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On some parabolic problems with measure sources



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)=μin Q:=(0,T)×Ω,u=0on(0,T)×Ω,b(u)=b(u0)inΩ,\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>2-1N+1p > 2 - {1 \over {N + 1}}, μ is a Radon measure with bounded total variation on Q, b is a C1increasing function which satisfies 0 < b0b′(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.