This paper is concerned with investigating the global existence of a solution to a nonlinear wave equation with damping and source terms.

In this article we consider an initial--boundary value problem for a wave equation in high dimensions with a nonlinear damping term that is not Lipschitz in $u_t$. We establish the existence and uniqueness of a global solution by using a compactness method and by exploiting the monotonicity property of the nonlinearity.

We study a model proposed by E.F. Keller and L.A. Segel in two dimensions. Under some assumptions on the initial datum we prove the global existence and uniqueness of the solution and for other initial values we show that there exists a radial symmetric solution exploding in finite time.

We consider the Dirichlet problem for a class of linear elliptic equations, whose model is \begin{align*} & -\Delta u-\sum _{i=1}^{n}\left( b_{i}(y)u\right) _{x_{i}}-\sum _{j=1}^{m}\left( \widetilde{b}_{j}(y)u\right) _{y_{j}}+ \sum _{i=1}^{n}d_{i}(y)u_{x_{i}} \\ & +\sum\limits_{j=1}^{m}\widetilde{d} _{j}(y)u_{y_{j}}+c(y)u=f(x,y) \ \ \text{ in }G, \end{align*} where $G=G^{\prime }\times G^{\prime \prime }$ is an open, bounded and connected subset of ${\mathbb R}^{N}={\mathbb R}^{n}\times {\mathbb R}^{m}$, the coefficients $b_{i}(y),$ $\widetilde{b}_{j}(y),$ $d_{i}(y)$, $\widetilde{d} _{j}(y)$ and $c(y)$ are in $L^{\infty }(G)$ and the datum $f(x,y)$ belongs to $ L^{p}(G)$ with $p>\frac{N}{2}$. We prove some comparison results by using Steiner symmetrization.

We study radially symmetric oscillatory solutions of semilinear elliptic equations of the form $$ \Delta u + \phi(|x|,u)=0\quad\hbox{in } \ R^n \ (n\geq 2)$$ where $\phi(r,u)$ is a nonnegative function having the form $\sum_i \! c_i r^{\nu_i} |u|^{p_i-1} u$ with $c_i>0$. Under certain resrictions on the exponents $\nu_i$ and $p_i$ (roughly speaking, $2\nu_i+n+2\geq (2-n)p_i$ for all $i$ where a strict inequality holds for at least one $i$), we show that all radial solutions must oscillate, i.e., change their signs infinitely many times. Moreover, we provide accurate estimates on the frequencies and amplitudes of these oscillatory solutions. These results are sharp in the sense that positive solutions exist when restrictions on these exponents are removed.