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 Weak Solutions for Nonlinear Parabolic Equations with Variable Exponents

2017 ◽  
Vol 25 (1) ◽  
pp. 55-70 ◽  
Author(s):  
Lingeshwaran Shangerganesh ◽  
Arumugam Gurusamy ◽  
Krishnan Balachandran
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Variable Exponent ◽  
Degenerate Parabolic Equation ◽  
Nonlinear Parabolic Equations ◽  
Variable Exponents ◽  
Nonlinear Parabolic ◽  
Order Degenerate ◽  
Degenerate Parabolic

Abstract In this work, we study the existence and uniqueness of weak solu- tions of fourth-order degenerate parabolic equation with variable exponent using the di erence and variation methods.

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 the Analytical and Numerical Study of a Two-Dimensional Nonlinear Heat Equation with a Source Term

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 921
Author(s):  
Alexander Kazakov ◽  
Lev Spevak ◽  
Olga Nefedova ◽  
Anna Lempert
Keyword(s):  
Source Term ◽  
High Efficiency ◽  
Numerical Study ◽  
Nonlinear Heat Equation ◽  
Two Dimensional ◽  
Dual Reciprocity Method ◽  
Reciprocity Method ◽  
Nonlinear Parabolic ◽  
Solution Existence And Uniqueness ◽  
Dimensional Boundary

The paper deals with two-dimensional boundary-value problems for the degenerate nonlinear parabolic equation with a source term, which describes the process of heat conduction in the case of the power-law temperature dependence of the heat conductivity coefficient. We consider a heat wave propagation problem with a specified zero front in the case of two spatial variables. The solution existence and uniqueness theorem is proved in the class of analytic functions. The solution is constructed as a power series with coefficients to be calculated by a proposed constructive recurrent procedure. An algorithm based on the boundary element method using the dual reciprocity method is developed to solve the problem numerically. The efficiency of the application of the dual reciprocity method for various systems of radial basis functions is analyzed. An approach to constructing invariant solutions of the problem in the case of central symmetry is proposed. The constructed solutions are used to verify the developed numerical algorithm. The test calculations have shown the high efficiency of the algorithm.

Quasilinear parabolic problem with variable exponent: Qualitative analysis and stabilization

2018 ◽  
Vol 20 (08) ◽  
pp. 1750065 ◽  
Author(s):  
Jacques Giacomoni ◽  
Vicenţiu Rădulescu ◽  
Guillaume Warnault
Keyword(s):  
Weak Solution ◽  
Existence And Uniqueness ◽  
Parabolic Problem ◽  
Variable Exponent ◽  
Global Behavior ◽  
Variable Exponents ◽  
Nonlinear Parabolic ◽  
Quasilinear Parabolic Problem ◽  
Quasilinear Elliptic ◽  
Quasilinear Parabolic

We discuss the existence and uniqueness of the weak solution of the following nonlinear parabolic problem: [Formula: see text] which involves a quasilinear elliptic operator of Leray–Lions type with variable exponents. Next, we discuss the global behavior of solutions and in particular the convergence to a stationary solution as [Formula: see text].

scholarly journals Asymptotic behavior of solutions for a semibounded nonmonotone evolution equation

2003 ◽  
Vol 2003 (9) ◽  
pp. 521-538
Author(s):  
Nikos Karachalios ◽  
Nikos Stavrakakis ◽  
Pavlos Xanthopoulos
Keyword(s):  
Dynamical System ◽  
Asymptotic Behavior ◽  
Parabolic Equation ◽  
Evolution Equation ◽  
Existence And Uniqueness ◽  
Nonlinear Parabolic Equation ◽  
Evolution Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Uniqueness Of Solutions ◽  
Nonlinear Parabolic

We consider a nonlinear parabolic equation involving nonmonotone diffusion. Existence and uniqueness of solutions are obtained, employing methods for semibounded evolution equations. Also shown is the existence of a global attractor for the corresponding dynamical system.

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.

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