Affiliation:
1. University of Fez, Faculty of Sciences Dhar El Mahraz , Laboratory LAMA , Department of Mathematics , B.P. 1796, Atlas Fez, Morocco .
Abstract
Abstract
One 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
{
b
(
u
)
t
-
div
(
a
(
t
,
x
,
u
,
∇
u
)
)
+
h
(
u
)
=
μ
in
Q
:
=
(
0
,
T
)
×
Ω
,
u
=
0
on
(
0
,
T
)
×
∂
Ω
,
b
(
u
)
=
b
(
u
0
)
in
Ω
,
\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 ℝ
N
, 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
N
+
1
p > 2 - {1 \over {N + 1}}
, μ is a Radon measure with bounded total variation on Q, b is a C
1
−increasing function which satisfies 0 < b
0 ≤ b′(s) ≤ b
1 (for positive constants b
0 and b
1). We assume that b(u
0) is an element of L
1(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.
Subject
Applied Mathematics,Control and Optimization,Numerical Analysis,Analysis
Reference43 articles.
1. [AA1] M. Abdellaoui, E. Azroul, Renormalized solutions for nonlinear parabolic equations with general measure data, Electron. J. Differential Equations, Vol. 2018, No. 132, pp. 1–21.
2. [AA2] M. Abdellaoui, E. Azroul, Nonlinear parabolic equations with soft measure data, arXive:submit\2570936 Submitted.
3. [AA3] M. Abdellaoui, E. Azroul, H. Redwane, Nonlinear parabolic equations of porous medium type with unbounded term and general measure data, Submitted.
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