In this paper we study the regularity theory for the Poisson equation in $\mathbb {R}^n$ under proper conditions. Furthermore, it will be verified that these conditions are optimal.

In this paper, the Jacobi elliptic function expansion technique is used to obtain the exact solutions of the sixth order (3+1)-dimensional Kadomtsev–Petviashvili–Sawada–Kotera–Ramani equation. Modulation instability is also discussed for this equation. The main purpose is to find novel exact solutions to this equation by means of a finite series expansion of degree n in terms of Jacobi elliptic functions. Single and combined Jacobi elliptic function solutions are obtained. The JEFE method is found to be highly effective for exact analytical solutions of nonlinear partial differential equations and its flexibility permits the development of several variations for specific problem types. The studied equation is reduced to nonlinear ordinary differential equation of integer order by using the traveling wave transformation. We observe that the solutions obtained are precise, and include periodic wave solutions, quasi-periodic wave solutions and solitary waves. Oscillatory phenomena in systems such as plasma physics and optics can be described by periodic wave solutions. Quasi periodic solutions occur in complex systems with multiple interacting frequencies, which are important in turbulence and nonlinear resonance. Solitary waves (solitons) are stable, localized waves that are critical to fluid dynamics, nonlinear optics, and plasma physics, and that model stable wave propagation in many applications. In addition, graphical representations of some solutions are presented to show the direct viewing analysis of the solutions. The results confirm that the proposed technique is a powerful tool for solving a large variety of NPDEs in mathematical physics, and may have applications to other nonlinear evolution equations.

In this paper we obtain the local Hölder regularity of the gradient of weak solutions for the general parabolic p(x, t)-Laplacian equations $$u_t-\text{div}\ \mathcal{A}\left( \nabla u, x, t \right)\,=\,\text{div} \left(|\mathrm{\bf f}|^{p(x, t)-2} \mathrm{\bf f}\right),$$ provided p(x, t), \({\mathcal{A}}\) and \({\mathrm{\bf f}}\) satisfy some proper conditions. More precisely, we shall prove that $$\nabla u \in C_{loc}^{0;\alpha,\alpha/2}(\Omega_T)\,\mbox{for some} \, \, \alpha \in (0, 1). $$