Regularizing effect and decay results for a parabolic problem with repulsive superlinear first order terms

• Published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Físicas y Naturales. Serie A, Matemáticas, 2021-04, Vol.115 (2), Article 77
• Main Author: Magliocca, Martina
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Decay; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Original Paper; Theoretical
• ISSN: 1578-7303; 1579-1505
• DOI: 10.1007/s13398-021-01010-w

We want to analyze both regularizing effect and long, short time decay concerning a class of parabolic equations having first order superlinear terms. The model problem is the following: u t - div ( A ( t , x ) | ∇ u | p - 2 ∇ u ) = γ | ∇ u | q in ( 0 , T ) × Ω , u = 0 on ( 0 , T ) × ∂ Ω , u ( 0 , x ) = u 0 ( x ) in Ω , where Ω is an open bounded subset of R N , N ≥ 2 , 0 < T ≤ ∞ , 1 < p < N and q < p . We assume that A ( t ,  x ) is a coercive, bounded and measurable matrix, the growth rate q of the gradient term is superlinear but still subnatural, γ is a positive constant, and the initial datum u 0 is an unbounded function belonging to a well precise Lebesgue space L σ ( Ω ) for σ = σ ( q , p , N ) .