The classical Huygens' principle asserts that the initial data of a wave equation determines the wave propagation in the domain of dependence of the support of the data. We provide a converse version of this theorem.

In this note, we give a new proof of subcritical Trudinger-Moser inequality on $R^n$. All the existing proofs on this inequality are based on the rearrangement argument with respect to functions in the Sobolev space $W^{1,n}(R^n)$. Our method avoids this technique and thus can be used in the Riemannian manifold case and in the entire Heisenberg group.

The Cauchy problem for a linear 2mth-order Schrödinger equation $$u_t=-i(-Δ)^mu, in\ R^N×R_+, u|_{t=0}=u_0; m≥ 1$$ is an integer, is studied, for initial data u0 in the weighted space $L^2_{ρ^*}(R^N)$, with $ρ^*(x)=e^{|x|^α}$ and $α=\frac{2m}{2m-1}∈(1,2]$. The following five problems are studied: (I) A sharp asymptotic behaviour of solutions as $t →+∞$ is governed by a discrete spectrum and a countable set Φ of the eigenfunctions of the linear rescaled operator $$B=-i(-Δ)^m+\frac{1}{2m}y\cdot∇+\frac{N}{2m}I, with\ the\ spectrum\ σ(B)={λ_β=-\frac{|β|}{2m}, |β|≥ 0}.$$ (II) Finite-time blow-up local structures of nodal sets of solutions as t?^- and a formation of "multiple zeros" are described by the eigenfunctions, being generalized Hermite polynomials, of the "adjoint" operator $$B^*=-i(-Δ)^m-\frac{1}{2m}y\cdot∇, with\ the\ spectrum\ σ(B^*)=σ(B).$$ Applications of these spectral results also include: (III) a unique continuation theorem, and (IV) boundary characteristic point regularity issues. Some applications are discussed for more general linear PDEs and for the nonlinear Schrödinger equations in the focusing ("+") and defocusing ("-") cases $$u_t=-i(-Δ)^mu ± i|u|^{p-1}u, in\ R^N×R_+, where\ P>1,$$ as well as for: (V) the quasilinear Schrödinger equation of a "porous medium type" $$u_t=-i(-Δ)^mu (|u|^nu), in\ R^N×R_+, where\ n>0.$$ For the latter one, the main idea towards countable families of nonlinear eigenfunctions is to perform a homotopic path $n→ 0^+$ and to use spectral theory of the pair ${B,B^*}$.

In this paper, we prove the existence of nonnegative solutions to the initial boundary value problems for the pseudo-parabolic type equation with weakly nonlinear sources. Further, we discuss the asymptotic behavior of the solutions as the viscous coefficient k tends to zero.

The objective of this paper is to apply the improved generalized Riccati equation mapping method to find many families of exact traveling wave solutions for the general nonlinear dynamic system in a new double-chain model of DNA. This model consists of two long elastic homogeneous strands connected with each other by an elastic membrane. Hyperbolic and trigonometric function solutions of this model are obtained. Comparison between our results and the well-known results are given.