We prove a maximum principle for local solutions of quasilinear stochastic PDEs with obstacle (in short OSPDE). The proofs are based on a version of Itô’s formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary. Our method is based on a version of Moser’s iteration scheme developed first by Aronson and Serrin [2] in the context of non-linear...

We extend to the parabolic setting some of ideas originated with Xiao Zhong's proof in [31] Hölder regularity $p-$harmonic functions Heisenberg group $\mathbb{H}^n$. Given a number $p\ge 2$, this paper we establish $C^{\infty}$ smoothness weak solutions class quasilinear PDE $\mathbb{H}^n$ modeled on equation $$?_t u = \sum_{i 1}^{2n} X_i \bigg((1+|\nabla_0 u|^2)^{\frac{p-2}{2}} u\bigg).$$

In this paper we analyze the abstract parabolic evolutionary equations Dt ðu xÞ þ AðuÞu 1⁄4 f ðuÞ þ hðtÞ; uð0Þ 1⁄4 x; in continuous interpolation spaces allowing a singularity as tk0: Here Dt denotes the timederivative of order aAð0; 2Þ: We first give a treatment of fractional derivatives in the spaces Lðð0;TÞ;XÞ and then consider these derivatives in spaces of continuous functions having (at m...

Existence of weak solutions for systems of quasilinear degenerate parabolic equations with non-diagonal main part and nonlinear boundary conditions is proved. Under some restrictions we find also L∞ bounds for the solutions.

In this paper we consider a quasilinear parabolic equation a(ux)ut = uxx (a(s) a0 > 0) subject to appropriate initial and boundary conditions. This equation can be used to describe the uni-directional motion of uid in soft tissue. A criterion is found to ensure the global solvability or nite time quenching (i.e. ux becomes unbounded in nite time). A concrete example is that for a(s) = (1 + s), ...

A maximum principle for C1 solutions of quasilinear parabolic differential inequalities which retains the strong conclusion of Nirenberg's well-known result [2] is established. The case of strongly differentiable solutions rather than of class C1 is also discussed.