We establish uniqueness of a solution to the initial-value problem for the integro-differential equation $$ \frac{d}{dt} \int_0^1J(x)\mu_t(dx) = \frac1{2}\int_0^1\int_0^1 \frac{J'(y)-J'(x)}{y -x} \cdot\frac{\mu_t(dx)\mu_t(dy)}{|y-x|^{\gamma}}, \quad\;\; t>0 $$ where the equality is required to hold for every smooth testing function $J$ with $J'(0) =J'(1) = 0$, and the solution $\mu_t=\mu_t(dx)$ is a finite measure on the unit interval $[0,1]$ for each $t$ and ${\gamma}$ a constant from the open interval $(-1,1).$ Stationary solutions are given explicitly and the convergence to them of general time-dependent solutions is proved.

Using the method of Bourgain (actually we use techniques developed in the paper of Ginibre, Tsutsumi, Velo (cf. [9])) we prove well-posedness of a generalized Zakharov system, describing the spontaneous generation of magnetic field in a cold plasma, in the framework of classical Sobolev spaces. Using the conservation laws of the system we extend these solutions globally in the energy space.

In this work we prove the existence of solutions for an elliptic system between lower and upper solutions when the nonlinearities are Hölder continuous functions without a Lipschitz condition. Specifically, under appropriate conditions of monotony on the nonlinear reaction terms we introduce two monotone sequences which converge to a minimal and a maximal solution respectively. Finally, we apply these results to a dynamical population problem with "slow" diffusion.

We introduce Conley index theory for non-smooth dynamical systems, and by using this theory, we obtain a global bifurcation theorem for such systems.

We consider the semilinear non-autonomous evolution equation $\frac{d}{dt}u(t)=Au(t)+G(t,u(t))$, $t\geq s\geq 0,$ where $(A,D(A))$ is a Hille-Yosida operator on a Banach space $X$ and $G$ is a continuous function on $\mathbb R_+\times \overline{D(A)}$ with values in the extrapolated Favard class corresponding to $A$. In our main results we present principles of linearized stability and instability for a solution of such an equation. Our approach is based on the theory of extrapolation spaces. We apply the results to non-autonomous semilinear retarded differential equations.

This paper considers optimal control problems for an abstract linear system with a quadratic cost functional in a Hilbert space on a fixed time interval $[0,T],$ $ 0 < T < +\infty$. The controls are unbounded and the state weight operators are indefinite and not necessarily smoothing. Some new inequalities are established and applied to prove that the optimal control and the optimal trajectory are continuous. The equivalences among the solvability of the LQ problems, the two-point boundary value problem and the Fredholm integral equation are proved and the state feedback representation for the optimal control is also given in terms of the solution to the Fredholm integral equation. Finally, we derive the closed-loop synthesis of the optimal control via the solution to the Riccati integral equation which exists under some mild conditions. An explanatory example is given at the end. The results are complementary to those of [15, 1}] and [17].

A model Boltzmann equation (see formulas (1.1.6) -- (1.1.9) below) without Grad's angular cutoff assumption is considered. One proves: 1) the instantaneous smoothing in both position and velocity variables by the evolution semigroup associated to the Cauchy problem for this model; 2) the derivation of the analogue of the Landau-Fokker-Planck equation in the limit when grazing collisions prevail.

We consider the following quasilinear elliptic problem \begin{equation} \begin{cases} -\mbox{div } |\nabla u|^{p-2}\nabla u)+c|u|^{p-2}u=|u|^ {p^*-2}u+f(x,u)+h(x)\\ u\in W^{1, p}(R^N), \ \ \ N>p\geq 2, \end{cases} \tag*{*} \end{equation} where $c>0, $ $ p^*=\frac{Np}{N-p},$ $ h(x)\in W^{-1, \frac{p}{p-1}}(R^N)$ (i.e., the dual space of $W^{1, p}(R^N)$), $f(x,0)=0 $ and $ f(x,u)$ is a lower-order perturbation of $|u|^ {p^*-2}u$ in the sense that $\lim_{u\rightarrow \infty}\frac{f(x,u)}{|u|^ {p^*-2}u}=0$. It is well known that (*) has only a trivial solution if $f(x,u)\equiv h(x)\equiv 0$ by a Pohozaev type identity, but (*) has a nontrivial solution if there is a subcritical perturbation, e.g., $h(x){\equiv} 0$ and $f(x,u){\not\equiv} 0$. In this paper, we prove that (*) has at least two distinct solutions if there are two perturbations, i.e., $f(x,u) {\not\equiv} 0 $ and $ h(x){\not\equiv} 0$ (inhomogeneous term) with $\| h\|$ small enough.

In this work we give a result of exact controllability for a thermoelastic system in which the control term is placed solely in the thermal equation. With such an indirect control input, one is able to control exactly the displacement of the plate, as well as the temperature. This exact controllability occurs in arbitrarily small time. In the case that the moment of inertia parameter for the plate is absent (i.e., $\gamma =0$ below), then one is provided here with a result of exact controllability for a thermoelastic system which is modelled by the generator of an analytic semigroup. The proof here depends upon a multiplier method so as to attain the associated observability inequality. The particular multiplier invoked is of an operator theoretic nature, and has been used previously by the author in deriving stability results for this partial differential equation model.

Global in time existence and uniqueness of classical solutions to a certain nonlinear parabolic partial differential equation, containing an integral term, are proved. Smoothness regularity and time-independent estimates for all partial derivatives are also obtained. Such an equation is of a non-standard type, and governs the time evolution of certain populations of infinitely many nonlinearly coupled random oscillators, described by a model first proposed by Kuramoto and Sakaguchi.

In this paper we consider a class of semilinear elliptic boundary value problems with nonlinear boundary conditions. The continuation method or the implicit function theorem is used to prove the existence of smooth branches of positive solutions. The characterization of the branches, and the uniqueness and asymptotic behavior of positive solutions are also studied by using some comparison principles with semilinear elliptic boundary value problems with linear boundary conditions.

We prove some existence and regularity results for solutions of equations in the form $ -\mathrm{div}(a(x,u) \nabla u) = f$, where $a(x,s) : \Omega \times {\mathbb R} \rightarrow {\mathbb R}$ is a bounded Carath\'eodory function satisfying the inequality $a(x,s)\ge (1+|s|)^{-\theta}$ with $0 \leq \theta \leq1$ and $\Omega$ is a bounded open set of ${\mathbb R}^2$.

This paper deals with a class of nonlinear Dirac equations. We relate their solutions to those of nonlinear Schrödinger equation by a perturbation parameter. It is demonstrated that the nondegenerate solution (ground state) of Schrödinger equation generates solutions for Dirac equation.

A strong comparison principle (SCP, for brevity) is obtained for nonnegative weak solutions $u\in W_0^{1,p}(\Omega)$ of the following class of quasilinear elliptic boundary value problems, \begin{equation} -\mbox{div }( {\bf a}(x,\nabla u) ) - b(x,u) = f(x) \;\hbox{ in } \Omega ; \quad u = 0 \;\hbox{ on } \partial\Omega . \tag*{(P)} \end{equation} Here, $p\in (1,\infty)$ is a given number, $\Omega$ is a bounded domain in $\mathbb R^N$ with a connected $C^2$-boundary, ${\bf a}(x,\nabla u)$ and $b(x,u)$ are slightly more general than the functions $a_0(x) |\nabla u|^{p-2}\nabla u$ and $b_0(x) |u|^{p-2} u$, respectively, with $a_0\geq $const$ > 0$ and $b_0\geq 0$ in $L^\infty (\Omega)$, and $0\leq f\in L^\infty (\Omega)$. Validity of the SCP is investigated also in the case when $b_0\leq 0$ depending upon whether $p\leq 2$ or $p>2$. The methods of proofs are new.

We study some time independent Schrödinger equations on complete noncompact manifolds. By showing that these equations have global positive solutions bounded away from zero, we prove that a large class of manifolds are conformal to complete manifolds with zero scalar curvature. They include those with Ricci curvature being nonnegative and the scalar curvature satisfying $V(x) \le C/d^{2}(x, x_{0})$ for an arbitrary $C>0$. As we have shown that there are noncompact manifolds with positive scalar curvature, which are not conformal to manifolds with positive constant scalar curvature, the issue of prescribing zero scalar curvature becomes interesting. The current result gives a partial answer to the Yamabe problem on noncompact manifolds with nonnegative Ricci curvatures.

We derive estimates for solutions to the equations like $$-{\operatorname{div}}(|\nabla u|^{p-2}\nabla u)=f\,,$$ where $f$ belongs to weak $L^q$ spaces. As applications of our results we show that the entropy solutions of $$-{\operatorname{div}}(|\nabla u|^{p-2}\nabla u)=|u|^{a-1}u$$ are regular provided that $0\le a < n(p-1)/(n-p)$.