We study the Hölder regularity of the spatial gradient of solutions for nonlinear parabolic systems of $p$-Laplacian type in the degenerate and singular cases $ \frac{{2m}}{{m + 2}} < p < \infty$. We obtain an optimal like criterion to gradient Hölder continuity for the right-hand side terms.

The validity of higher-order weighted Rellich inequalities with respect to $L^p$-norm in $\mathbb R^N$ ($\||x|^{\alpha-2m}u\|_{L^p}\leq C\||x|^{\alpha}\Delta u\|_{L^p}$) for $N\in \mathbb N$, $1\leq p\leq \infty$, $m\in\mathbb N$ and $\alpha\in\mathbb R$ is investigated. The result generalizes the validity of weighted Rellich inequalities obtained in [6].

In this paper, we consider a nonlinear evolution equation for surface waves in shallow water over uneven bottom. First, the local well-posedness is obtained in Besov space $B^{ \frac 32 }_{2,1}$. Then, persistence properties on strong solutions are also investigated.

In this paper, we consider the Majda-Biello system, a coupled KdV-type system, on the torus. In the first part of the paper, it is shown that, given initial data in a Sobolev space, the difference between the linear and the nonlinear evolution almost always resides in a smoother space. The smoothing index depends on number-theoretic properties of the coupling parameter in the system which control the behavior of the resonant sets. In the second part of the paper, we consider the forced and damped version of the system and obtain similar smoothing estimates. These estimates are used to show the existence of a global attractor in the energy space. We also show that when the damping is large in relation to the forcing terms, the attractor is trivial.

We consider a generalization of wave maps based on the Adkins-Nappi model of nuclear physics. In particular, we show that solutions to this equation remain continuous at the origin, which is a first step towards establishing a regularity theory for this equation.

In this article, we consider a model describing hospital acquired infections. The model derived is a system of delay differential equations. The state variable is formed by the patients and the health care workers components. The system is a slow-fast system where the fast equation corresponds to the health care workers equation. The question addressed in this paper is the convergence to the so-called reduced equations which is a single equation for patients. We investigate both finite time convergence and infinite time convergence (uniformly for all positive time) of the original system to the reduced equation.

This paper is concerned with a parabolic-parabolic Keller-Segel-type system in a bounded domain $\Omega \subset \mathbb{R}^N$ (with $N=2$ or $N=3$) presenting source and damping terms. We impose Neumann and Robin boundary conditions to each one of the two unknowns of the problem and study the non-negative solutions which blow up in finite time $t^*$. In this way, it is possible to derive explicit lower bounds for $t^*$, under appropriate conditions on the data of the problem.

In this paper, we are concerned with questions of existence of solution for a nonlocal and non-homogeneous Neumann boundary value problems involving the $p(x)$-Laplacian in which the non-linear terms have critical growth. The main tools we will use are the generalized Sobolev spaces and the Mountain Pass Theorem.