Related papers: Weak extinction versus global exponential growth o…
Let $L$ be a positive self-adjoint operator on $L^2(X)$, where $X$ is a $\sigma$-finite metric measure space. When $\alpha \in (0,1)$, the subordinated semigroup $\{\exp(-tL^{\alpha}):t \in \mathbb{R}^+\}$ can be defined on $L^2(X)$ and…
In this paper, we study unique, globally defined uniformly bounded weak solutions for a class of semilinear reaction-diffusion-advection systems. The coefficients of the differential operators and the initial data are only required to be…
In the past few decades, much attention has been paid to the bubbling problem for semilinear Neumann elliptic equation with the critical and subcritical polynomial nonlinearity, much less is known if the polynomial nonlinearity is replaced…
We study a class of semilinear elliptic equations on spaces of tempered ultradistributions of Beurling and Roumieu type. Assuming that the linear part of the equation is an elliptic pseudodifferential operator of infinite order with a…
In the present paper, we are concerned with the semilinear viscoelastic wave equation subject to a locally distributed dissipative effect of Kelvin-Voigt type, posed on a bounded domain with smooth boundary. We begin with an auxiliary…
We develop strong and weak maximum principles for boundary-degenerate elliptic and parabolic linear second-order partial differential operators, $Au := -\mathrm{tr}(aD^2u)-<b, Du> + cu$, with partial Dirichlet boundary conditions. The…
We prove a quantitative unique continuation principle for infinite dimensional spectral subspaces of Schr\"odinger operators. Let $\Lambda_L = (-L/2,L/2)^d$ and $H_L = -\Delta_L + V_L$ be a Schr\"odinger operator on $L^2 (\Lambda_L)$ with a…
A mathematical model for the discrete nonlinear fragmentation (collision-induced breakage) equation with diffusion is studied. The existence of global weak solutions is established in arbitrary spatial dimensions without assuming a strictly…
In this paper we study the behavior of the energy of solutions of the wave equation with localized damping in exterior domain. We assume that the damper is positive at infinity. Under the Geometric Control Condition of Bardos et al (1992),…
We study nonlinear diffusion problems of the form $u_t=u_{xx}+f(u)$ with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For…
Following the well-known theory of Beurling and Roumieu ultradistributions, we investigate new spaces of ultradistributions as dual spaces of test functions which correspond to associated functions of logarithmic-type growth at infinity. In…
We consider the growth of the norms of transfer matrices of ergodic discrete Schr\"odinger operators in one dimension. It is known that the set of energies at which the rate of exponential growth is slower than prescribed by the Lyapunov…
The global existence for semilinear wave equations with space-dependent critical damping $\partial_t^2u-\Delta u+\frac{V_0}{|x|}\partial_t u=f(u)$ in an exterior domain is dealt with, where $f(u)=|u|^{p-1}u$ and $f(u)=|u|^p$ are in mind.…
We study the variational structure of the biased infinity Laplacian by introducing a notion of the $\beta$\textit{-Exponential Absolute Minimizing Extension} ($\beta$--AM) on arbitrary length space, which absolutely minimizing the…
A numerical and analytical study of the role of exponentially truncated L\'evy flights in the superdiffusive propagation of fronts in reaction-diffusion systems is presented. The study is based on a variation of the Fisher-Kolmogorov…
Extinction of a long-lived isolated stochastic population can be described as an exponentially slow decay of quasi-stationary probability distribution of the population size. We address extinction of a population in a two-population system…
We consider elliptic differential operators on either the entire Euclidean space $\mathbb{R}^d$ or on subsets consisting of a cube $\Lambda_L$ of integer length $L$. For eigenfunctions of the operator, and more general solutions of elliptic…
On a complete Riemannian manifold $(M,g)$, we consider $L^{p}_{loc}$ distributional solutions of the the differential inequality $-\Delta u + \lambda u \geq 0$ with $\lambda >0$ a locally bounded function that may decay to $0$ at infinity.…
In this article, we study semi-linear $\sigma$-evolution equations with double damping including frictional and visco-elastic damping for any $\sigma\ge 1$. We are interested in investigating not only higher order asymptotic expansions of…
We revisit a classical continuum model for the diffusion of multiple species with size-exclusion constraint, which leads to a degenerate nonlinear cross-diffusion system. The purpose of this article is twofold: first, it aims at a…