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We study a system of semilinear wave equations satisfying the weak null condition, which can be regarded as a simplified model for the Einstein vacuum equations. The main objective is to establish precise pointwise decay estimates, as both…

We provide upper and lower bounds on the semileptonic weak decay form factors for $B \to D^(*)$ and $\Lambda_b \to \Lambda_c$ decays by utilizing inclusive heavy quark effective theory sum rules. These bounds are calculated to second order…

We consider a class of semilinear parabolic evolution equations subject to a hysteresis operator and a Bochner-Lebesgue integrable source term. The underlying spatial domain is allowed to have a very general boundary. In the first part of…

We consider on Riemannian manifolds the non-linear evolution equation $$\rho \partial _{t}u=\Delta _{p}u^{q}.$$ Assuming that the manifold satisfies a \textit{(weighted) Sobolev inequality} and under certain assumptions on $p, q$ and…

We investigate the homogeneous Dirichlet problem for the Fast Diffusion Equation $u_t=\Delta u^m$, posed in a smooth bounded domain $\Omega\subset \mathbb{R}^N$, in the exponent range $m_s=(N-2)_+/(N+2)<m<1$. It is known that bounded…

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $\rho_1>0$, $\beta\ge\frac{m\rho_1}{n-2-nm}$ and $\alpha=\frac{2\beta+\rho_1}{1-m}$. For any $\lambda>0$, we will prove the existence and uniqueness (for $\beta\ge\frac{\rho_1}{n-2-nm}$) of radially…

We describe non-standard contributions to semileptonic processes in a model independent way in terms of an SU(2)_L X U(1)_Y invariant effective lagrangian at the weak scale, from which we derive the low-energy effective lagrangian governing…

We consider a self-adjoint two-dimensional Schr\"odinger operator $H_{\alpha\mu}$, which corresponds to the formal differential expression \[ -\Delta - \alpha\mu, \] where $\mu$ is a finite compactly supported positive Radon measure on…

Let $T:X\to X$ be a linear power bounded operator on Banach space. Let $X_0$ is a subspace of vectors tending to zero under iterating of $T$. We prove that if $X_0$ is not equal to $X$ then there exists $\lambda$ in Sp(T) such that, for…

We investigate rates of decay for $C_0$-semigroups on Hilbert spaces under assumptions on the resolvent growth of the semigroup generator. Our main results show that one obtains the best possible estimate on the rate of decay, that is to…

We study the scattering problem for the nonlinear Schr\"odinger equation $i\partial_t u + \Delta u = |u|^p u$ on $\mathbb{R}^d$, $d\geq 1$, with a mass-subcritical nonlinearity above the Strauss exponent. For this equation, it is known that…

Let $H=-\Delta+V$, where $V$ is a real valued potential on $\R^2$ satisfying $|V(x)|\les \la x\ra^{-3-}$. We prove that if zero is a regular point of the spectrum of $H=-\Delta+V$, then $$ \|w^{-1} e^{itH}P_{ac}f\|_{L^\infty(\R^2)}\les…

For a large class of non-negative initial data, the solutions to the quasilinear viscous Hamilton-Jacobi equation $\partial\_t u-\Delta\_p u+|\nabla u|^q=0$ in $(0,\infty)\times\real^N$ are known to vanish identically after a finite time…

Schr\"odinger operators with periodic potential have generally been shown to exhibit ballistic transport. In this work, we investigate if the propagation velocity, while positive, can be made arbitrarily small by a suitable choice of the…

We study the spatial decay behaviour of resolvent kernels for a large class of non-local L\'evy operators and bound states of the corresponding Schr\"odinger operators. Our findings naturally lead us to proving results for L\'evy measures,…

In this paper we study a class of double phase problems involving critical growth, namely $-\text{div}\big(|\nabla u|^{p-2} \nabla u+ \mu(x) |\nabla u|^{q-2} \nabla u\big)=\lambda|u|^{\vartheta-2}u+|u|^{p^*-2}u$ in $\Omega$ and $u= 0$ on…

We prove that the operator $L_0=-(1+|x|)^\beta(-\Delta)^{\alpha/2}$ with $\alpha\in(0,2)$, $d>\alpha$ and $\beta\ge0$ generates a compact semigroup or resolvent on $L^2(\R^d;(1+|x|)^{-\beta}\,dx)$, if and only if $\beta>\alpha$. When…

This paper deals with the long-term behavior of positive solutions for the following parabolic-elliptic chemotaxis competition system with weak singular sensitivity and logistic source \begin{equation} \label{abstract-eq} \begin{cases}…

We consider the fractional unforced Burgers equation in the one-dimensional space-periodic setting: $$\partial u/\partial t+(f(u))_x +\nu \Lambda^{\alpha} u= 0, t \geq 0,\ \mathbb{x} \in \mathbb{T}^d=(\mathbb{R}/\mathbb{Z})^d.$$ Here $f$ is…

In this note we show that for analytic semigroups the so-called Weiss condition of uniform boundedness of the operators $Re(\lambda)^\einhalb C(\lambda+A)^{-1}, \qquad Re(\lambda)>0$ on the complex right half plane and weak Lebesgue…