No abstract

Abstract. The aim of this paper is to establish a higher integrability result for very weak solutions of certain parabolic systems whose model is the parabolic p(x, t)-Laplacian system. Under assumptions on the exponent function p :n+2 , 2 , it is shown that any very weak solution u :, provided ε > 0 is small enough. This extends the main result of [V. Bögelein and Q. Li, Nonlinear Anal., 98 (2014), pp. 190-225] to the subquadratic case.

We establish weak Harnack inequalities for positive, weak supersolutions to certain doubly degenerate parabolic equations. The prototype of this kind of equations isOur proof is based on Caccioppoli inequalities, De Giorgi's estimates and Moser's iterative method.This equation models the filtration of a polytropic non-Newtonian fluid in a porous medium (see for example [17]). Porzio and Vespri [25] and Ivanov [12] independently proved that the weak solutions to (1.1) are Hölder continuous. The Harnack inequality of weak solutions to doubly degenerate parabolic equations has been established by Vespri [?] and the case of equations with general quasi-linear structure was treated by Fornaro and Sosio 2010 Mathematics Subject Classification. Primary 35K65, 35K92, 35B65; Secondary 35K59, 35B45.

Motivated by the work of Grujić and Kalisch, [Z. Grujić and H. Kalisch, Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions, Differential and Integral Equations 15 (2002) 1325-1334], we prove the local well-posedness for the periodic KdV equation in spaces of periodic functions analytic on a strip around the real axis without shrinking the width of the strip in time.

This paper considers a certain doubly singular parabolic equations with one singularity occurs in the time derivative, whose model iswhere Ω ⊂ R N and N ≥ 3. We show that the bounded weak solutions are locally continuous in the range 2 − ε 0 ≤ p < 2, provided ε 0 > 0 is small enough, and the continuity is stable as p → 2.