Related papers: Large-time rescaling behaviors for large data to t…

This manuscript is concerned with the one-dimensional system \[ \begin{array}{l} \tau u_{ttt} + \alpha u_{tt} = b \big(\gamma(\Theta) u_{xt}\big)_x + \big( \gamma(\Theta) u_x\big)_x, \\[1mm] \Theta_t = D \Theta_{xx} + b\gamma(\Theta)…

In this paper we consider the problem: $\partial_{t} u- \Delta u=f(u),\; u(0)=u_0\in \exp L^p(\R^N),$ where $p>1$ and $f : \R\to\R$ having an exponential growth at infinity with $f(0)=0.$ We prove local well-posedness in $\exp L^p_0(\R^N)$…

Motivated by the study of quantum fields in a Friedman-Robertson-Walker (FRW) spacetime, the one-loop effective action for a scalar field defined in the ultrastatic manifold $R\times H^3/\Gamma$, $H^3/\Gamma$ being the finite volume,…

We study the dynamics of roots of f'(z,t), where f(z,t) is a locally univalent polynomial solution of the Polubarinova-Galin equation for the evolution of the conformal map onto a Hele-Shaw blob subject to injection at one point. We give…

In this paper, we investigate the two-dimensional incompressible primitive equations with fractional horizontal dissipation. Specifically, we establish global well-posedness of strong solutions for arbitrarily large initial data when the…

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…

We will look at reaction-diffusion type equations of the following type, $$\partial^\beta_tV(t,x)=-(-\Delta)^{\alpha/2} V(t,x)+I^{1-\beta}_t[V(t,x)^{1+\eta}].$$ We first study the equation on the whole space by making sense of it via an…

We present a gauge-invariant formalism to study the evolution of the curvature and entropy perturbations in the case in which spatial and time variations of the inflaton decay rate into ordinary matter are present. During the reheating…

We derive the large time asymptotics of initially regular and localized solutions of the Teukolsky equation on the exterior of a subextremal Kerr black hole for any half integer spin. More precisely, we obtain the leading order term…

The semiclassical backreaction equations are solved in closed Robertson-Walker spacetimes containing a positive cosmological constant and a conformally coupled massive scalar field. Renormalization of the stress-energy tensor results in…

In this paper we prove global regularity for the full water waves system in 3 dimensions for small data, under the influence of both gravity and surface tension. This problem presents essential difficulties which were absent in all of the…

In a bounded domain $\Omega$, we consider a positive solution of the problem $\Delta u+f(u)=0$ in $\Omega$, $u=0$ on $\partial\Omega$, where $f:\mathbb{R}\to\mathbb{R}$ is a locally Lipschitz continuous function. Under sufficient conditions…

We prove a result of existence of positive solutions of the Dirichlet problem for $-\Delta_p u=\mathrm{w}(x)f(u,\nabla u)$ in a bounded domain $\Omega\subset\mathbb{R}^N$, where $\Delta_p$ is the $p$-Laplacian and $\mathrm{w}$ is a weight…

Let the set $\Omega_\varepsilon$ be obtained from the bounded domain $\Omega$ by removing a family of $\varepsilon$-periodically distributed identical balls. In $\Omega_\varepsilon$ one considers the standard Steklov spectral problem. It is…

We study non-univalent solutions of the Polubarinova-Galin equation, describing the time evolution of the conformal map from the unit disk onto a Hele-Shaw blob of fluid subject to injection at one point. In particular, we tackle the…

We obtain new a priori estimates for the nonnegative solutions of the equation \[ u_{t}-\Delta u+|\nabla u|^{q}=0 \] in $Q_{\Omega,T}=\Omega\times\left( 0,T\right) ,$ $T\leqq\infty,$ where $q>0,$ and $\Omega=\mathbb{R}^{N},$ or $\Omega$ is…

A novel perturbative method, proposed by Panda {\it et al.} [1] to solve the Helmholtz equation in two dimensions, is extended to three dimensions for general boundary surfaces. Although a few numerical works are available in the literature…

We study the large time behavior of solutions of first-order convex Hamilton-Jacobi Equations of Eikonal type set in the whole space. We assume that the solutions may have arbitrary growth. A complete study of the structure of solutions of…

We study the initial value problem of the thermal-diffusive combustion system: $u_{1,t} = u_{1,x,x} - u_1 u^2_2, u_{2,t} = d u_{2,xx} + u_1 u^2_2, x \in R^1$, for non-negative spatially decaying initial data of arbitrary size and for any…

We consider the initial value problem for the 2D quasi-geostrophic equation with weak dissipation term $\kappa(-\Delta)^{\alpha/2}\theta\ (0<\alpha\leqslant 1)$ and dispersive forcing term $Au_2$. We establish a unique global solution for a…