Related papers: Global and touchdown behaviour of the generalized …

We prove existence of solutions to problems whose model is $$\begin{cases} \displaystyle -\Delta_p u + u^q = \frac{f}{u^\gamma} & \text{in}\ \Omega, \newline u\ge0 &\text{in}\ \Omega,\newline u=0 &\text{on}\ \partial\Omega, \end{cases}$$…

We consider the Euler alignment system with mildly singular interaction kernels. When the local repulsion term is of the fractional type, global in time existence of smooth solutions was proved…

We consider solutions satisfying the Neumann zero boundary condition and a linearized mean field game system in $\Omega \times (0,T)$, where $\Omega$ is a bounded domain in $\mathbb{R}^d$ and $(0,T)$ is the time interval. We prove two kinds…

We continue our study of bounded solutions of the semilinear parabolic equation $u_t=u_{xx}+f(u)$ on the real line, where $f$ is a locally Lipschitz function on $\mathbb{R}.$ Assuming that the initial value $u_0=u(\cdot,0)$ of the solution…

We consider solutions to linear parabolic SPDEs of the form \[ \mathrm{d} u(t) + A u(t)\, \mathrm{d} t = g(t)\, \mathrm{d} \beta, \qquad u(0)=0, \] where $A$ is a positive, invertible, and self-adjoint operator on a Hilbert space $X$,…

We consider the solution of the problem $$ -\Delta u=f(u) \ \mbox{ and } \ u>0 \ \ \mbox{ in } \ \Omega, \ \ u=0 \ \mbox{ on } \ \Gamma, $$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with boundary $\Gamma$ of class $C^{2,\tau}$,…

Here we introduce a new notion of renormalized solution to nonlinear parabolic problems with general measure data whose model is $$ \begin{cases} u_t-\Delta_{p} u =\mu & \text{in}\ (0,T)\times\Omega, u=u_0 & \text{on}\ \{0\} \times \Omega,…

This paper is devoted to global existence of weak solutions to the following degenerate kinetic model of chemotaxis \begin{equation} \begin{cases}\label{chemo0} u_t=\Delta (\gamma (v)u) \tau v_{t}=\Delta v-v+u \end{cases} \end{equation}in a…

We address a generalised three-dimensional $\alpha$-Muskat model that comes from the fluid interface problem given by two incompressible fluids with different densities in the stable regime. We establish local-in-time wellposedness when…

Bounds for the area of general closed marginally trapped surfaces (MTSs) are presented. They do not require any stability condition, and are determined by a constant that depends on a particular component of the Einstein tensor on the…

We consider the initial value problem associated to the neural field equation of Amari type with plasticity \[ u_t(x,t)=-u(x,t)+\int_{\Omega}w(x,y)[1+\gamma g( u(x,t) - u(y,t) )] f(u(y,t))\; dy, \;(x,t) \in \Omega \times (0, \infty), \]…

In this paper, we study the global stability of classical solutions to a Keller--Segel equations in scaling-invariant spaces. We prove that for any given $0<\mathcal{M}<1+\lambda_1$ with $\lambda_1$ being the first eigenvalue of Neumann…

In this paper, we study the behavior of multiple continua of solutions to the semilinear elliptic problem \begin{equation*} \begin{cases} -\Delta u = \lambda f(u) &\text{ in } \Omega, u=0 &\text{ on } \partial\Omega, \end{cases}…

The Generalized Method of Moments (GMM) is a partition of unity based technique for solving electromagnetic and acoustic boundary integral equations. Past work on the GMM for electromagnetics was confined to geometries modeled by piecewise…

The global boundedness and asymptotic behavior are investigated for the solutions of a nonlocal time fractional reaction-diffusion equation (NTFRDE) $$ \frac{\partial^{\alpha }u}{\partial t^{\alpha }}=\Delta u+\mu u^{2}(1-kJ*u)-\gamma u,…

We consider the 1D transport equation with nonlocal velocity field: \begin{equation*}\label{intro eq} \begin{split} &\theta_t+u\theta_x+\nu \Lambda^{\gamma}\theta=0, \\ & u=\mathcal{N}(\theta), \end{split} \end{equation*} where…

Using a calibration method we prove that, if $\Gamma\subset \Omega$ is a closed regular hypersurface and if the function $g$ is discontinuous along $\Gamma$ and regular outside, then the function $u_{\beta}$ which solves $$ \begin{cases}…

The global regularity problem for the periodic Navier-Stokes system asks whether to every smooth divergence-free initial datum $u_0: (\R/\Z)^3 \to \R^3$ there exists a global smooth solution u. In this note we observe (using a simple…

We are concerned with the nodal set of solutions to equations of the form \begin{equation*} -\Delta u = \lambda_+ \left(u^+\right)^{q-1} - \lambda_- \left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where $\lambda_+,\lambda_- >…

We investigate stable solutions of elliptic equations of the type \begin{equation*} \left \{ \begin{aligned} (-\Delta)^s u&=\lambda f(u) \qquad {\mbox{ in $B_1 \subset \R^{n}$}} \\ u&= 0 \qquad{\mbox{ on $\partial B_1$,}}\end{aligned}\right…