We prove that if $n = 2$ or $3$, the problem $- \Delta u = e^u$ on $R^n$ has no stable negative solution. We then use this to remove self-adjointness conditions in a paper of Crandall and Rabinowitz on the primary branch of positive solutions of a nonlinear boundary-value problem.

We present invariants for nonlinear Dirac equations in space-time ${\mathbb R}^{n+1}$, by which we prove that a special choice of the Cauchy data yields free solutions. Our argument works for Klein-Gordon-Dirac equations with Yukawa coupling as well. Related problems on the structure of Dirac matrices are studied.

For a class of semilinear parabolic equations, we prove both global existence and finite-time blow-up depending on the initial datum. The proofs involve tools from the potential-well theory, from the critical-point theory, and from classical comparison principles.

Motivated by a model of a system of many particles at low densities, we consider a lattice differential equation with two uniform steady states and we investigate the existence of travelling fronts connecting them. This leads to a two-point boundary-value problem for a nonlinear delay-differential equation. We replace the original parabolic nonlinearity by a piecewise-linear function, where explicit computations are possible. We find monotone and nonmonotone fronts. Finally we also describe all the fronts such that the $\alpha$-limit is the unstable uniform state. For different values of the wave speed $c$ of the front we find bounded and unbounded as well as eventually periodic orbits, i.e., orbits $u_c (x)$ that are periodic for $x\geqslant x_{\text{per}}(c)$ for some $x_{\text{per}}(c)\in{{\mathbb {R}}}$.

We consider a system of nonlinear wave equations $$ ({\partial}_t^2-c_i^2 \Delta_x)u_i=F_i(u, {\partial} u, {\partial}_x {\partial} u) \text{ in $(0, \infty)\times \mathbb R^3$} $$ for $ i=1, \dots, m $, where $ F=(F_1, \dots, F_m) $ is a smooth function satisfying $$ F(u, {\partial} u, {\partial}_x {\partial} u)=O(|u|^3+|{\partial} u|^2+|{\partial}_x {\partial} u|^2) \quad \text{near the origin,} $$ $ u=(u_1, \dots, u_m) $, while $ {\partial} u $ and $ {\partial}_x {\partial} u $ represent the first and second derivatives of $ u $, respectively. We assume $ 0 < c_1\le c_2 \le \cdots \le c_m $. In this paper, we show global existence of classical solutions to the above system with small initial data under the ``null condition'' for systems with different propagation speeds. We also show ``almost-global'' existence for the above system for the case where the null condition is not satisfied.

We deal with the Cauchy problem for a $2$-evolution operator of Schrödinger type with $C^N$ coefficients in the time variable, $N>2$. We find the Levi conditions for well-posedness in Gevrey classes of index $1/2 + N/4$, which is the best possible, as we show by means of counterexamples.

We consider nonautonomous parabolic variational inequalities having the strong formulation \begin{align*} \langle { u'(t) }{ v - u(t) } \rangle + \langle { A(t) u(t) }{ v - u(t) }\rangle + \Phi(t, v ) - \Phi(t,u(t)) \geq 0, \\ \forall \, v \in V^{**}, \ a.e. \ t \geq s, \end{align*} where $u(s) = u_{s}$ for some admissible initial datum, $V$ is a separable Banach space with separable dual $V^{*}$, $A(t) : V^{**} \rightarrow V^{*}$ is an appropriate family of monotone operators, and $\Phi(t,\cdot) : V^{**} \rightarrow \mathbb R \cup \{ \infty \}$ is a family of convex, weak* lower-semicontinuous functionals. Well-posedness follows from an explicit construction of the related evolution family $\{ U(t,s) : t \geq s \}$. Illustrative applications are given.

We study the Dirichlet problem for the degenerate elliptic equations \[ P_0\Delta\,u+\gamma(\nabla\,P_0,\nabla\,u)=f(x,u,\nabla\,u)\quad \mbox{in $\Omega$},\qquad \mbox{$u=0$ on $\partial \Omega$}, \] where $\gamma\geq 0$ is a given parameter, $\Omega\subset \mathbb{R}^n$ is an annular region, the given function $P_0(x)$ is such that $|\nabla\,P_0|+P_0\geq \epsilon> 0$ in $\overline\Omega$, and $P_0=0$ on the outer boundary of $\Omega$. The equation is degenerate elliptic when $\gamma>0$, while for $\gamma=0$ it transforms into the classical Poisson equation. We introduce the weighted Hölder spaces suitable for the study of the problem throughout the range of the parameter $\gamma\geq 0$. We derive the Schauder-type estimates and prove the existence of a unique classical solution. It is shown that in the case $\gamma>0$ the solution of the degenerate equation and the given function $P_0$ possess the same regularity properties. In the case $\gamma=0$ (the Poisson equation) the regularity of $u$ is better than the regularity of $P_0$. The proof is based on a new method of estimating the derivatives of solutions of the Poisson equation near the boundary of the problem domain which requires neither differentiation of the equation, nor straightening the boundary.

This is the first of two papers on boundary optimal control problems with linear state equation and convex cost arising from boundary control of PDEs and the associated Hamilton--Jacobi--Bellman equation. In this paper we study necessary and sufficient conditions of optimality (Pontryagin maximum principle), and study the properties of a family of approximating problems that will be useful both in this paper and in the sequel. In the second paper we will apply dynamic programming to show that the value function of the problem is a solution of an integral version of the HJB equation, and moreover that it is the pointwise limit of classical solutions of approximating equations.

In this paper we prove the existence of a solution for a nonhomogeneous bi-layer shallow-water model in depth-mean velocity formulation. In [7] the homogeneous case was studied. The main difficulties in the nonhomogeneous case arise from the treatment of the boundary terms.