This paper is concerned with the Cauchy problem for the semilinear wave equation: $$ u_{tt}-\Delta u=F(u) \quad \mbox{in}\quad \mathbb R^n\times[0,\infty), $$ where the space dimension $n\geq 2$, $F(u)=|u|^p$ or $F(u)=|u|^{p-1}u$ with $p > 1$. Here, the Cauchy data are non-zero and non-compactly supported. Our results on the blow-up of positive radial solutions (not necessarily radial in low dimensions $n=2, 3$) generalize and extend the results of Takamura [19] for zero initial position and Takamura, Uesaka and Wakasa [21] for zero initial velocity. The main technical difficulty in the paper lies in obtaining the lower bounds for the free solution when both initial position and initial velocity are non-identically zero in even space dimensions.

In this paper, we are concerned with a general singular Dirichlet boundary value problem whose model is the following: $$ \begin{cases} -\Delta u = \frac{\mu}{u^{\gamma}} & \text{in}\ \Omega,\\ u=0 &\text{on}\ \partial\Omega,\\ u>0 &\text{on}\ \Omega\,. \end{cases} $$ Here, $\mu$ is a nonnegative bounded Radon measure on a bounded open set $\Omega\subset\mathbb R^N$, and $\gamma>0$.

We study a hyperbolic problem in the framework of periodic homogenization assuming a high contrast between the diffusivity coefficients of the two components $M_{\varepsilon}$ and $ B_{\varepsilon}$ of the heterogeneous medium. There are three regimes depending on the ratio between the size of the period and the amplitude ${\alpha_{\varepsilon}}$ of the diffusivity in $ B_{\varepsilon}$. For the critical regime $ \alpha_{\varepsilon} \simeq {\varepsilon}$, the limit problem is a strongly coupled system involving both the macroscopic and the microscopic variables. We also include the results in the non critical case.

We study the initial-boundary value problem for the cubic nonlinear Klein-Gordon equation \[ \Bigg \{ \begin{array} [c]{c} v_{tt}+v-v_{xx}=F ( v ) ,\text{ } ( t,x ) \in \mathbb{R}^{+}\times\mathbb{R}^{+}\mathbf{,}\\ v ( 0,x ) =v_{0}(x),v_{t} ( 0,x ) =v_{1}(x),x\in \mathbb{R}^{+}{\mathbf{,}}\\ v ( t,0 ) =h(t),t\in\mathbb{R}^{+} \end{array} \] where \[ F ( v ) :=\sum_{\alpha+\beta+\gamma=3}C_{\alpha,\beta,\gamma } ( i\partial_{t}v ) ^{\alpha} ( -i\partial_{x}v ) ^{\beta}v^{\gamma}, \] with real constants $C_{\alpha,\beta,\gamma},$ with inhomogeneous Dirichlet boundary conditions. We prove the global in time existence of solutions of IBV problem for cubic Klein-Gordon equations with inhomogeneous Dirichlet boundary conditions. We obtain the asymptotic behavior of the solution. Our approach is based on the estimates of the integral equation in the Sobolev spaces. We propose a new method of the decomposition of the critical cubic nonlinearity, into a resonant, nonresonant and remainder terms, in order to obtain the smoothness of the solutions.

We discuss evolution variational inequalities for a wave equation with unilateral constraint on the velocity. Both deterministic and stochastic obstacle problems will be considered. For deterministic obstacles, we will focus on the case which is directly related to stochastic equations. This makes it necessary to formulate a new definition of a solution. Based on this definition, we will establish existence and uniqueness of a solution to both deterministic and stochastic problems.

We obtain a one-parameter family $$ (u_{\mu}(x,t),\eta_{\mu}(x,t))_{\mu\geq \mu_0} =(\phi_{\mu}(x-\omega_{\mu} t),\psi_{\mu}(x-\omega_{\mu} t))_{\mu\geq \mu_0} $$ of traveling-wave solutions to the Boussinesq system $$ \left\{\begin{array}{llll} u_t+\eta_x+uu_x+c\eta_{xxx}=0\qquad (x,t)\in\mathbb R^2 \\ \eta_t+u_x+(\eta u)_x+au_{xxx}=0, \end{array}\right. $$ in the case $a,c < 0$, with non-null speeds $\omega_{\mu}$ arbitrarily close to $0$ ($\omega_{\mu}\xrightarrow[\mu\to+\infty]{} 0$). We show that the $L^2$-size of such traveling-waves satisfies the uniform (in $\mu$) estimate $\|\phi_{\mu}\|_2^2+\|\psi_{\mu}\|_2^2\leq C\sqrt{|a|+|c|},$ where $C$ is a positive constant. Furthermore, $\phi_{\mu}$ and $-\psi_{\mu}$ are smooth, non-negative, radially decreasing functions which decay exponentially at infinity.

In this paper, we establish a local gradient estimate for a $p$-Laplacian equation with a fast growing gradient nonlinearity. This estimate allows us to prove a parabolic Liouville theorem for ancient solutions (i.e., defined for $t < 0$) satisfying some growth restriction near infinity.

Let $\Omega \subset \mathbb R^d$ be a bounded domain and suppose that for each $x\in \Omega$ a radius $r = r(x)$ is given so that the ball $B_x = B(x,r)$ is contained in $\Omega$. For $0 \leq \alpha < 1 $, we consider the following operator in $\mathcal{C}(\overline{\Omega})$ $$ T_{\alpha}u(x) = \frac{\alpha}{2}\big ( \sup_{B_x} u + \inf_{B_x} u \big ) + (1-\alpha ) \int_{B_x} u, $$ and show that, under certain assumptions on $\Omega$ and the radius function $r(x)$, the Dirichlet problem with continuous boundary data has a unique solution $u\in \mathcal{C}(\overline{\Omega})$ satisfying $T_{\alpha}u = u$. The motivation comes from the study of so called $p$-harmonious functions and certain stochastic games.

In this work, we will show the global existence of the strong solutions of the inhomogeneous incompressible Navier-Stokes equations in a thin three dimensional domain $\Omega=\mathbb R^2\times [0,\epsilon]$, with Dirichlet boundary condition on the top and bottom boundary: the global well-posedness may hold for large initial data when the vertical size $\epsilon$ is sufficiently small. Furthermore, when $\epsilon\rightarrow 0$ the velocity tends to vanish away from the initial time. The analysis relies on the a priori $H^2$-estimate for the solutions (similar as in [4, 5, 21]) and one pays attention to the dependence on the vertical size $\epsilon$.

We prove that the solution map associated with the $1D$ half-wave cubic equation in the periodic setting cannot be uniformly continuous on bounded sets of the periodic Sobolev spaces $H^s$ with $s\in ( \frac 14 , \frac 12 )$.