We consider the Cauchy problem for the generalized Thirring model $(\partial _t \pm \partial _x ) U_{\pm} = i |U_{\pm}|^k |U_{\mp}|^{m-k} U_{\pm}$ in one spatial dimension which was introduced in [4]. Several results on well-posedness and ill-posedness have been obtained. Since the nonlinearity is not smooth if $k$ or $m$ is odd, an upper bound of $s$ to be well-posed appears. We prove that the upper bound is essential. More precisely, we show ill-posedness in $H^s(\mathbb{R})$ for sufficiently large $s$ which is a novel feature of this paper.

We derive fast decay estimates of the total energy for wave equations with localized variable damping coefficients, which are dealt with in the one dimensional half line $(0,\infty)$. The variable damping coefficient vanishes near the boundary $x = 0$, and is effective critically near spatial infinity $x = \infty$.

We consider a competitive system of differential equations describing the behavior of two biological species ``$u$" and ``$v$". The system is weakly coupled and one of the species has the capacity to diffuse and moves toward the higher concentration of the second species following its gradient, the density function satisfies a second order parabolic equation with chemotactic terms. The second species does not have motility capacity and satisfies an ordinary differential equation. We prove that the solutions are uniformly bounded and exist globally in time. The asymptotic behavior of solutions is also studied for a range of parameters and initial data. If the chemotaxis coefficient $\chi$ is small enough the quadratic terms drive the solutions to the constant steady state.

We show a sharp fractional Moser-Trudinger type inequality in dimension $1$, i.e., for any interval $I\Subset \mathbb R $ and $p\in (1,\infty)$ there exists $\alpha_p>0$ such that $$ \sup_{u\in \tilde H^{\frac 1p,p}(I): \|(-\Delta)^\frac{1}{2p}u\|_{L^p(I)}\le 1} \int_I e^{\alpha_p |u|^\frac{p}{p-1}}dx = C_p|I|, $$ and $\alpha_p$ is optimal in the sense that $$ \sup_{u\in \tilde H^{\frac 1p,p}(I): \|(-\Delta)^\frac{1}{2p}u\|_{L^p(I)}\le 1} \int_I h(u) e^{\alpha_p |u|^\frac{p}{p-1}}dx = +\infty, $$ for any function $h:[0,\infty) \to [0,\infty)$ with $\lim_{t\to\infty}h(t)=\infty$. Here, $\tilde H^{\frac 1p,p}(I)=\{u\in L^p( \mathbb R ): (-\Delta)^\frac{1}{2p}u\in L^p( \mathbb R ), \mathrm{supp}(u)\subset \bar I\}$. Restricting ourselves to the case $p=2$, we further consider for $\lambda>0$ the functional $$ J(u):=\frac{1}{2}\int_{ \mathbb R }|(-\Delta)^\frac14 u|^2 dx - \lambda\int_I \left (e^{\frac12 u^2}-1 \right )dx, \quad u\in \tilde H^{\frac{1}{2},2}(I), $$ and prove that it satisfies the Palais-Smale condition at any level $c\in (-\infty,\pi)$. We use these results to show that the equation $$ (-\Delta)^\frac12 u =\lambda u e^{\frac{1}{2}u^2}\quad \text{in }I, $$ has a positive solution in $\tilde H^{\frac12,2}(I)$ if and only if $\lambda\in (0,\lambda_1(I))$, where $\lambda_1(I)$ is the first eigenvalue of $(-\Delta)^\frac12$ on $I$. This extends to the fractional case for some previous results proven by Adimurthi for the Laplacian and the $p$-Laplacian operators. Finally, with a technique by Ruf, we show a fractional Moser-Trudinger inequality on $ \mathbb R $.

In this paper, we compare the solutions of two Poisson PDE's in cylinders with Neumann boundary conditions, one with given initial data and one with data arranged decreasing in the $y-$direction. When the solutions are normalized to have zero mean, we show that the solution with symmetrized data is itself symmetrized and exhibits larger convex means. The main tools used are the $\star-$function introduced by Baernstein and a new subharmonicity result. As a consequence, we give a new proof of a conjecture of Kawohl for temperature gaps in rectangles.

In this paper, we deal with a Kirchhoff type problem driven by a nonlocal fractional integrodifferential operator $\mathcal L_K$, that is, $$ -M( \|u \|^2)\mathcal L_Ku=\lambda f(x,u) \Big [\int_\Omega F(x,u(x))dx \Big ]^r+ \left |u \right |^{2^* -2}u \quad \mbox{in }\Omega, $$ $u=0$ in $ \mathbb{R}^{n}\setminus\Omega, $ where $\Omega$ is an open bounded subset of $\mathbb{R}^n$, $M$ and $f$ are continuous functions, $ \left \|\cdot \right \|$ is a functional norm, $$ F(x,u(x))=\int^{u(x)}_0 f(x,\tau)d\tau , $$ $2^*$ is a fractional Sobolev exponent, $\lambda$ and $r$ are real parameters. For this problem, we prove the existence of infinitely many solutions, through a suitable truncation argument and exploiting the genus theory introduced by Krasnoselskii.

In this paper, we deal with elliptic problems having terms singular in the variable $u$ which represents the solution. The problems are posed in cylinders $\Omega_n^\varepsilon$ of height $2n$ and perforated according to a parameter $\varepsilon$. We study existence, uniqueness and asymptotic behavior of the solutions $u_n^\varepsilon$ as the cylinders become infinite ($n\rightarrow +\infty$) and the size of the holes decreases while the number of the holes increases ($\varepsilon\rightarrow 0$).

Monge-Ampère equation $\det(D^2u)=f$ in two dimensional spaces is different in nature from their counterparts in higher dimensional spaces. In this article we employ new ideas to establish two main results for the Monge-Ampère equation defined either globally in $\mathbb{R}^2$ or outside a convex set. First, we prove the existence of a global solution that satisfies a prescribed asymptotic behavior at infinity, if $f$ is asymptotically close to a positive constant. Then we solve the exterior Dirichlet problem if data are given on the boundary of a convex set and at infinity.

We study a Dynamic Programming Principle related to the $p$-Laplacian for $1 < p < \infty.$ The main results are existence, uniqueness and continuity of solutions.