Existence and Uniqueness of Renormalized Solutions to Nonlinear Parabolic Equations with Lower Order Term and Diffuse Measure Data

2018 ◽  
Vol 15 (4) ◽  
Author(s):  
A. Bouajaja ◽  
H. Redwane ◽  
A. Marah
Keyword(s):  
Lower Order ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Order Term ◽  
Nonlinear Parabolic Equations ◽  
Lower Order Term ◽  
Measure Data ◽  
Renormalized Solutions ◽  
Nonlinear Parabolic ◽  
Diffuse Measure

scholarly journals Existence of a renormalized solution of nonlinear parabolic equations with lower order term and general measure data

2021 ◽  
Vol 39 (3) ◽  
pp. 93-114
Author(s):  
A. Marah ◽  
Abdelkader Bouajaja ◽  
H. Redwane
Keyword(s):  
Lower Order ◽  
Parabolic Equations ◽  
Order Term ◽  
Nonlinear Parabolic Equations ◽  
Lower Order Term ◽  
Measure Data ◽  
General Measure ◽  
Renormalized Solution ◽  
Nonlinear Parabolic ◽  
The Right

We give an existence result of a renormalized solution for a classof nonlinear parabolic equations@b(u)/@t div(a(x; t;grad(u))+ H(x; t;ru) = ,where the right side is a general measure, b is a strictly increasing C1-function,div(a(x; t;grad(u)) is a Leray{Lions type operator with growth  in grad(u)and H(x; t;grad(u) is a nonlinear lower order term which satisfy the growth condition with respect to grad(u).

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 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.

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