Under some conditions on the initial data, we show the global existence of solutions of the Cauchy problem of the quasilinear wave equation: $\partial_{t}^2 u = c(u)^2 \partial^2 _x u+\lambda c(u)c'(u)(\partial_x u)^2$ for $0 \leq \lambda\leq 2$, which has richly physical background. In [16], P. Zhang and Y. Zheng show the global existence of solutions to this equation with $\lambda =1$. It is difficult to apply their method to this global existence problem of this equation with $\lambda =0$ directly. As an application of the theorem of the global existence of solutions to $\partial_{t}^2 u = c(u)^2 \partial^2 _x u$, we construct solutions to $\partial_{t}^2 u = (u+1)^2 \partial^2 _x u$ which is going to $-1$ in finite time with $u(0,x)+1\geq \delta $ for some $\delta>0 $.

This paper is devoted to the study of the structure of positive radial solutions for the following semi-linear equation: $$\Delta u + f(u,|x|)=0 .$$ We require $f$ to be nonnegative and to exhibit both subcritical and supercritical behavior with respect to the Sobolev critical exponent. More precisely we assume that $f$ is subcritical for $u$ small and $|x|$ large and supercritical for $u$ large and $|x|$ small, and we give existence and non-existence results for ground states regular and singular, with either fast or slow decay. We find a surprisingly rich structure, which is characterized by two different patterns of bifurcations. We perform a Fowler transformation and we use a dynamical approach, exploiting some ideas borrowed from Bamon, Del Pino, and Flores, combining them with the use of the translation of the Pohozaev function for this dynamical context.

This paper studies a class of weighted non-Newtonian filtration equations with slow diffusion. By using the method introduced by Galaktionov and Levine for the classical non-Newtonian filtration equation, we establish the blow-up theorems of Fujiita type for the extended model, where more difficult and complicated estimates are required to treat the additional degeneracy and singularity. In particular, we prove via a delicate analysis that the critical situation of $p=p_c$ belongs to the blow-up case. The conclusions of this paper quantitatively show the influence of the degeneracy and singularity to the critical Fujita exponents of non-Newtonian filtration equations.

In this paper, we consider discretizations of systems of differential equations on manifolds that admit a strict Lyapunov function. We study the long-time behavior of the discrete solutions. In the continuous case, if a solution admits an accumulation point for which a Lojasiewicz inequality holds then its trajectory converges. Here we continue the work started in [18] by showing that discrete solutions have the same behavior under mild hypotheses. In particular, we consider the $\theta$-scheme for systems with solutions in $\mathbf{R}^d$ and a projected $\theta$-scheme for systems defined on an embedded manifold. As illustrations, we show that our results apply to existing algorithms: 1) Alouges' algorithm for computing minimizing discrete harmonic maps with values in the sphere, and 2) a discretization of the Landau--Lifshitz equations of micromagnetism.

We consider the nonlinear periodic-Dirichlet heat equation on a polygonal domain $\Omega$ of the plane in weighted $L^p$-Sobolev spaces \begin{align} \partial_t u -\Delta u = f(x,t,u), is & \mbox{in } \Omega \times {(-\pi,\pi)}, \notag \\ u=0,is & \mbox{on } \partial\Omega \times {(-\pi,\pi)}, \tag*{(0.1)} \\ u(\cdot, -\pi)=u(\cdot, \pi)is & \mbox{in } \Omega. \notag \end{align} Here $f$ is $L^p(0,T;L^p_\mu(\Omega))$-Carath\'eodory, where $L^p_\mu(\Omega)=\{v \in L^p_{loc}(\Omega): r^\mu v\in L^p(\Omega)\},$ with a real parameter $\mu$ and $r(x)$ the distance from $x$ to the set of corners of $\Omega$. We prove some existence results of this problem in presence of lower and upper solutions well-ordered or not. We first give existence results in an abstract setting obtained using degree theory. We secondly apply them for polygonal domains of the plane under geometrical constraints.

We consider second-order linear elliptic equations with lower-order terms whose coefficients lay in Marcinkiewicz spaces. We prove existence, uniqueness and regularity of a solution to the Dirichlet problem.

This paper deals with the Neumann eigenvalue problem for the Hermite operator defined in a convex, possibly unbounded, planar domain $\Omega$, having one axis of symmetry passing through the origin. We prove a sharp lower bound for the first eigenvalue $\mu_1^{odd}(\Omega)$ with an associated eigenfunction odd with respect to the axis of symmetry. Such an estimate involves the first eigenvalue of the corresponding one-dimensional problem.

This paper deals with approximation problems in a multidimensional setting for a family of convolution integral operators of homothetic type of the form $$ (T_wf)({s}) = \int_{\mathcal R^N} K_w ({ t}) f({ st}) \,d{ t},~~w>0,~ { s} \in \mathcal R^N; $$ here $\{K_w\}_{w>0}$ are approximate indentities, $\mathcal R:=\mathbb R^+_0$, and $f$ belongs to $BV^{\varphi}(\mathcal R^N)$, the space of functions with bounded $\varphi$-variation on $\mathcal R^N$. The main result of the paper establishes that, if $f$ is $\varphi$-absolutely continuous, there exists $\mu >0$ such that $$ V^{\varphi}[\mu(T_w f-f)]\longrightarrow 0,\ \ \hbox{as}\ w\rightarrow +\infty, $$ where $V^{\varphi}$ denotes the multidimensional $\varphi$-variation introduced in [8]. The rate of approximation is also investigated, by means of suitable Lipschitz classes.