Boundary Value Problems For Parabolic Equations Research Articles

Internal and Startup Controls of the Solutions of Boundary-Value Problem for Parabolic Equations with Degenerations

For a second-order parabolic equation with degenerations, we construct the solution of the problem of optimal control for systems described by the first boundary-value problem with internal and startup controls. The coefficients of parabolic equation have power singularities of any order in time and space variables on a certain set of points.

Internal and start controls of solutions of the boundary-value problem for parabolic equations with degenerations

Internal and start controls of solutions of the boundary-value problem for parabolic equations with degenerations

Coercive Solvability of Nonlocal Boundary-Value Problems for Parabolic Equations

In an arbitrary Banach space E, we consider the nonlocal problem $$ {\displaystyle \begin{array}{l}\upsilon^{\prime }(t)+A(t)\upsilon (t)=f(t)\kern1em \left(0\le t\le 1\right),\\ {}\upsilon (0)=\upsilon \left(\uplambda \right)+\mu \kern1em \left(0<\uplambda \le 1\right)\end{array}} $$ for an abstract parabolic equation with a linear unbounded strongly positive operator A(t) such that its domain D = D(A(t)) is independent of t and is everywhere dense in E. This operator generates an analytic semigroup exp{−sA(t)}(s ≥ 0). We prove the coercive solvability of the problem in the Banach space $$ {C}_0^{\alpha, \alpha}\left(\left[0,1\right],E\right)\left(0<\alpha <1\right) $$ with weight (t + τ)α. Earlier, this result was known only for constant operators. We consider applications in the class of parabolic functional differential equations with transformation of spatial variables and in the class of parabolic equations with nonlocal conditions on the boundary of the domain. Thus, this describes parabolic equations with nonlocal conditions both with respect to time and with respect to spatial variables.

Boundary-Value Problem for Parabolic Equations with Impulsive Conditions and Degenerations

By using the maximum principle and a priori estimates, we study the first boundary-value problem for a linear parabolic equation with power singularities in coefficients with respect to the space variables and impulsive conditions with respect to the time variable. In Holder spaces with power weights, we establish the existence and uniqueness of the solution of the posed problem.

Positivity of a differential operator with nonlocal conditions

In the present paper, the positivity of the differential operator with nonlocal boundary conditions is established. The structure of fractional spaces is investigated. In applications, we will obtain new coercive inequalities for the solution of local and nonlocal boundary value problems for parabolic equations.

The boundary-value problems for parabolic equations with a nonlocal condition and degenerations

With the help of the principle of maximum and the a priori estimates, we consider the first boundary-value problem and the problem with directional derivative under a nonlocal condition in the time variable for parabolic equations with power singularities of the coefficients in the time and space variables. The existence and the uniqueness of solutions of the posed problems are proved for the Holder spaces with power weight.

Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point

The Dirichlet problem for the heat equation in a bounded domain $$ \mathcal{G} $$ ⊂ ℝ n+1 is characteristic because there are boundary points at which the boundary touches a characteristic hyperplane t = c, where c is a constant. For the first time, necessary and sufficient conditions on the boundary guaranteeing that the solution is continuous up to the characteristic point were established by Petrovskii (1934) under the assumption that the Dirichlet data are continuous. The appearance of Petrovskii’s paper was stimulated by the existing interest to the investigation of general boundary-value problems for parabolic equations in bounded domains. We contribute to the study of this problem by finding a formal solution of the Dirichlet problem for the heat equation in a neighborhood of a cuspidal characteristic boundary point and analyzing its asymptotic behavior.

Necessary and Sufficient Conditions of Stabilization of Solutions of the First Boundary-Value Problem for a Parabolic Equation

Necessary and sufficient conditions are established for the stabilization of the solution of the first boundary-value problem for a divergent second-order parabolic equation with no lower-order terms. For an arbitrary (possibly unbounded) domain Q ⊂ ℝN, N ≥ 3, with an initial function u 0(x) defined in Q for t = 0, one studies the influence of Q on the stabilization of the solution in the cylinder Z = Q × [0,∞) with the homogeneous boundary conditions on the lateral surface of Z.

On the problem of determining the parameter of a parabolic equation

We study the boundary-value problem of determining the parameter p of a parabolic equation $$ v^{\prime}(t) + Av(t) = f(t) + p,\quad 0 \leqslant t \leqslant 1,\quad v(0) = \varphi, \quad v(1) = \psi, $$ with strongly positive operator A in an arbitrary Banach space E. The exact estimates are established for the solution of this problem in Hölder norms. In applications, the exact estimates are obtained for the solutions of the boundary-value problems for parabolic equations.

Solutions in Hölder spaces of boundary-value problems for parabolic equations with nonconjugate initial and boundary data

The first and second one-dimensional boundary-value problems for parabolic equations are investigated in the case where the conjugation conditions for all required orders are not satisfied. The existence and uniqueness are proved. Estimates of solutions in classical and weighted Holder spaces are obtained. We prove that the violation of conjugation for the given functions on the boundary of the domain at the initial-time moment causes the appearance of singular solutions. The order of singularity (as a power of t) is found for the singular solutions for t = 0.

Nonlocal boundary-value problems for abstract parabolic equations: well-posedness in Bochner spaces

The well-posedness of the nonlocal boundary-value problem for abstract parabolic differential equations in Bochner spaces is established. The first and second order of accuracy difference schemes for the approximate solutions of this problem are considered. The coercive inequalities for the solutions of these difference schemes are established. In applications, the almost coercive stability and coercive stability estimates for the solutions of difference schemes for the approximate solutions of the nonlocal boundary-value problem for parabolic equation are obtained.

Boundary-Value Problem for Linear Parabolic Equations with Degeneracies

In spaces of classical functions with power weight, we prove the correct solvability of a boundary-value problem for parabolic equations with an arbitrary power order of degeneracy of coefficients with respect to both time and space variables.

On the Asymptotic Behavior of Solutions of the First Initial Boundary-Value Problems for Parabolic Equations

We consider the first initial boundary-value problem for a strongly parabolic system on an infinite cylinder with nonsmooth boundary. We prove some results on the existence, uniqueness, and asymptotic behavior of solutions as t → ∞.

Neumann–Lame–Clapeyron–Stefan Problem and Its Solution Using Fractional Differential-Integral Calculus

A method is proposed to solve boundary-value problems for parabolic equations with unknown boundaries (where the solutions join) moving according to time-dependent laws. This method is based on fractional differential-integral calculus. An integral relation between the temperature and velocity of the moving interface is put forward. Necessary examples are given.

One-Sided Nonlocal Boundary-Value Problem for Singular Parabolic Equations

In spaces of classical functions with power weight, we prove the existence and uniqueness of a solution of a one-sided nonlocal boundary-value problem for parabolic equations with an arbitrary power order of degeneracy of coefficients. We obtain an estimate for the solution of this problem in the corresponding spaces.

A High-Order Difference Scheme for a Nonlocal Boundary-Value Problem for the Heat Equation

Abstract This paper is concerned with a high order difference scheme for a non- local boundary-value problem of parabolic equation. The integrals in the boundary equations are approximated by the composite Simpson rule. The unconditional solv- ability and L_∞ convergence of the difference scheme is proved by the energy method. The convergence rate of the difference scheme is second order in time and fourth order in space. Some numerical examples are provided to illustrate the convergence.

A boundary-value problem for parabolic equations with general nonlocal conditions

We study a boundary-value problem with general two-point conditions with respect to the time coordinate, and periodic conditions on the spatial coordinates for Shilov-parabolic equations with constant coefficients. We construct the solution in the form of a Fourier series. We establish conditions for existence and uniqueness of a classical solution of the problem. We prove quantitative theorems on a lower bound for the small denominators that arise in solving the problem.

Nonlocal boundary-value problem for parabolic equations with variable coefficients

We study the boundary-value problem for Petrovskii parabolic equations of arbitrary order with variable coefficients with conditions nonlocal in time. We establish conditions for the existence and uniqueness of a classical solution of this problem and prove metric theorems on lower bounds of small denominators appearing in the construction of a solution of the problem.

Nonlocal boundary-value problem for parabolic equations

We study the problem for Shilov parabolic equations of arbitrary order with constant coefficients with conditions nonlocal in time and periodic in space variables. We establish conditions for the existence and uniqueness of a classical solution of the problem and prove metric theorems on lower bounds of small denominators appearing in the construction of a solution of the problem.

Representations of solutions of nonlocal boundary-value problems for parabolic equations

We study two-point boundary-value problems for parabolic equations whose solutions are representable in terms of Green's functions of the Cauchy problem.