Let $X, Y$ be real Banach spaces and $B_r(0)$ the open ball in $Y$ with center at zero and radius $r> 0.$ Let $G$ be an open subset of $X.$ One of the problems that constantly arise in the study of perturbations of maximal monotone and $m$-accretive operators is the verification of the invariance of a degree function on parameter-dependent domains. For example, these domains can be of the type $A_tG, $ $t\in [0,1],$ where $A_t:X\to 2^Y$ has a continuous inverse $A_t^{-1}:Y\to X$ and $D(A_t)\cap G \neq \emptyset.$ Browder has shown that such degrees are invariant in various situations where the operator $A_t$ is a local homeomorphism, but no such result is known (or can be easily derived from known results) when $A_t$ is a discontinuous (possibly densely defined and/or multi-valued) mapping. The main purpose of this paper is to show that, under certain natural conditions, the Leray-Schauder degree $d(I-H(t),A_tG\cap B_m(0),0)$ is well defined and constant for all $t\in [0,1]$ and all sufficiently large $m,$ where $H(t):Y\to Y$ is an appropriate compact homotopy. This fundamental result can be used to improve various results in the perturbation theory of $m$-accretive and maximal monotone operators. In particular, results by Hirano and Kartsatos, involving saddle point conditions in real Hilbert spaces, are improved. Ideas from the main theory are also used to create new degree theories for perturbed $m$-accretive operators as well as demicontinuous accretive operators defined on closures of bounded open subsets of $X$ when $X^*$ is uniformly convex.

Suppose $0\le u_0\in L^1(R^2)\cap L^p(R^2)$ for some constant $p>1$ is a radially symmetric and monotone decreasing function of $r=|x|\ge 0$ such that either supp $u_0$ is compact or $0\le u_0(x)\le C\min (1,|x|^{-(2+\delta)})$ in $R^2$ for some constants $C>0$, $\delta>0$. We will show that if $u$ is the solution of the equation $u_t=\Delta\text{ log }u$, $u>0$, in $R^2\times (0,T)$, $u(x,0)=u_0(x)$ in $R^2$, which satisfies the conditions $r\partial ($log $u(x,t)) /\partial r\to -4$ uniformly on any compact subset of $(0,T)$ as $r=|x|\to\infty$ and $\int_{R^2}u(x,t)dx =\int_{R^2}u_0(x)dx-8\pi t$ $\forall 0\le t < T$, where $T=\int_{R^2}u_0dx/8\pi$, then the function $v(x,s)=u(x,t)/(T-t)$ where $s=-\text{ log }(T-t)$ will converge uniformly on every compact subsets of $R^2$ to a solution of the equation $\Delta\text{ log }v+v=0$ in $R^2$ as $s\to\infty$. In other words $u(x,t)\approx 8 \lambda (T-t)/(\lambda +|x|^2)^2$ as $t$ tends to $T$ for some $\lambda >0$.

Paper [11] showed exact null controllability of two non-classical abstract parabolic systems with distributed control: (i) a structurally damped second-order abstract equation, and (ii) an abstract thermoelastic system. We now complement [11] by providing optimal blow-up estimates of norms of fast controls.

We address a viscous Cahn-Hilliard equation describing the phase separation process in a binary alloy. Such a material is subject to the influence of internal and external mechanical stresses, whose contribution is assumed to be known. The physical modelling refers to a recent work by Dreyer and Müller. The main features and difficulties of this model are given by a highly nonlinear fourth order elliptic term, a strong constraint imposed by the presence of a double obstacle energy potential, and the dependence of the mobility matrix on the concentration variable (i.e., the unknown of the problem). We are able to prove global existence of solutions of the corresponding initial boundary value problem by using approximation and compactness tools. Hence, we perform some asymptotic analysis of the problem and obtain, at the limit, two models of independent physical interest.