We consider the stochastic nonlinear pseudoparabolic equations on thehalf-line with Dirichlet noise boundary conditions. We establish the localexistence and uniqueness of the solution to the initial-boundary valueproblem with values in an appropriate weighted space. We are alsointerested in the regularity behavior of the solution, especially near theorigin, where the boundary data are highly irregular.

In this paper, we prove existence of entropy solutions to the time-fractional porous medium type equation,$$\partial_t[k\ast(u-u_0)]-{\text{div}} (A(t,x)\nabla\varphi(u))=f\text{ in }Q_T=(0,T)\times\Omega,$$ with Dirichlet boundary condition, initial condition $u(0,\cdot)=u_0$ in $\Omega$, and $L^1$-data $f\in L^1((0,T)\times\Omega), u_0\in L^1(\Omega)$. To this end, we approximate the data by $L^\infty$-functions, use a known existence result of weak solutions for these more regular data, and additionally a known contraction principle for weak solutions, which can be adopted to the entropy solutions.

We extend the Poincaré--Birkhoff Theorem to a Hamiltonian system which couples two systems with fairly different behaviors; the first one involves a twist assumption, while the second one is generated from a nonresonant isochronous center. By a suitable change of variables we modify the second system into a perturbation of a nonresonant linear one, and then prove that there exist multiple periodic solutions.

We consider the cubic defocusing nonlinear Schrödinger equation on product manifold $\mathbb{R}\times \mathbb{T}$. In this paper, we obtain polynomial bounds on the growth in time of high Sobolev norms of the solutions, which measure the transfer of energy from low to high modes as time grows on. The main ingredient of the proof is to establish an iteration bound, which is based on the idea of Bourgain in [3].

In this paper, we study the following quasilinear Schrödinger-Poisson system\begin{equation} \begin{cases}-\Delta_{p}u+V(x)|u|^{p-2}u+\phi|u|^{p-2}u=f(u)\qquad & x\in\mathbb{R}^{3},\\-\Delta\phi=|u|^{p} & x\in\mathbb{R}^{3}. \end{cases}\notag\end{equation}By using the method of invariant sets of descending flow, we obtain the existence of multiple sign-changing solutions. Our results extend and enrich existing results of Schrödinger-Poisson system.

We consider wave equations with a special type of log-fractional damping. We study the Cauchy problem for this model in ${\bf R}^{n}$, and we obtain an asymptotic profile and optimal estimates of solutions as $t \to \infty$ in the $L^{2}$-sense. A maximal discovery of this note is that under the effective damping, in case of $n=1$ the $L^{2}$-norm of the solution blows up in infinite time, and in case of $n = 2$ the $L^{2}$-norm of the solution never decays and never blows up in infinite time. The latter phenomenon seems to be a rare case in the community.