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 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.

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 A reaction infiltration problem: classical solutions

1997 ◽  
Vol 40 (2) ◽  
pp. 275-291 ◽  
Author(s):  
John Chadam ◽  
Xinfu Chen ◽  
Roberto Gianni ◽  
Riccardo Ricci
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Parabolic Equation ◽  
Elliptic Equation ◽  
Boundary Conditions ◽  
Classical Solution ◽  
Existence And Uniqueness ◽  
Classical Solutions ◽  
Bounded Domains ◽  
Global Classical Solution

In this paper, we consider a reaction infiltration problem consisting of a parabolic equation for the concentration, an elliptic equation for the pressure, and an ordinary differential equation for the porosity. Existence and uniqueness of a global classical solution is proved for bounded domains Ω⊂RN, under suitable boundary conditions.

scholarly journals Well-Posedness and Numerical Study for Solutions of a Parabolic Equation with Variable-Exponent Nonlinearities

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Jamal H. Al-Smail ◽  
Salim A. Messaoudi ◽  
Ala A. Talahmeh
Keyword(s):  
Parabolic Equation ◽  
Operator Theory ◽  
Existence And Uniqueness ◽  
Variable Exponent ◽  
Nonlinear Operator ◽  
Numerical Study ◽  
Well Posedness ◽  
Two Dimensional ◽  
Nonlinear Parabolic ◽  
Decay Of Solutions

We consider the following nonlinear parabolic equation: ut-div(|∇u|p(x)-2∇u)=f(x,t), where f:Ω×(0,T)→R and the exponent of nonlinearity p(·) are given functions. By using a nonlinear operator theory, we prove the existence and uniqueness of weak solutions under suitable assumptions. We also give a two-dimensional numerical example to illustrate the decay of solutions.

scholarly journals On the Estimates in Various Spaces to the Control Function of the Extremum Problem for Parabolic Equation

2021 ◽  
Vol 16 ◽  
pp. 187-192
Author(s):  
Irina Astashova ◽  
Alexey Filinovskiy ◽  
Dmitriy Lashin
Keyword(s):  
Parabolic Equation ◽  
Free Convection ◽  
Existence And Uniqueness ◽  
Control Function ◽  
Extremum Problem ◽  
Convection Term ◽  
Temperature Deviation ◽  
One Dimensional ◽  
Control Functions ◽  
Quadratic Cost

For the minimization problem with pointwise observation governed by a one-dimensional parabolic equation with a free convection term and a depletion potential, we formulate a result on the existence and uniqueness of a minimizer from a prescribed set. We use a weight quadratic cost functional showing the temperature deviation. We obtain estimates for the norm of control functions in terms of the value of the quality functional in different functional spaces. It gives us a possibility to estimate the required internal energy of the system. To prove these results we establish the positivity principle.

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 Pullback Attractors for a Nonautonomous Retarded Degenerate Parabolic Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Fahe Miao ◽  
Hui Liu ◽  
Jie Xin
Keyword(s):  
Weak Solution ◽  
Parabolic Equation ◽  
Galerkin Method ◽  
Existence And Uniqueness ◽  
Degenerate Parabolic Equation ◽  
Pullback Attractors ◽  
Asymptotic Compactness ◽  
Absorbing Sets ◽  
Degenerate Parabolic ◽  
Standard Galerkin Method

This paper is devoted to a nonautonomous retarded degenerate parabolic equation. We first show the existence and uniqueness of a weak solution for the equation by using the standard Galerkin method. Then we establish the existence of pullback attractors for the equation by proving the existence of compact pullback absorbing sets and the pullback asymptotic compactness.

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.

Existence and decay rates of smooth solutions for a non-uniformly parabolic equation

2002 ◽  
Vol 132 (6) ◽  
pp. 1477-1491 ◽  
Author(s):  
Jinghua Wang ◽  
Hui Zhang
Keyword(s):  
Weak Solution ◽  
Parabolic Equation ◽  
Classical Solution ◽  
Initial Value Problem ◽  
Decay Rates ◽  
The Other ◽  
Successive Approximations ◽  
Smooth Solutions ◽  
Initial Value ◽  
Set Up

We obtain the existence and decay rates of the classical solution to the initial-value problem of a non-uniformly parabolic equation. Our method is to set up two equivalent sequences of the successive approximations. One converges to a weak solution of the initial-value problem; the other shows that the weak solution is the classical solution for t > 0. Moreover, we show how bounds of the derivatives to the classical solution depend explicitly on the interval with compact support in (0, ∞). Then we study decay rates of this classical solution.

scholarly journals On reflection principles for parabolic equations in one space variable

1978 ◽  
Vol 21 (2) ◽  
pp. 143-147 ◽  
Author(s):  
David Colton
Keyword(s):  
Parabolic Equation ◽  
Classical Solution ◽  
Parabolic Equations ◽  
Boundary Data ◽  
Space Variable ◽  
Reflection Principles ◽  
Image Position

In this note we shall consider the problem of uniquely continuing solutions of the parabolic equationacross an analytic arc σ: x=s1(t) satisfies the boundary dataWe assume that u(x,t) is a classical solution of (1) in the domain D ={(x,t): s1(t)< x < s 2(t), 0 < t < t0}, continuously differentiate in D ∪ σ and define the “reflection” of D across σ by

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