In this article, we analyze solitary waves in nonlinear left-handed transmission line with nonlinear diodes (Schottkys) which is an important issue, especially for soliton devices. By applying the Kirchhoffs laws and reductive direct method, the voltage in the spectral domain was obtained. Considering the Taylor series around a certain modulation frequency, we obtained one dimensional Nonlinear Schrödinger Equation (NSE), which support envelops soliton, and bright soliton solutions. Using sine-cosine mathematical method, soliton solutions of the standard Nonlinear Schrödinger equation are obtained. The method used is straightforward and concise and can be applied to solve further of nonlinear PDEs in mathematical physics.

The well-posedness and stability properties of a stochastic viscoelastic equation with multiplicative noise, Lipschitz and locally Lipschitz nonlinear terms are investigated. The method of Lyapunov functions is used to investigate the asymptotic dynamics when zero is not a solution of the equation by using an appropriate cocycle and random dynamical system. The stability of mild solutions is proved in both cases of Lipschitz and locally Lipschitz nonlinear terms. Furthermore, we investigate the existence of a non-trivial stationary solution which is exponentially stable, by using a general random fixed point theorem for general cocycles. In this case, the stationary solution is generated by the composition of random variable and Wiener shift. In addition, the theory of random dynamical system is used to construct another cocycle and prove the existence of a random fixed point exponentially attracting every path.

An existence and uniqueness result of a renormalized solution for a class of doubly nonlinear parabolic equations with singular coefficient with respect to the unknown and with diffuse measure data is established. A comparison result is also proved for such solutions.

Let $\mathbb{B}_1$ be a unit disc of $\mathbb{R}^2$, and $\mathscr{H}$ be a completion of $C_0^\infty(\mathbb{B}_1)$ under the norm  $$\|u\|_{\mathscr{H}}^2=\int_{\mathbb{B}_1}\le(|\nabla u|^2-\frac{u^2}{(1-|x|^2)^2}){\rm d}x.$$ Using blow-up analysis, Wang-Ye [1] proved existence of extremals for a Hardy-Trudinger-Moser inequality. In particular, the supremum $$\sup_{u\in \mathscr{H},\,\|u\|_{\mathscr{H}}\leq 1}\int_{\mathbb{B}_1}e^{4\pi u^2}{\rm d}x$$ can be attained by some function $u_0\in\mathscr{H}$ with $\|u_0\|_{\mathscr{H}}= 1$. This was improved by the author and Zhu [2] to a version involving the first eigenvalue of the Hardy-Laplacian operator $-\Delta-1/(1-|x|^2)^2$. In this note, the results of [1, 2] will be reproved by the method of energy estimate, which was recently developed by Malchiodi-Martinazzi [3] and Mancini-Martinazzi [4].

In this paper, we study a class of stochastic differential equations with additive noise that contains a non-stationary multifractional Brownian motion (mBm) with a Hurst parameter as a function of time and a Poisson point process of class (QL). The differential equation of this kind is motivated by the reserve processes in a general insurance model, in which between the claim payment and the past history of liability present the long term dependence. By using the variable order fractional calculus on the fractional Wiener-Poisson space and a multifractional derivative operator, and employing Girsanov theorem for multifractional Brownian motion, we prove the existence of weak solutions to the SDEs under consideration, As a consequence, we deduce the uniqueness in law and the pathwise uniqueness.

In this paper, we consider a nonlinear wave equation with nonlocal damping and nonlinear boundary damping. We prove a general energy decay property for solutions in terms of coefficient of the frictional boundary damping by using of the multiplier technique from the idea of Martinez [1]. Our result extends and improves the result in the literature such as the work by Lourêdo, Ferreira de Araújo and Mirandain [2] in which only exponential energy decay is considered. Furthermore, we get also the energy decay for the equation with nonlocal damping only but without nonlinear boundary damping.