Evolution equations of p-Laplace type with absorption or source terms\n and measure data

2014/09/04 by Marie‐Françoise Bidaut‐Véron, Bidaut-Véron, Marie-Françoise, Quoc‐Hung Nguyen +1
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1409.1520

openalex publication_date 2014/09/04 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

Let \Ω be a bounded domain of \ℝ^N, and Q=\Ω\n\×(0,T). We consider problems\ of the type % \
left
\n% u_t-
Delta_pu
pm
mathcalG(u)=
mu
qquad
textin\nQ,

u=0
qquad
texton
partial
Omega
times(0,T),

ν(0)=u₀
qquad
textin
Omega, \
right. where \Δ_p\nis the p-Laplacian, \μ is a bounded Radon measure, u₀\∈\nL¹(\Ω), and \±\G(u) is an absorption or a source term. In\nthe model case \G(u)=\± \vert u \vert ^q-1u (q>p-1),\nor \G has an exponential type. We prove the existence of\nrenormalized solutions for any measure \μ in the subcritical case, and give\nsufficient conditions for existence in the general case, when \μ is good in\ntime and satisfies suitable capacitary conditions.\n

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