We consider the Cauchy-Neumann problem for the heat equation in the exterior domain $\Omega$ of a compact set in ${\bf R}^N$ ($N\ge 2$). In this paper we give an estimate of the $L^\infty$-norm of the gradient of the solutions.

In this paper, we study the existence and uniqueness of mild solutions, and also the almost sure exponential stability of mild solutions of stochastic neutral partial functional differential equations in real separable Hilbert spaces using Hölder type conditions on the nonlinear terms. The results obtained here generalize the main results from [Govindan, Stochastics 77 (2005), 139--154].

We establish the local well posedness of the generalized Benjamin-Ono equation $\partial_tu+\mathcal{H}\partial_x^2u\pm u^k\partial_xu=0$ in $H^s({\mathbb{R}})$, $s > 1/2-1/k$ for $k\geq 12$ and without smallness assumption on the initial data. The condition $s > 1/2-1/k$ is known to be sharp since the solution map $u_0\mapsto u$ is not of class $\mathcal{C}^{k+1}$ on $H^s({\mathbb{R}})$ for $s < 1/2-1/k$. On the other hand, in the particular case of the cubic Benjamin-Ono equation, we prove the ill posedness in $H^s({\mathbb{R}})$, $s < 1/3$.

The Cauchy problem for the nonperiodic KdV equation is shown by the iteration method to be locally well-posed in $H^{-3/4}(\mathbb R )$. In particular, solutions are unique in the whole Banach space for the iteration. This extends the previous well-posedness result in $H^s$, $s>-3/4$ obtained by Kenig, Ponce and Vega (1996) to the limiting case, and improves the existence result in $H^{-3/4}$ given by Christ, Colliander and Tao (2003). Our result immediately yields global well-posedness for the KdV equation in $H^{-3/4}(\mathbb R )$ and for the modified KdV equation in $H^{1/4}(\mathbb R )$, combined with the argument of Colliander, Keel, Staffilani, Takaoka and Tao (2003).

We give a few existence results of a weak-renormalized solution for a class of nonlinear Boussinesq systems: \begin{eqnarray*} & \dfrac{\partial u}{\partial t}+(u\cdot\nabla)u- 2 \textrm{ div } (\mu(\theta) D u)+\nabla p= F(\theta) & \textrm{ in } \Omega\times(0,T),\\ & \dfrac{\partial b(\theta)}{\partial t}+u\cdot\nabla b(\theta)-\Delta \theta = 2 \mu(\theta) |D u |^2 & \textrm{ in } \Omega\times(0,T),\\ & \textrm{div }u = 0 & \textrm{ in } \Omega\times(0,T), \end{eqnarray*} where $u$ is the velocity field of the fluid, $p$ is the pressure and $\theta$ is the temperature. The function $\mu(\theta)$ is the viscosity of the fluid and the function $F(\theta)$ is the buoyancy force which satisfies a growth assumption in dimension $2$ and is bounded in dimension $3$. The usual techniques for Navier-Stokes equations are mixed with the tools involved for renormalized solutions.

We consider a regularity estimate for a solution of the homogeneous Dirichlet problem for $N$-Laplace equations in a bounded domain $\Omega\subset{\mathbb R}^N$ with external force $f\in L^1(\Omega)$. Introducing the generalized Lorentz-Zygmund space, we show the multiple exponential integrability of the Brezis-Merle type for an entropy solution of the Dirichlet problem of the $N$-Laplace equation. We also discuss conditions on $f$ that guarantee the solutions are bounded.

Let $A$ be a hyperbolic bisectorial operator on a Banach space. In this paper we study the optimal regularity of the solutions of the abstract first-order evolution equation $u' (t) = Au(t) + f (t) $ on the whole line, depending on the regularity of the inhomogeneity $f.$

In this paper we study a stochastic Cahn-Hilliard equation driven by Lévy space-time white noise with Neumann boundary conditions. We establish the global existence and uniqueness of a mild solution under some regularity and boundedness on the coefficients.

Anisotropic Maxwell equations with electric conductivity are considered. Electromagnetic waves propagate in the exterior of a bounded connected obstacle with Lipschitz boundary. Our main result says that we can obtain uniform rates of decay of the total energy as $t \rightarrow + \infty$. No special requirements on the geometry of the obstacle are required. Previous results of this type were only given in the isotropic case. We use multipliers and properties of an associated evolution coupled system of first order.

We prove, by using bifurcation theory, the existence of at least two positive solutions for the quasilinear problem $-\Delta_p u = f(x,u)$ in $\Omega$, $u=0$ on $\partial \Omega$, where $N>p>1$ and $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $N\geq2,$ and the non-linearity $f$ is a locally Lipschitz continuous function, among other assumptions.

This paper concerns the formation of a coincidence set for the positive solution of the boundary-value problem $-{\varepsilon} (\phi_p(u_x))_x= \phi_q(u)f(c(x)-u)$ in $I=(-1,1)$ with $u(\pm 1)=0$, where ${\varepsilon}$ is a positive parameter, $\phi_p(s)=|s|^{p-2}s,\ 1 < q \le p < \infty$, $f(s) \sim \phi_{\theta+1}(s)\ (s \to 0)$, $0 < \theta < \infty$ and $c(x)$ is a positive smooth function satisfying $(\phi_p(c_x))_x=0$ in $I$. The positive solution $u_{\varepsilon}(x)$ converges to $c(x)$ uniformly on any compact subset of $I$ as ${\varepsilon} \to 0$. It is known that if $c(x)$ is constant and $\theta < p-1$, then the solution coincides with $c(x)$ somewhere in $I$ for sufficiently small ${\varepsilon}$ and the coincidence set $I_{\varepsilon}=\{x\in I : u_{\varepsilon}(x)=c(x)\}$ converges to $I$ as $|I \setminus I_{\varepsilon}| \sim {\varepsilon}^{1/p}\ ({\varepsilon} \to 0)$. It is proved in this paper that even if $c(x)$ is variable and $\theta < 1$, then $I_{\varepsilon}$ has a positive measure and converges to $I$ with order $O({\varepsilon}^{\kappa})$ as ${\varepsilon} \to 0$, where $\kappa=\min\{1/p,1/2\}$. Moreover, it is also shown that, if $\theta \ge 1$, then $I_{\varepsilon}$ is empty for every ${\varepsilon}$. The proofs rely on comparison principles and an energy method for obtaining local comparison functions.