We investigate dispersive estimates for the Schrödinger operator $H=-\Delta +V$ with $V$ is a real-valued decaying potential when there are zero energy resonances and eigenvalues in four spatial dimensions. If there is a zero energy obstruction, we establish the low-energy expansion \begin{align*} e^{itH}\chi(H) P_{ac}(H) & =O(\frac 1{\log t}) A_0+ O(\frac 1 t )A_1+O((t\log t)^{-1})A_2 \\ & + O(t^{-1}(\log t)^{-2})A_3. \end{align*} Here, $A_0,A_1:L^1(\mathbb R^4)\to L^\infty (\mathbb R^4)$, while $A_2,A_3$ are operators between logarithmically weighted spaces, with $A_0,A_1,A_2$ finite rank operators, further the operators are independent of time. We show that similar expansions are valid for the solution operators to Klein-Gordon and wave equations. Finally, we show that under certain orthogonality conditions, if there is a zero energy eigenvalue one can recover the $|t|^{-2}$ bound as an operator from $L^1\to L^\infty$. Hence, recovering the same dispersive bound as the free evolution in spite of the zero energy eigenvalue.

In this paper, we prove the existence of infinitely many nontrivial solutions of the following equations driven by a nonlocal integro-differential operator $\mathcal{L}_K$ with concave-convex nonlinearities and homogeneous Dirichlet boundary conditions \begin{align*} \mathcal{L}_{K} u + \mu |u|^{q-1}u + \lambda |u|^{p-1}u &= 0 \quad\mbox{in}\quad \Omega, \\ u&=0 \quad\mbox{in}\quad\mathbb{R}^N\setminus\Omega, \end{align*} where $\Omega$ is a smooth bounded domain in $ \mathbb R^N $, $N > 2s$, $s\in(0, 1)$, $0 < q < 1 < p\leq \frac{N+2s}{N-2s}$. Moreover, when $\mathcal{L}_K$ reduces to the fractional laplacian operator $-(-\Delta)^s $, $p=\frac{N+2s}{N-2s}$, $\frac{1}{2} (\frac{N+2s}{N-2s}) < q < 1$, $N > 6s$, $ \lambda =1$, we find $\mu^*>0$ such that for any $\mu\in(0,\mu^*)$, there exists at least one sign changing solution.

We consider reaction-diffusion systems where some components react and diffuse on the boundary of a region, while other components diffuse in the interior and react with those on the boundary through mass transport. We establish criteria guaranteeing that solutions are uniformly bounded in time.

This paper is concerned with a class of wave equations with acoustic boundary condition subject to non-autonomous external forces. Under some general assumptions, the problem generates a well-posed evolution process. Then, we establish the existence of a minimal pullback attractor within a universe of tempered sets defined by the forcing terms. We also, study the upper semicontinuity of attractors as the non-autonomous perturbation tends to zero. Our results allow unbounded external forces and nonlinearities with critical growth.

This paper is devoted to investigate the existence and multiplicity of radial nodal solutions for the following Dirichlet problem with mean curvature operator in Minkowski space \begin{eqnarray} \begin{cases} -\text{div} \Big (\frac{\nabla v}{\sqrt{1-\vert \nabla v\vert^2}} \Big ) = \lambda f(\vert x\vert,v)\,\, &\text{in}\,\, B_R(0),\\ v=0~~~~~~~~~~~~~~~~~~~~~~\,\,&\text{on}\,\, \partial B_R(0). \end{cases} \nonumber \end{eqnarray} By bifurcation approach, we determine the interval of parameter $\lambda$ in which the above problem has two or four radial nodal solutions which have exactly $n-1$ simple zeros in $(0,R)$ according to linear/sublinear/ superlinear nonlinearity at zero. The asymptotic behaviors of radial nodal solutions as $\lambda \to +\infty$ and $n \to +\infty$ are also studied.