Renormalized Solutions of Nonlinear Parabolic Equations in Weigthed Variable-Exponent Space

2015 ◽  
Vol 28 (3) ◽  
pp. 225-252
Author(s):  
Youssef Akdim
Keyword(s):  
Parabolic Equations ◽  
Variable Exponent ◽  
Nonlinear Parabolic Equations ◽  
Renormalized Solutions ◽  
Nonlinear Parabolic

scholarly journals On some parabolic problems with measure sources

2019 ◽  
Vol 5 (1) ◽  
pp. 1-21 ◽  
Author(s):  
Mohammed Abdellaoui
Keyword(s):  
Parabolic Equations ◽  
Stability Result ◽  
Initial Boundary ◽  
Nonlinear Parabolic Equations ◽  
Point Of View ◽  
Parabolic Problems ◽  
Measure Data ◽  
Renormalized Solutions ◽  
Renormalized Solution ◽  
Nonlinear Parabolic

AbstractOne of the recent advances in the investigation of nonlinear parabolic equations with a measure as forcing term is a paper by F. Petitta in which it has been introduced the notion of renormalized solutions to the initial parabolic problem in divergence form. Here we continue the study of the stability of renormalized solutions to nonlinear parabolic equations with measures but from a different point of view: we investigate the existence and uniqueness of the following nonlinear initial boundary value problems with absorption term and a possibly sign-changing measure data\left\{ {\matrix{ {b{{\left( u \right)}_t} - {\rm{div}}\left( {a\left( {t,x,u,\nabla u} \right)} \right) + h\left( u \right) = \mu } \hfill & {{\rm{in}}Q: = \left( {0,T} \right) \times {\rm{\Omega }},} \hfill \cr {u = 0} \hfill & {{\rm{on}}\left( {0,T} \right) \times \partial {\rm{\Omega }},} \hfill \cr {b\left( u \right) = b\left( {{u_0}} \right)} \hfill & {{\rm{in}}\,{\rm{\Omega }},} \hfill \cr } } \right.where Ω is an open bounded subset of ℝN, N ≥ 2, T > 0 and Q is the cylinder (0, T) × Ω, Σ = (0, T) × ∂Ω being its lateral surface, the operator is modeled on the p−Laplacian with p > 2 - {1 \over {N + 1}}, μ is a Radon measure with bounded total variation on Q, b is a C1−increasing function which satisfies 0 < b0 ≤ b′(s) ≤ b1 (for positive constants b0 and b1). We assume that b(u0) is an element of L1(Ω) and h : ℝ ↦ ℝ is a continuous function such that h(s) s ≥ 0 for every |s| ≥ L and L ≥ 0 (odd functions for example). The existence of a renormalized solution is obtained by approximation as a consequence of a stability result. We provide a new proof of this stability result, based on the properties of the truncations of renormalized solutions. The approach, which does not need the strong convergence of the truncations of the solutions in the energy space, turns out to be easier and shorter than the original one.

Renormalized solutions of nonlinear parabolic equations with diffuse measure data

2012 ◽  
Vol 141 (3-4) ◽  
pp. 601-635 ◽  
Author(s):  
Dominique Blanchard ◽  
Francesco Petitta ◽  
Hicham Redwane
Keyword(s):  
Parabolic Equations ◽  
Nonlinear Parabolic Equations ◽  
Measure Data ◽  
Renormalized Solutions ◽  
Nonlinear Parabolic ◽  
Diffuse Measure

scholarly journals Renormalized solutions for nonlinear parabolic problems with L1−data in orlicz-sobolev spaces

2017 ◽  
Vol 35 (1) ◽  
pp. 57 ◽  
Author(s):  
Youssef El hadfi ◽  
Abdelmoujib Benkirane ◽  
Mostafa El moumni
Keyword(s):  
Sobolev Spaces ◽  
Lower Order ◽  
Parabolic Equations ◽  
Existence Result ◽  
Nonlinear Parabolic Equations ◽  
Parabolic Problems ◽  
Renormalized Solutions ◽  
L1 Data ◽  
Nonlinear Parabolic ◽  
Lower Order Terms

In this work, we prove an existence result of renormalized solutions in Orlicz-Sobolev spaces for a class of nonlinear parabolic equations with two lower order terms and L1-data. 

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.

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