The existence of smooth families of solutions bifurcating from the trivial solution for a two-parameter bifurcation problem for a class of variational inequalities is proved. As an example, a model of an elastic beam compressed by a force $\lambda$ and supported by a unilateral connected fixed obstacle at the height $h$ is studied. In the language of this example, we show that nontrivial solutions touching the obstacle on connected intervals bifurcate from the trivial solution and form smooth families parametrized by $\lambda$ and $h$. In particular, the corresponding contact intervals depend smoothly on $\lambda$ and $h$.

The goal of this paper is to study the symmetry properties of nonnegative solutions of elliptic equations involving a nonuniformly elliptic operator. We consider on a ball the solutions of \[ \Delta_pu+f(u)=0 \] with zero Dirichlet boundary conditions, for $p>2$, where $\Delta_p$ is the $p$-Laplace operator and $f$ a continuous nonlinearity. The main tools are a comparison result for weak solutions and a local moving-plane method which has been previously used in the $p=2$ case. We prove local and global symmetry results when $u$ is of class $C^{1,\gamma}$ for $\gamma$ large enough, under some additional technical assumptions.

In this work, we analyze a nonlinear population-dynamics model with age dependence and spatial diffusion, and where we are assuming the influence of a reaction term. Using a sub-supersolution method we derive existence and uniqueness results. We apply this method to study the existence and uniqueness of a positive solution of a generalized logistic and of a Holling-Tanner-type age-dependent model as well as its large-time behaviour.

We study the global existence and asymptotic behavior in time of small solutions to nonlinear Schrödinger equations with quadratic nonlinearities, \begin{eqnarray*} \left\{ \begin{array}{ll} i \partial_{t} u + \frac{1}{2}\Delta u = \mathcal{N}(u,\bar u), & (t, x) \in {\mbox{\bf{R}}} \times {\mbox{\bf{R}}}^3, \\ u(0, x) = u_{0}, & x \in {\mbox{\bf{R}}}^3, \end{array} \right. \label{0.1} \end{eqnarray*} where the initial data $u_{0}$ are sufficiently small in a suitable norm, $\bar{u}$ is the complex conjugate of $u$. The nonlinear term $\mathcal{N}$ is a smooth quadratic function in the neighborhood of the origin with respect to $u$ and $\bar u$ and does not contain the term $|u|^2$. Our purpose in this paper is to show there exists a unique final state $u_{+}$ such that $$ \big \|u(t)-e^{\frac{i t}{2} \Delta} u_{+} \big \|_{L^2} \le C t^{-\frac{5}{4}}, \ \ \text{ for small $u_{0}.$}

We study the global existence and large-time asymptotic behavior of solutions to the initial/boundary-value problem for the nonlinear nonlocal Whitham equation on a segment $\left( 0,a\right), $ \begin{equation} \left\{ \begin{array}{c} u_{t}+uu_{x}+\mathbb{K}u=0,\text{ }t>0,x\in \left( 0,a\right) \\ u(x,0)=u_{0}(x),\text{ }x\in \left( 0,a\right) , \end{array} \right. \tag*{(0.1)} \end{equation} where the pseudodifferential operator $\mathbb{K}u$ on a segment $\left[ 0,a \right] $ is defined by \begin{align} \mathbb{K}u = \theta _{a}(x)\frac{1}{2\pi i}\int_{-i\infty }^{i\infty }e^{px}K(p) \Big ( \widehat{u}(p,t)-\frac{u(0,t)-e^{-pa}u(a,t)}{p}\Big) dp, \tag*{(0.2)} \end{align} where $K(p)=C_{\alpha }p^{\alpha },$ $\alpha \in ( \frac{3}{2},2 ) ,$ and $C_{\alpha }$ is chosen by the dissipation conditions. We prove that if the initial data $u_{0}\in \mathbf{L}^{\infty }(0,a)$ have a small norm $ \left\Vert u_{0}\right\Vert _{\mathbf{L}^{\infty }} < \varepsilon ,$ then there exists a unique solution $u\in \mathbf{C}\left( \left[ 0,\infty \right) ;\mathbf{L}^{2}(0,a)\right) $ $\cap \mathbf{C}\left( \left( 0,\infty \right) ;\mathbf{H}^{1}(0,a)\right) $ to problem (0.2). Moreover, there exists a function $A\left( x\right) \in \mathbf{L}^{\infty }(0,a)$ such that the solution has the following asymptotics for large time $ t\rightarrow \infty $: \begin{equation*} u(x,t)=A\left( x\right) Bt^{-\frac{1}{\alpha }}+O ( t^{-\frac{1+\delta }{ \alpha }} ) , \end{equation*} uniformly with respect to $x\in \left( 0,a\right) ,$ where $\delta \in \left( 0,2-\alpha \right) .$

We consider asymptotic behaviors of star-shaped curves expanding by $V=1-K$, where $V$ denotes the outward-normal velocity and $K$ curvature. In this paper, we show the followings. The difference of the radial functions between an expanding curve and circle has its asymptotic shape as $t\rightarrow+\infty$. For two curves, if the asymptotic shapes are identical, then the curves are also. The set of all asymptotic shapes is dense in $C(S^1)$.

We investigate the following question: Given a set of $k$ partial differential equations for $m$ unknowns, where $k < m$, can we find $m-k$ additional equations in such a way that the full set of equations forms an elliptic system? We formulate a maximal rank condition which is obviously necessary. In general, however, the maximal rank condition is sufficient only if we allow the introduction of additional variables, not just additional equations. In particular, the equation ${\rm div}\,u=0$ can be part of an elliptic system for the components of the vector field $u$ only if the space dimension is 1,2,4, or 8.