General decay and blow up of solutions for a variable-exponent viscoelastic double-Kirchhoff type wave equation with nonlocal degenerate damping

In this paper we consider a viscoelastic double-Kirchhoff type wave equation of the form $$ u_{tt}-M_{1}(\|\nabla u\|^{2})\Delta u-M_{2}(\|\nabla u\|_{p(x)})\Delta_{p(x)}u+(g\ast\Delta u)(x,t)+\sigma(\|\nabla u\|^{2})h(u_{t})=\phi(u), $$ where the functions $M_{1},M_{2}$ and $\sigma, \phi$ are real valued functions and $(g\ast\nabla u)(x,t)$ is the viscoelastic term which are introduced later. Under appropriate conditions for the data and exponents, the general decay result and blow-up of solutions are proved with positive initial energy. This study extends and improves the previous results in the literature to viscoelastic double-Kirchhoff type equation with degenerate nonlocal damping and variable-exponent nonlinearities.

D’Alembert wave and soliton molecule of the generalized Nizhnik–Novikov–Veselov equation

Traveling wave solution is one of the effective methods for solving nonlinear partial differential equations. D’Alembert solution is a special kind of traveling wave solution. There have been many studies about D’Alembert solution. In this paper, we will solve D’Alembert-type wave solutions for (2+1)-dimensional generalized Nizhnik–Novikov–Veselov equation. Based on the Hirota bilinear transformation and velocity resonance mechanism, the states of soliton molecules composed of two solitons, three solitons and four solitons are studied. It is concluded that D’Alembert-type wave is closely related to soliton molecules.