Existence of a renormalized solution to an anisotropic parabolic problem with variable nonlinearity exponents

2018 ◽  
Vol 209 (5) ◽  
pp. 714-738 ◽  
Author(s):  
F. Kh. Mukminov
Keyword(s):  
Parabolic Problem ◽  
Renormalized Solution

scholarly journals Existence of Renormalized Solutions for p(x)-Parabolic Equation with three unbounded nonlinearities

2016 ◽  
Vol 34 (1) ◽  
pp. 225-252 ◽  
Author(s):  
Youssef Akdim ◽  
Nezha El Gorch ◽  
Mounir Mekkour
Keyword(s):  
Parabolic Equation ◽  
Sobolev Space ◽  
Growth Condition ◽  
Parabolic Problem ◽  
Variable Exponent ◽  
Critical Growth ◽  
Renormalized Solutions ◽  
Renormalized Solution ◽  
Coercivity Condition ◽  
Sign Condition

In this article, we study the existence of a renormalized solution for the nonlinear $p(x)$-parabolic problem associated to the equation: $$\frac{\partial b(x,u)}{\partial t} - \mbox{div} (a(x,t,u,\nabla u)) + H(x,t,u,\nabla u) = f - \mbox{div}F \;\mbox{in }\;Q= \Omega\times(0,T)$$with $ f $ $ \in L^{1} (Q),$\; $b(x,u_{0}) \in L^{1} (\Omega)$ and $ F \in (L^{P'(.)}(Q))^{N}. $The main contribution of our work is to prove the existence of a renormalized solution in the Sobolev space with variable exponent. The critical growth condition on $ H(x,t,u,\nabla u)$\; is with respect to$ \nabla u$, no growth with respect to $u$ and no sign condition or the coercivity condition.

ON A HOMOGENEOUS PARABOLIC PROBLEM IN AN INFINITE ANGULAR DOMAIN

2019 ◽  
Vol 7 (1) ◽  
pp. 38-52
Author(s):  
Jenaliyev M.T. ◽  
◽  
Ramazanov M.I. ◽  
Iskakov S.A. ◽  
Keyword(s):  
Parabolic Problem ◽  
Angular Domain

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

scholarly journals Smoothing properties in multistep backward difference method and time derivative approximation for linear parabolic equations

2005 ◽  
Vol 2005 (4) ◽  
pp. 523-536
Author(s):  
Yubin Yan
Keyword(s):  
Difference Method ◽  
Parabolic Equations ◽  
Parabolic Problem ◽  
Approximation Scheme ◽  
Time Derivative ◽  
Nonsmooth Data ◽  
Smoothing Property ◽  
Linear Parabolic Equations ◽  
Linear Parabolic Problem ◽  
Data Error

A smoothing property in multistep backward difference method for a linear parabolic problem in Hilbert space has been proved, where the operator is selfadjoint, positive definite with compact inverse. By using the solutions computed by a multistep backward difference method for the parabolic problem, we introduce an approximation scheme for time derivative. The nonsmooth data error estimate for the approximation of time derivative has been obtained.

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