Global existence of solutions to a singular parabolic/Hamilton–Jacobi coupled system with Dirichlet conditions

A Generalization of a Logarithmic Sobolev Inequality to the Hölder Class

“…where W 2m,m 2 is the parabolic Sobolev space (we refer to [11] for the definition and further properties), and BM O is the parabolic bounded mean oscillation space (defined via parabolic balls instead of Euclidean ones [5, Definition 2.1]). The above inequality reflects a limiting case of Sobolev embeddings in the parabolic framework (see [6,7] for similar type inequalities, and [1,2,3,8,9,10,12] for various elliptic versions). By considering functions f ∈ W 2m,m 2 (Ø T ) defined on the bounded domain Ø T = (0, 1) n × (0, T ), T > 0, we have the following estimate (see [5,Theorem 1.2]):…”

On a parabolic logarithmic Sobolev inequality

“…However, as is well known, it is important, say for parabolic partial differential equations to consider spaces that are anisotropic. Motivated by the study of the long-time existence of a certain class of singular parabolic coupled systems (see [8,9]), we show in this paper an analogue of the Kozono-Taniuchi inequality (1.1) but of the parabolic (anisotropic) type. Due to the parabolic anisotropy, we consider functional spaces on R n+1 = R n × R with the generic variable z = (x, t), where each coordinate x i , i = 1, .…”

A critical parabolic Sobolev embedding via Littlewood-Paley decomposition

“…was successfully applied in order to obtain some a priori bounds on the gradient of the solution of particular parabolic equations leading eventually to the long-time existence (see [13,Proposition 3.7] or [12,Theorem 1.3]). The bounded version of (1.1) (see [13,Theorem 1.2]) reads: if f ∈ W 2m,m 2 (Ω T ) with 2m > n+2 2 , then:…”

“…Remark 1.4 Inequalities (1.1) and (1.9) have the same order of the higher regular term. As a consequence, inequality (1.9) can also be applied in order to establish the long-time existence of solutions of the parabolic problems studied in [12,13].…”