We give necessary and sufficient conditions for the existence and uniqueness of bounded or almost-periodic solutions of the first-order differential system: $u' + \nabla \Phi (u) = e(t)$, when $\nabla \Phi$ denotes the gradient of a convex function on $\mathbb R^N$. We also study the relations of continuity between the forcing term $e$ and the solution $u$. Then we give similar results for the second-order differential system: $u'' = \nabla \Phi (u) + e(t)$.

We study simultaneous and nonsimultaneous blow-up for solutions of the following system $$\left\{\begin{array}{l} u_t = \Delta u + u^r + v^p, \\ v_t = \Delta v + v^s + u^q, \end{array} \quad\mbox{ in } \Omega \times (0,T), \right. $$ with Dirichlet boundary conditions. We show that, in the range of exponents where either component may blow up alone, there also exist initial data for which both components blow up simultaneously. The proof is based on a continuity argument, which requires upper and lower blow-up estimates, independent of initial data, and continuous dependence of the existence time. In turn, we prove a result of continuous dependence of the existence time, under the assumption of uniform upper blow-up estimates, in the framework of general abstract semiflows.

In this paper we prove a relaxation result for a class of degenerate Hamilton--Jacobi equations; i.e., equations which do not admit a strict subsolution, extending in some directions the results in [6] and [7].

We study standing wave solutions of the form $e^{i(\omega t+m\theta)}\phi_\omega(r)$ to the nonlinear Schrödinger equation $$iu_t+\Delta u+|u|^{p-1}u=0\quad\text{for $x\in \mathbb{R}^2$ and $t>0$,}$$ where $(r,\theta)$ are polar coordinates and $m\in\mathbb N\cup\{0\}$. We prove that standing waves which have no node are unique for each $m$ and that they are unstable if $p>3$.

In this paper, we discuss almost periodicity of solutions for evolution equations in Banach space. We introduce the concept of difference-variation equation. For a special Hilbert space, we establish some results on almost periodicity of all solutions for evolution equations. In particular, we extend Haraux's result on $R^2$ to $R^n.$