqueness. A solution to the problem is constructed in the formof the sum of a series. In substantiation of existence of a solution to a problem small denominators appear. We obtain the estimates about a separation from zero of denominators with the corresponding asymptotics. They allow to prove existence of a solution in a class of regular solutions.