Related papers: From Gaussian estimates for nonlinear evolution eq…

Quasistatic evolutions of critical points of time-dependent energies exhibit piecewise smooth behavior, making them useful for modeling continuum mechanics phenomena like elastic-plasticity and fracture. Traditionally, such evolutions have…

We study the Cauchy problem associated to a family of nonautonomous semilinear equations in the space of bounded and continuous functions over R^d and in L^p-spaces with respect to tight evolution systems of measures. Here, the linear part…

We study evolution equations governed by an averaging operator on a directed tree, showing existence and uniqueness of solutions. In addition we find conditions of the initial condition that allows us to find the asymptotic decay rate of…

The stability of $q$-Gaussian distributions as particular solutions of the linear diffusion equation and its generalized nonlinear form, $\pderiv{P(x,t)}{t} = D \pderiv{^2 [P(x,t)]^{2-q}}{x^2}$, the \emph{porous-medium equation}, is…

We consider a 1-dimensional Brownian motion whose diffusion coefficient varies when it crosses the origin. We study the long time behavior and we establish different regimes, depending on the variations of the diffusion coefficient:…

We consider the one-dimensional nonlinear Schr\"odinger equation $$ iu_t + u_{xx} + \mathcal{N}(u)u=0, \quad x,t \in \mathbb R, $$ with the nonlinearity term that is expressed as a sum of powers, possibly infinite: $$ \mathcal{N}(u) = \sum…

This is a continuation, and conclusion, of our study of bounded solutions $u$ of the semilinear parabolic equation $u_t=u_{xx}+f(u)$ on the real line whose initial data $u_0=u(\cdot,0)$ have finite limits $\theta^\pm$ as $x\to\pm\infty$. We…

We propose a new method for constructing exact solutions to nonlinear delay reaction--diffusion equations of the form $$ u_t=ku_{xx}+F(u,w), $$ where $u=u(x,t)$, $w=u(x,t-\tau)$, and $\tau$ is the delay time. The method is based on…

We present an existence theory for martingale and strong solutions to doubly nonlinear evolution equations in a separable Hilbert space in the form $$d(Au) + Bu\,dt \ni F(u)\,dt + G(u)\,dW$$ where both $A$ and $B$ are maximal monotone…

In this article we deal with the stability and convergence of numerical solutions of nonlinear evolution equations of the form $A(u(t))+f(u(t))=u'(t)$, the numerical analysis of solutions to this problems will be performed using some…

We investigate some regularity properties of a class of doubly nonlinear anisotropic evolution equations whose model case is \begin{align*} \partial_t \big(|u|^{\alpha -1}u \big) - \sum^N_{i=1} \partial_i \big( |\partial_i u|^{p_i - 2}…

We introduce state-space models where the functionals of the observational and the evolutionary equations are unknown, and treated as random functions evolving with time. Thus, our model is nonparametric and generalizes the traditional…

We consider the natural time-dependent fractional $p$-Laplacian equation posed in the whole Euclidean space, with parameters $p>2$ and $s\in (0,1)$ (fractional exponent). We show that the Cauchy Problem for data in the Lebesgue $L^q$ spaces…

Motivated by the oscillations that were seen at the Tacoma Narrows Bridge, we introduce the notion of solutions with a prevailing mode for the nonlinear evolution beam equation $$ u_{tt} + u_{xxxx} + f(u)= g(x, t) $$ in bounded space-time…

We study the large time behavior of solutions to the dissipative Korteweg-de Vrie equations $u_t+u_{xxx}+|D|^{\alpha}u+uu_x=0$ with $0<\alpha<2$. We find $v$ such that $u-v$ decays like $t^{-r(\alpha)}$ as $t\to\infty$ in various Sobolev…

In the paper, we consider the large time behavior of solutions to the convection-diffusion equation u_t - Delta u + nabla cdot f(u) = 0 in R^n times [0,infinity), where f(u) ~ u^q as u --> 0. Under the assumption that q >= 1+1/(n+beta) and…

We study fractional parabolic equations with indefinite nonlinearities $$ \frac{\partial u} {\partial t}(x,t) +(-\Delta)^s u(x,t)= x_1 u^p(x, t),\,\, (x, t) \in \mathbb{R}^n \times \mathbb{R}, $$ where $0<s<1$ and $1<p<\infty$. We first…

Consider the following class of conformable time-fractional stochastic equation $$T_{\alpha,t}^a u(x,t)=\lambda\sigma(u(x,t))\dot{W}_t,\,\,\,\,x\in\mathbb{R},\,t\in[a,\infty), \,\,0<\alpha<1,$$ with a non-random initial condition…

Let $N(t)$ be the collection of particles alive at time $t$ in a branching Brownian motion in $\mathbb{R}^d$, and for $u\in N(t)$, let $\mathbf{X}_u(t)$ be the position of particle $u$ at time $t$. For $\theta\in \mathbb{R}^d$, we define…

We study a class of stochastic evolution equations with a dissipative forcing nonlinearity and additive noise. The noise is assumed to satisfy rather general assumptions about the form of the covariance function; our framework covers…