Hypothesis on the Solvability of Parabolic Equations with nonlocal Condition

2002 ◽  
Vol 7 (1) ◽  
pp. 93-104 ◽  
Author(s):  
Mifodijus Sapagovas
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Nonlocal Condition ◽  
Nonlocal Conditions ◽  
Numerical Algorithms ◽  
Uniqueness Of The Solution

Numerous and different nonlocal conditions for the solvability of parabolic equations were researched in many articles and reports. The article presented analyzes such conditions imposed, and observes that the existence and uniqueness of the solution of parabolic equation is related mainly to ”smallness” of functions, involved in nonlocal conditions. As a consequence the hypothesis has been made, stating the assumptions on functions in nonlocal conditions are related to numerical algorithms of solving parabolic equations, and not to the parabolic equation itself.

scholarly journals The Cauchy Problem for Parabolic Equations with Degeneration

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Ivan Pukal’skii ◽  
Bohdan Yashan
Keyword(s):  
Boundary Condition ◽  
Cauchy Problem ◽  
Parabolic Equation ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Arbitrary Order ◽  
Power Weight ◽  
The Cauchy Problem ◽  
Uniqueness Of The Solution ◽  
Set Of Points

Annotation. For a second-order parabolic equation, the multipoint in time Cauchy problem is considered. The coefficients of the equation and the boundary condition have power singularities of arbitrary order in time and space variables on a certain set of points. Conditions for the existence and uniqueness of the solution of the problem in Hölder spaces with power weight are found.

scholarly journals On the existence and uniqueness of the solution of the coefficient inverse problem for a semilinear parabolic equation

2021 ◽  
Vol 2092 (1) ◽  
pp. 012008
Author(s):  
A L Sugezhik
Keyword(s):  
Inverse Problem ◽  
Parabolic Equation ◽  
Existence And Uniqueness ◽  
Input Data ◽  
Source Function ◽  
Semilinear Parabolic Equation ◽  
Coefficient Inverse Problem ◽  
Uniqueness Of The Solution ◽  
Bounded Functions ◽  
Semilinear Parabolic

Abstract In this paper, we consider the problem of determining the source function and the coefficient by the derivative with respect to time in a semilinear parabolic equation with overdetermination conditions defined on two different hyperplanes. The existence and uniqueness theorems of the classical solution of the posed coefficient inverse problem in the class of smooth bounded functions were proved. An example of input data satisfying the conditions of the proved theorems is given.

scholarly journals Well posedness for a class of ultra-parabolic equations with discontinuous flux

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 793-800
Author(s):  
Jela Susic
Keyword(s):  
Weak Solution ◽  
Parabolic Equation ◽  
Classical Solution ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Well Posedness ◽  
Convection Term ◽  
Discontinuous Flux ◽  
Parabolic Term

We prove existence and uniqueness of a weak solution to an ultra-parabolic equation with discontinuous convection term. Due to degeneracy in the parabolic term, the equation does not admit the classical solution. Equations of this type describe processes where transport is negligible in some directions.

scholarly journals Time discretization of nonlinear Cauchy problems applying to mixed hyperbolic-parabolic equations

1996 ◽  
Vol 19 (3) ◽  
pp. 481-494 ◽  
Author(s):  
Pierluigi Colli ◽  
Angelo Favini
Keyword(s):  
Hilbert Space ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Maximal Monotone ◽  
Time Discretization ◽  
Cauchy Problems ◽  
Initial Value ◽  
Selfadjoint Operators ◽  
Convergence Results ◽  
Uniqueness Of The Solution

In this paper we deal with the equationL(d2u/dt2)+B(du/dt)+Au∋f, whereLandAare linear positive selfadjoint operators in a Hilbert spaceHand from a Hilbert spaceV⊂Hto its dual spaceV′, respectively, andBis a maximal monotone operator fromVtoV′. By assuming some coerciveness onL+BandA, we state the existence and uniqueness of the solution for the corresponding initial value problem. An approximation via finite differences in time is provided and convergence results along with error estimates are presented.

scholarly journals On a nonlocal problem for parabolic equation with time dependent coefficients

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nguyen Duc Phuong ◽  
Ho Duy Binh ◽  
Le Dinh Long ◽  
Dang Van Yen
Keyword(s):  
Parabolic Equation ◽  
Mild Solution ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theory ◽  
Nonlocal Problem ◽  
Point Theory ◽  
Time Dependent ◽  
Well Posedness ◽  
Conformable Derivative

AbstractThis paper is devoted to the study of existence and uniqueness of a mild solution for a parabolic equation with conformable derivative. The nonlocal problem for parabolic equations appears in many various applications, such as physics, biology. The first part of this paper is to consider the well-posedness and regularity of the mild solution. The second one is to investigate the existence by using Banach fixed point theory.

scholarly journals The second boundary value problem in a half-strip for a B-parabolic equation with the Gerasimov–Caputo time derivative

2020 ◽  
pp. 37-50
Author(s):  
Ф.Г. Хуштова
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Integral Transform ◽  
Boundary Value ◽  
Time Derivative ◽  
Half Strip ◽  
Uniqueness Of The Solution ◽  
Solution Representation ◽  
Class Of Functions

В работе исследуется вторая краевая задача в полуполосе для параболического уравнения с оператором Бесселя, действующим по пространственной переменной, и частной производной Герасимова–Капуто по времени. Доказаны теоремы существования и единственности решения рассматриваемой задачи. Представление решения найдено в терминах интегрального преобразования с функцией Райта в ядре. Единственность решения доказана в классе функций быстрого роста. При частных значениях параметров, содержащихся в рассматриваемом уравнении, последнее совпадает с классическим уравнением диффузии. In the present paper, we investigate the second boundary value problem in a half-strip for a parabolic equation with the Bessel operator acting with respect to the spatial variable and the Gerasimov–Caputo partial time derivative. Theorems of existence and uniqueness of the solution of the problem under consideration are proved.The solution representation is found in terms of an integral transform with the Wright function in the kernel. The uniqueness of the solution is proved in the class of functions of rapid growth. The considered equation for particular values of the parameters coincides with the classical diffusion equation.

scholarly journals On weak solutions of semilinear hyperbolic-parabolic equations

1996 ◽  
Vol 19 (4) ◽  
pp. 751-758 ◽  
Author(s):  
Jorge Ferreira
Keyword(s):  
Parabolic Equation ◽  
Continuous Function ◽  
Weak Solutions ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
A Priori ◽  
A Priori Estimate ◽  
Dirichlet Boundary Conditions ◽  
Priori Estimate

In this paper we prove the existence and uniqueness of weak solutions of the mixed problem for the nonlinear hyperbolic-parabolic equation(K1(x,t)u′)′+K2(x,t)u′+A(t)u+F(u)=fwith null Dirichlet boundary conditions and zero initial data, whereF(s)is a continuous function such thatsF(s)≥0,∀s∈Rand{A(t);t≥0}is a family of operators ofL(H01(Ω);H−1(Ω)). For the existence we apply the Faedo-Galerkin method with an unusual a priori estimate and a result of W. A. Strauss. Uniqueness is proved only for some particular classes of functionsF.

scholarly journals M-MATRICES AND CONVERGENCE OF FINITE DIFFERENCE SCHEME FOR PARABOLIC EQUATION WITH INTEGRAL BOUNDARY CONDITION

2020 ◽  
Vol 25 (2) ◽  
pp. 167-183
Author(s):  
Regimantas Čiupaila ◽  
Mifodijus Sapagovas ◽  
Kristina Pupalaigė
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Nonlocal Condition ◽  
Nonlocal Conditions ◽  
Stability And Convergence ◽  
Semilinear Parabolic Equation ◽  
Difference Problem ◽  
Integral Boundary ◽  
The Stability ◽  
Semilinear Parabolic

In the paper, the stability and convergence of difference schemes approximating semilinear parabolic equation with a nonlocal condition are considered. The proof is based on the properties of M-matrices, not requiring the symmetry or diagonal predominance of difference problem. The main presumption is that all the eigenvalues of the corresponding difference problem with nonlocal conditions are positive.

scholarly journals A Mixed Problem with an Integral Two-Space-Variables Condition for Parabolic Equation with The Bessel Operator

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Bouziani Abdelfatah ◽  
Oussaeif Taki-Eddine ◽  
Ben Aoua Leila
Keyword(s):  
Parabolic Equation ◽  
Mixed Problem ◽  
Existence And Uniqueness ◽  
Weighted Sobolev Space ◽  
A Priori ◽  
Energy Inequality ◽  
A Priori Estimate ◽  
Bessel Operator ◽  
Priori Estimate ◽  
Uniqueness Of The Solution

We study a mixed problem with an integral two-space-variables condition for parabolic equation with the Bessel operator. The existence and uniqueness of the solution in functional weighted Sobolev space are proved. The proof is based on a priori estimate “energy inequality” and the density of the range of the operator generated by the problem considered.

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