### A compact evolution operator generated by a nonlinear time-dependent m-accretive operator in a Banach space.

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Given a nonlinear autonomous system of ordinary or partial differential equations that has at least local existence and uniqueness, we offer a linear condition which is necessary and sufficient for existence to be global. This paper is largely concerned with numerically testing this condition. For larger systems, principals of computations are clear but actual implementation poses considerable challenges. We give examples for smaller systems and discuss challenges related to larger systems. This...

We prove existence and uniqueness of solutions for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth.

We prove existence and uniqueness of entropy solutions for the Neumann problem for the quasilinear elliptic equation $u-\mathrm{div}\phantom{\rule{0.166667em}{0ex}}\mathbf{a}(u,Du)=v$, where $v\phantom{\rule{-0.166667em}{0ex}}\in \phantom{\rule{-0.166667em}{0ex}}{L}^{1}$, $\mathbf{a}(z,\xi )={\nabla}_{\xi}f(z,\xi )$, and $f$ is a convex function of $\xi $ with linear growth as $\parallel \xi \parallel \to \infty $, satisfying other additional assumptions. In particular, this class includes the case where $f(z,\xi )=\varphi \left(z\right)\psi \left(\xi \right)$, $\varphi \>0$, $\psi $ being a convex function with linear growth as $\parallel \xi \parallel \to \infty $. In the second part of this work, using Crandall-Ligget’s iteration scheme, this result will permit us to prove existence and uniqueness of entropy solutions for the...

In the theory of elliptic equations, the technique of Schwarz symmetrization is one of the tools used to obtain a priori bounds for classical and weak solutions in terms of general information on the data. A basic result says that, in the absence of lower-order terms, the symmetric rearrangement of the solution $u$ of an elliptic equation, that we write ${u}^{*}$, can be compared pointwise with the solution of the symmetrized problem. The main question we address here is the modification of the method to...

Si studiano le proprietà delle soluzioni dell'equazione semilineare astratta ${u}^{\prime}(t)=\mathrm{\Lambda}u(t)+\phi (t,u(t))$ quando $\mathrm{\Lambda}$ è il generatore infinitesimale di un semigruppo analitico in uno spazio di Banach. Vengono provati nuovi teoremi di regolarità anche nel caso in cui $\phi $ non è continuo in tutto lo spazio.

Based on the notion of A - monotonicity, a new class of nonlinear variational inclusion problems is presented. Since A - monotonicity generalizes H - monotonicity (and in turn, generalizes maximal monotonicity), results thus obtained, are general in nature.

In this paper we consider a second order differential equation involving the difference of two monotone operators. Using an auxiliary equation, a priori bounds and a compactness argument we show that the differential equation has a local solution. An example is also presented in detail.