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Existence results of renormalized solutions to nonlinear parabolic equations in Musielak-Orlicz spaces
Resultados de existencia de soluciones renormalizadas para ecuaciones parabólicas no lineales en espacios de Musielak-Orlicz
DOI:
https://doi.org/10.15446/recolma.v59n2.125952Palabras clave:
Inhomogeneous Musielak-Orlicz-Sobolev spaces, Parabolic problems, Musielak-Orlicz function, Renormalized solutions (en)Espacios inhomog´eneos de Musielak-Orlicz-Sobolev (es)
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In this paper, we prove the existence of renormalized solutions for a nonlinear parabolic problem of the type ∂u/∂t − div a(x, t, u,∇u) + Φ(u) + g(u) φ(x, |∇u|) = f in Q (a bounded subset of RN), in the setting of Musielak-Orlicz, where −div(a(x, t, u,∇u)) is a Leray-Lions operator, Φ ∈ C0 (R, RN). The term g(u) φ(x, |∇u|) represents a nonlinear lower-order term with natural growth with respect to |∇u| and satisfies a sign condition. Note that the Δ2-condition is not assumed on the Musielak function and the source term f belongs to L1(Q).
En este artículo, probamos la existencia de soluciones renormalizadas para el problema no lineal parabólico del tipo ∂u/∂t − div a(x, t, u,∇u) + Φ(u) + g(u) φ(x, |∇u|) = f en Q (un subconjunto acotado de RN), en el espacio
de Musielak-Orlicz, donde −div(a(x, t, u,∇u)) es un operador de tipo Leray-Lions, Φ ∈ C0 (R, RN). El término g(u) φ(x, |∇u|) representa un término no lineal de menor orden con crecimiento natural con respecto a |∇u| y satisface una condición de signo. Note que no se asume la condición Δ2 en la función de Musielak function y la función fuente f pertenece a L1(Q).
Referencias
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