This note deals with the initial value problem for the abstract nonlinear nonlocal equation $ (\mathcal A u)' + (\mathcal B u) \ni f$, where $ \mathcal A $ is a possibly degenerate maximal monotone operator from the Hilbert space $ V $ to its dual space $ V ^* $, while $ \mathcal B $ is a nonlocal maximal monotone operator from $ L^2(0,T,V) $ to $ L^2(0,T;V^*)$. Assuming suitable boundedness and coerciveness conditions and letting $ \mathcal A $ be a subgradient, existence of a solution is established by making use of an approximation procedure. Applications to various classes of degenerate nonlinear integrodifferential equations are discussed.

We consider an autonomous system of a differential and a functional equation for one-dimensional motion of an object which attempts to regulate its distance from a given point by means of reflected signals. In a suitable, compact state space the forward initial value problem is well-posed. For certain configurations of the parameters involved we prove that there exist periodic orbits which are exponentially stable with asymptotic phase.

In this paper we are concerned with the existence and multiplicity of nodal solutions to the Dirichlet problem associated to the elliptic equation $\Delta u+q(|x|)g(u)=0$ in the unit ball in ${\bf R}^N$. The nonlinearity $g$ has a linear growth at infinity and zero, while the weight function $q$ is nonnegative in $[0,1]$ and strictly positive in some interval $[r_1,r_2]\subset [0,1]$. By means of a topological degree approach, we are able to prove the existence of solutions with prescribed nodal properties, depending on the behaviour of the ratio $g(u)/u$ at infinity and zero.

In this paper we discuss weak solutions of the multi-dimensional two-phase Stefan problem with the boundary condition including the subdifferential operator of the convex function on $\mathbb R$ so that the boundary condition is nonlinear and contains a multi-valued operator, in general. Kenmochi and Pawlow already established the uniqueness and existence of a solution to our problem in the sense of the vanishing viscosity solution. The purpose of this paper is to prove the uniqueness theorem for a solution defined in the usual variational sense. Our proof is due to the standard method in which the dual problem of the original problem plays a very important role.

In this paper we study the Cauchy problem $$ u_t = \Delta u^m-u^m \ \hbox{ in } \mathbb R^N\times (0,\infty) , $$ $u(x,0) = u_0(x),$ with $\frac{N-2}{N} < m < 1$ and $N\ge 2$. If $u_0 \not\equiv 0$ is a non-negative, compactly supported function such that and $u_0^m \in H^1(\mathbb R^N)\cap L^\infty(\mathbb R^N)$, then the solution $u$ vanishes identically after a least finite time $T>0$. We prove the asymptotic formula $$ u(x,t) \sim (1-m)^{1\over 1-m} (T-t)^{1 \over 1-m}w_*^m (|x-\bar x| ) $$ as $t\uparrow T$, for certain uniquely determined $\bar x\in \mathbb R^N$. Here $w_*$ is the unique positive radial solution of $$\Delta w- w + w^p =0 \ \hbox{ in } \mathbb R^N , $$ $$ w(x)\to 0 \hbox{ as } |x|\to \infty , $$ where $p= 1/m$.