The objective of this paper is to study nonlinear partial differential systems like $$ \partial_t {\bf u}- \Delta {\bf u} +{\bf H}(x,t,{\bf u}, \nabla {\bf u})={\bf G}(x,t), $$ with applications to the solution of stochastic differential games with $N$ players, where $N$ is arbitrarily large. It is assumed that the Hamiltonian ${\bf H}$ of the nonlinear system satisfies a quadratic growth condition in $\nabla {\bf u}$ and has a positive definite Jacobian ${\bf H_u}$. An energy estimate and the uniqueness property for bounded weak solutions are proved. Moreover, applications to stochastic games and financial economics such as modern portfolio theory are discussed.

This paper is concerned with the study of the unique continuation property associated with the defocusing Schrödinger equation \begin{eqnarray*} iu_{t} +\Delta u - |u|^2u =0 ~\hbox{ in }\Omega \times (0,\infty), \end{eqnarray*} subject to Dirichlet boundary conditions, where $\Omega \subset \mathbb{R}^2$ is a bounded domain with smooth boundary $\partial \Omega=\Gamma$. In addition, we prove exponential decay rates of the energy for the damped problem \begin{eqnarray*} iu_{t} +\Delta u - |u|^2u +i a(x) u =0 \hbox{ in } \mathbb{R}^2 \times (0,\infty), \end{eqnarray*} provided that $a(x) \geq a_0 >0$ almost everywhere in $\Omega_{R}:=\{x\in \mathbb{R}^2 : |x| \geq R\}$, where $R>0$.

We continue our study of the well-posedness theory of a one-parameter family of coupled KdV-type systems in the periodic setting. When the value of a coupling parameter ${\alpha} \in (0, 4) \setminus \{1\}$, we show that the Gibbs measure is invariant under the flow and the system is globally well posed almost surely on the statistical ensemble, provided that certain Diophantine conditions are satisfied.

In this work, we prove a multiplicity result for a class of quasilinear elliptic equation involving the subcritical Hardy-Sobolev exponent, and singularities both in the operator and in the non-linearity. Precisely, we study the problem $$ \begin{cases} {-\operatorname{div} \big[ |x_N|^{-ap} | \nabla u |^{p-2} \nabla u \big] + \lambda|x_N|^{-(a+1-c)p} |u|^{p-2}u } & \\ \ \ \ = |x_N|^{-bq} |u|^{q-2} u + f & \mbox{in }\mathbb R_+^N \\ {u} = 0 &\mbox{on } \partial \mathbb R_+^N, \end{cases} $$ where we denote $ x=(x_1,x_2,\dots,x_N)=(x',x_N) \in \mathbb R^{N-1}\times \mathbb R $, $ \mathbb R_+^N= \left\{ x \in \mathbb R^N : x_N > 0 \right\} $, $ \partial \mathbb R_+^N= \left\{ x \in \mathbb R^N : x_N = 0 \right\} $, and we consider $ 1 < p < N $, $ 0 \leqslant a < (N-p)/p $, $ a < b < a+1 $, $c=0 $, $ d \equiv a+1-b $, $ q = q(a,b) \equiv Np/(N - pd) $ (the Hardy-Sobolev critical exponent), $ \lambda \in \mathbb R $ is a parameter, and $ f \in \big( L_b^q(\mathbb R_+^N) \big)^{*} $, the dual space of the weighted Lebesgue space. We prove an existence result for the case $ f \equiv 0 $ and a multiplicity result in the case $ \lambda = 0 $ for non-autonomous perturbations~$ f \not\equiv 0.$

We present an existence result for periodic solutions of autonomous differential delay equations with several fixed delays \begin{eqnarray*} x'(t) = \sum_{i=1}^D f_i (x(t - d_i)), \end{eqnarray*} where the functions $f_i$ are close to two-valued step functions $h_i$. We give conditions under which existence of periodic solutions of the more tractable problem $$ y'(t) = \sum_{i=1}^D h_i (y(t - d_i)) $$ implies the existence of similar periodic solutions of the original problem. Our approach is to show that an analog of a Poincaré map has non-trivial fixed-point index.

We consider the following quasilinear elliptic system: \begin{equation} \begin{cases} \Delta u+u(a-u-cv)=0 \ \ & \mbox{in} \ \ \Omega ,\\ \Delta \Big [ \Big ( 1+\dfrac{\gamma }{1+\beta u} \Big ) v \Big ]+v(b+du-v)=0 \ \ & \mbox{in} \ \ \Omega ,\\ u=v=0 \ \ & \mbox{on} \ \ \partial\Omega, \end{cases} \nonumber \end{equation} where $\Omega $ is a bounded domain in $\mathbb{R}^{N}$. This system is a stationary problem of a prey-predator model with non-linear diffusion $\Delta (\frac{v}{1+\beta u})$, and $u$ (respectively $v$) denotes the population density of the prey (respectively the predator). Kuto [15] has studied this system for large $\beta $ under the restriction $b>(1+\gamma )\lambda_{1}$, where $\lambda_{1}$ is the least eigenvalue of $-\Delta$ with homogeneous Dirichlet boundary condition. The present paper studies two {\it shadow systems} and gives the complete limiting characterization of positive solutions as $\beta\to\infty$ without any restriction on $b$.

In this work, we prove a multiplicity result for a class of quasilinear elliptic equation involving the subcritical Hardy-Sobolev exponent, and singularities both in the operator and in the non-linearity. Precisely, we study the problem \[ \left\{\begin{array}{rcll} -\mbox{div} & = & [|x_N|^{-ap}|\nabla_u|^{p-2}\nabla u]+\lambda|x_N|^{-(a+1-c)p}|u|^{p-2}u \\ & = & |x_N|^{-ap}|u|^{q-2}u+f & \mbox{in }\mathbb{R}^N_+ \\ u & = & 0 & \mbox{on } \partial\mathbb{R}^N_+ \end{array}\right. \]where we denote $ x=(x_1,x_2,\dots,x_N)=(x',x_N) \in \mathbb R^{N-1}\times \mathbb R $, $ \mathbb R_+^N= \left\{ x \in \mathbb R^N : x_N > 0 \right\} $, $ \partial \mathbb R_+^N= \left\{ x \in \mathbb R^N : x_N = 0 \right\} $, and we consider $ 1 < p < N $, $ 0 \leqslant a < (N-p)/p $, $ a < b < a+1 $, $c=0 $, $ d \equiv a+1-b $, $ q = q(a,b) \equiv Np/(N - pd) $ (the Hardy-Sobolev critical exponent), $ \lambda \in \mathbb R $ is a parameter, and $ f \in \big( L_b^q(\mathbb R_+^N) \big)^{*} $, the dual space of the weighted Lebesgue space. We prove an existence result for the case $ f \equiv 0 $ and a multiplicity result in the case $ \lambda = 0 $ for non-autonomous perturbations~$ f \not\equiv 0.$

We prove the growth rate of global solutions of the equation $u_t=\Delta u-u^{-\nu}$ in ${{\mathbb{R}}}^n\times (0,\infty)$, $u(x,0)=u_0>0$ in ${{\mathbb{R}}}^n$, where $\nu>0$ is a constant. More precisely for any $0 <u_0\in C({{\mathbb{R}}}^n)$ satisfying $A_1(1+|x|^2)^{\alpha_1}\le u_0\le A_2(1+|x|^2)^{\alpha_2}$ in ${{\mathbb{R}}}^n$ for some constants $1/(1+\nu)\le\alpha_1 <1$, $\alpha_2\ge\alpha_1$ and $A_2\ge A_1= (2\alpha_1(1-{\varepsilon})(n+2\alpha_1-2))^{-1/(1+\nu)}$ where $0 <{\varepsilon} <1$ is a constant, the global solution $u$ exists and satisfies $A_1(1+|x|^2+b_1t)^{\alpha_1}\le u(x,t)\le A_2(1+|x|^2+b_2t)^{\alpha_2}$ in ${{\mathbb{R}}}^n\times (0,\infty)$ where $b_1=2(n+2\alpha_1-2){\varepsilon}$ and $b_2=2n$ if $0 <\alpha_2\le 1$ and $b_2=2(n+2\alpha_2-2)$ if $\alpha_2>1$. When $0 <u_0\le A(T_1+|x|^2)^{1/(1+\nu)}$ in ${{\mathbb{R}}}^n$ for some constant $0 <A <((1+\nu)/2n)^{1/(1+\nu)}$, we prove that $u(x,t) \le A(b(T-t)+|x|^2)^{1/(1+\nu)}$ in ${{\mathbb{R}}}^n\times (0,T)$ for some constants $b>0$ and $T>0$. Hence, the solution becomes extinct at the origin at time $T$. We also find various other conditions for the solution to become extinct in a finite time and obtain the corresponding decay rate of the solution near the extinction time.

In this paper, we investigate the exact number of positive solutions of the Dirichlet boundary-value problem $u''-u^{-\gamma}+\beta=0$. We will show that the exact number of positive solutions can be 2, 1, 0, depending on the length of the interval and $\gamma$. This solves some open problems posed in [1].

We show that the condition $0\le \beta \le \frac{1}{2\ln 2}$ is necessary for the chaoticity of the solution of the cell population model \begin{equation}\label{eq1} \left\{\begin{array}{ll} \frac{\partial u(t,x)}{\partial t}=-\frac{\partial(x u(t,x))}{\partial x}+\gamma u(t,x)-\beta u(t,x)+4\beta u(t,2x)\chi_{(0,\frac{1}{2})}(x),\\ u(0,\cdot)= f \in L^1(0,1). \end{array} \right. \end{equation} (If $\gamma -3\beta >0$, then this condition is known to be sufficient.) The analysis depends on solving a forward delay equation.