In this article we prove existence of positive solutions for the nonlinear elliptic equation \begin{eqnarray*} \begin{array}{rcll}{{{\mathcal M}_{\lambda,\Lambda}^+}}(D^2u)-\gamma u+f(u) & = & 0 \quad\mbox{in} \quad \Omega ,\\ u & = & 0 \quad \mbox{on}\quad \partial \Omega,\\ \end{array} \end{eqnarray*} where ${{{\mathcal M}_{\lambda,\Lambda}^+}}$ denotes Pucci's extremal operator with parameters $0 s < \lambda\leq \Lambda $ and $\Omega$ is convex smooth domain in ${\hbox{\bf R}}^N$, $N\geq 3$. The result applies to a class of nonlinear functions $f$, including the model cases: i) $\gamma=1$ and $f(s)=s^p$, $1s < p \leq p^+$; and ii) $\gamma=0$, $f(s)=\alpha s+s^p$, $1s <p\leq p^+$, and $ 0\le\alpha s <\mu^+_1$. Here $p^+=\tilde{N}^+/(\tilde{N}^+-2)$, $\tilde{N}^+=\lambda(N-1)/\Lambda+1$, and $\mu^+_1$ is the first eigenvalue of ${{{\mathcal M}_{\lambda,\Lambda}^+}}$ in $\Omega$. Analogous results are obtained for the operator ${{{\mathcal M}_{\lambda,\Lambda}^-}}$.

The viscoelastic Euler-Bernoulli equation with nonlinear and nonlocal damping $$u_{tt}+\Delta^2u-\int_0^tg(t-\tau )\Delta^2u(\tau )\,d\tau +a(t )u_t=0\,\,\,\hbox{in}\,\,\,\Omega\times {\bf R}^{+},$$ where $a(t)=M\left(\int_{\Omega}\left|\nabla u(x,t)\right|^2dx\right )$, is considered in bounded or unbounded domains $\Omega$ of ${\bf R}^ n$. The existence of global solutions and decay rates of the energy are proved.

This work is concerned with the study of an initial and boundary value problem for a nonconserved system of phase field equations arising from the Penrose-Fife approach to phase transitions problems. Several works deal with variations of the same problem coupled with third type boundary conditions for the heat flux. On the contrary, our aim is to consider the case of the nonhomogeneous Neumann boundary condition for the heat flux, to find well-posedness for a weak formulation of this problem, and to prove a regularity result in case of smoother data and a slightly less general heat flux law.

This paper is concerned with a supercritical semilinear diffusion equation. We show the existence of a solution that undergoes a birth-and-death process of a single peak emerging at arbitrarily prescribed positions and heights. In particular the solution has no asymptotic center of radial symmetry as time approaches infinity. We also construct a solution with arbitrarily prescribed grow-up set.

Time-independent bounds for the classical solutions of a certain nonlinear integroparabolic equation of the Fokker-Planck type are established. Such an equation describes the statistical time evolution of large populations of nonlinearly coupled random oscillators with inertia. The basic tool is deriving ``energy-like" estimates. A comparison theorem is also proved to obtain estimates for the solutions in terms of some special solutions.

We consider weak solutions to the initial- boundary-value problem for the system $\frac{\partial u}{\partial t}- \mbox{div}(K(u)\nabla u) = \sigma(u)|\nabla \varphi|^2$, $\mbox{div}\left(\sigma(u)\nabla\varphi\right) =0$ in the case where $K(u)$ and $\sigma(u)$ may both tend to $0$ as $u\rightarrow \infty$. It is established that $u$ in the solution belongs to some Orlicz space under certain conditions. This implies that $u$ is exponentially integrable in some cases.

It is well-known that the pressure of a lubricating fluid filling the gap between two solid surfaces satisfies the Reynolds equation involving the distance function, $h$, between both planes, as a crucial coefficient. Nevertheless, in most of the applications the function $h$ is not known a priori. Here we consider the simple case in which the surfaces are two parallel planes and assume prescribed the total force applied upon one of the surfaces. We give some sufficient conditions on the total force in order to solve this inverse problem. We show that in the incompressible case, such a condition is also necessary.

In this paper we deal with systems of nonlinear wave equations in two space dimensions. When the system has common propagation speeds and cubic nonlinearity, the small data global existence result was obtained by Katayama [9], provided that the cubic part of Taylor's expansion for the nonlinearity satisfies the so-called null condition. The aim of this paper is to extend the result to the case where the system has multiple speeds of propagation. To realize this, we make use of a kind of Hardy's inequality given in Lemma 2.2 below, which creates the loss of decay but only with respect to $(1+||x|-c_i t|)$. Thus we are able to absorb such a loss by means of the decay estimates in Proposition 4.2 below.

The global Cauchy problem for nonlinear Dirac and Klein-Gordon equations in space--time $\mathbb R^{n+1}$ is studied in Sobolev and Besov spaces. Global existence of small solutions is proved under a scale-invariant setting when reduced to the corresponding massless case.

We study global existence in time of small solutions to the Cauchy problem for the nonlinear damped wave equation \begin{equation} \left\{ \begin{array}{c} \partial _{t}^{2}u+\partial _{t}u-\Delta u=\mathcal{N}\left( u\right) ,\quad x\in \mathbf{R}^{n},\text{ }t>0, \\ u(0,x)=\varepsilon u_{0}\left( x\right) ,\partial _{t}u(0,x)=\varepsilon u_{1}\left( x\right) , \quad x\in \mathbf{R}^{n}, \end{array} \right. \tag*{(0.1)} \end{equation} where $\varepsilon >0$. The nonlinearity $\mathcal{N}\left( u\right) \in \mathbf{C}^{k}\left( \mathbf{R}\right) $ satisfies the estimate \begin{equation*} \Big | \frac{d^{j}}{du^{j}}\mathcal{N}\left( u\right) \Big | \leq C\big | u\big | ^{\rho -j}, \quad 0\leq j\leq k\leq \rho . \end{equation*} The power $\rho >1+\frac{2}{n}$ is considered as super critical for large time. We assume that the initial data \begin{equation*} u_{0}\in \mathbf{H}^{\alpha ,0}\cap \mathbf{H}^{0,\delta },\text{ }u_{1}\in \mathbf{H}^{\alpha -1,0}\cap \mathbf{H}^{0,\delta }, \end{equation*} where $\delta >\frac{n}{2},$ $\left[ \alpha \right] \leq \rho ,\alpha \geq \frac{n}{2}-\frac{1}{\rho -1}$ for $n\geq 2$, and $\alpha \geq \frac{1}{2}- \frac{1}{2\left( \rho -1\right) }$ for $n=1.$ Weighted Sobolev spaces are \begin{equation*} \mathbf{H}^{l,m}= \Big \{ \phi \in \mathbf{L}^{2}; \big \| \left\langle x\right\rangle ^{m}\left\langle i\partial _{x}\right\rangle ^{l}\phi \left( x\right) \big \| _{\mathbf{L}^{2}}s < \infty \Big \} , \end{equation*} where $\left\langle x\right\rangle =\sqrt{1+x^{2}}.$ Then we prove that there exists a small $\varepsilon _{0}>0$ such that for any $\varepsilon \in \left( 0,\varepsilon _{0}\right] $ there exists a unique global solution $ u\in \mathbf{C}\left( \left[ 0,\infty \right) ;\mathbf{H}^{\alpha ,0}\cap \mathbf{H}^{0,\delta }\right) $ for the Cauchy problem (0.1) and solutions satisfy the time decay property \begin{equation*} \big \| u ( t ) \big \| _{\mathbf{L}^{p}}\leq Ct^{-\frac{n}{2 } ( 1-\frac{1}{p} ) } \end{equation*} for all $t>0$, where $2\leq p\leq $ $\frac{2n}{n-2\alpha }$ if $\alpha s < \frac{n}{2},$ $2\leq ps <$ $\infty $ if $\alpha =\frac{n}{2},$ and $2\leq p\leq $ $\infty $ if $\alpha \gt\frac{n}{2}.$

In this paper, we consider the Cauchy problem for linear first-order partial differential equations with discontinuous coefficients. We show that, under suitable assumptions, it has a unique continuous solution. Moreover, this solution is stable under perturbation of the coefficients. We also show that this solution can be expressed explicitly by integrating along generalized characteristics.

In this paper, the boundary stabilization of wave equations with variable coefficients and memory is considered, and the uniform energy decay rate for the problem is established by Riemannian geometry methods and sharp trace regularity.

In this paper, we consider the equation $\Delta ^2 u=K u^5$, $u>0$ in $\Omega$, $u=\Delta u=0$ on $\partial\Omega$, where $K$ is a positive function and $\Omega$ is a bounded and smooth domain in $\mathbb R^6$. Using the theory of critical points at infinity, we give some topological conditions on $K$ to ensure some existence results.

Let $N\ge 2,$ $ B_{R}\subset \mathbb R^{N}$ denote the open ball of radius $R$ about the origin. Let $f:[0,\infty)\rightarrow [0,\infty)$ be a continuous map which behaves like $e^{s^{p}}$ for some $p\in [1,\frac{N}{N-1}]$ as $s\rightarrow \infty$ and like $s^{q}$ for some $q\in (0, N-1)$ as $s\rightarrow 0$. Then we show that there exists $\Lambda \in (0,\infty)$ such that the following problem, $$(P_{\lambda})\hspace{3cm}\left\{ \begin{array}{cllll}\left. \begin{array}{rllll} -\Delta_{N}u & = & \lambda f(u)\\ u & > & 0 \end{array}\right\} \;\; \text{in} \; B_{1}(0), \\[2mm] u\;\;=\;\; 0 \;\;\text{on}\;\; \partial B_{1}(0), \end{array}\right. $$ admits at least two solutions for all $\lambda \in (0,\Lambda)$ and no solution for $\lambda >\Lambda$.