Existence of a Renormalized Solution of a Parabolic Problem in Anisotropic Sobolev–Orlicz Spaces

• Published in: Journal of mathematical sciences (New York, N.Y.) Vol. 258; no. 1; pp. 37 - 64
• Main Authors: Vorob’yov, N. A., Mukminov, F. Kh
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2021; Springer; Springer Nature B.V
• Subjects: Anisotropy; Domains; Mathematics; Mathematics and Statistics; 35K20; renormalized solution; anisotropic parabolic equation; nonpower nonlinearity; existence of solutions; function; 35K55; 35K65
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-021-05535-8

We consider the first mixed problem for a certain class of anisotropic parabolic equations of the form β x u t ′ − div a t x u ∇ u − b t x u ∇ u = u where μ is a measure and the coefficients contain nonpower nonlinearities in the cylindrical domain D T = (0 , T ) × Ω, where Ω ⊂ ℝ n is a bounded domain. We prove the existence of a renormalized solution of the problem for g t = 0 and a function β ( x, r ), which increases with respect to r and satisfies the Carath´eodory condition.