Related papers: The Snapshot Problem for the Euler-Poisson-Darboux…
A general class of probability density functions \[u(x,t)=Ct^{-\alpha d}\left (1-\left (\frac{\|x\|}{ct^{\alpha}}\right )^{\beta}\right )_+^{\gamma},\quad x\in \mathbb{R}^d,t>0,\] is considered, containing as particular case the Barenblatt…
Degenerate hyperbolic equations are dealing with many important issues for applied nature. While a variety of degenerate equations and boundary conditions, successfully matched to these differential equation, most in the characteristic…
When $P$ is the fractional Laplacian $(-\Delta )^a$, $0<a<1$, or a pseudodifferential generalization thereof, the Dirichlet problem for the associated heat equation over a smooth set $\Omega \subset{\Bbb R}^n$:…
We consider the equation $-\epsilon^{2}\Delta u + u = u^ {p}$ in a bounded domain $\Omega\subset\R^{3}$ with edges. We impose Neumann boundary conditions, assuming $1<p<5$, and prove concentration of solutions at suitable points of…
Consider a complete $d$-dimensional Riemannian manifold $(\mathcal M,g)$, a point $p\in\mathcal M$ and a nonlinearity $f(q,u)$ with $f(p,0)>0$. We prove that for any odd integer $N\ge3$, there exists a sequence of smooth domains…
In this note we prove uniqueness of the critical point for positive solutions of elliptic problems in bounded planar domains: we first examine the Poisson problem - Delta u = f(x,y) finding a geometric condition involving the curvature of…
In this paper we consider the problem $$(P)\qquad \{{array}{rclll} u_t-\D u^m&=&|\n u|^q +\,f(x,t),&\quad u\ge 0 \hbox{in} \Omega_T\equiv \Omega\times (0,T), u(x,t)&=&0 &\quad \hbox{on} \partial\Omega\times (0,T) u(x,0)&=&u_0(x),&\quad x\in…
The parabolic problem $u_t-\Delta u=\frac{\lambda f(x)}{(1-u)^2}+P$ on a bounded domain $\Omega$ of $R^n$ with Dirichlet boundary condition models the microelectromechanical systems(MEMS) device with an external pressure term. In this…
We consider the Dirichlet problem for the nonhomogeneous equation $-\Delta_p u -\Delta_q u = \alpha |u|^{p-2}u + \beta |u|^{q-2}u + f(x)$ in a bounded domain, where $p \neq q$, and $\alpha, \beta \in \mathbb{R}$ are parameters. We explore…
We are concerned with a global existence theory for finite-energy solutions of the multidimensional Euler-Poisson equations for both compressible gaseous stars and plasmas with large initial data of spherical symmetry. One of the main…
It is shown that for each finite number of Dirac measures supported at points $s_n$ in three-dimensional Euclidean space, with given amplitudes $a_n$, there exists a unique real-valued Lipschitz function $u$, vanishing at infinity, which…
Let $\Omega$ be a open bounded domain in $\mathbb{R}^n $ with smooth boundary $\partial\Omega$. We consider the equation $ \Delta u + u^{\frac{n-k+2}{n-k-2}-\varepsilon} =0\,\hbox{ in }\,\Omega $, under zero Dirichlet boundary condition,…
Structure and properties of families of critical points for classes of functions $W(z,\bar{z})$ obeying the elliptic Euler-Poisson-Darboux equation $E(1/2,1/2)$ are studied. General variational and differential equations governing the…
A complete family of solutions for the one-dimensional reaction-diffusion equation \[ u_{xx}(x,t)-q(x)u(x,t) = u_t(x,t) \] with a coefficient $q$ depending on $x$ is constructed. The solutions represent the images of the heat polynomials…
This paper investigates a weakly coupled system of semilinear Euler-Poisson-Darboux-Tricomi equations (EPDTS) with power-type nonlinear terms. More precisely, in the case where the damping terms dominate over the mass terms, the critical…
Here is one of the results obtained in this paper: Let $\Omega\subset {\bf R}^n$ be a smooth bounded domain, let $q>1$, with $q<{{n+2}\over {n-2}}$ if $n\geq 3$ and let $\lambda_1$ be the first eigenvalue of the problem $$\cases{-\Delta…
In this paper, we are concerned with the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping \begin{equation*} \partial_t\rho+\operatorname{div}(\rho u)=0, \quad…
Under conditions of Levinson-Smith type, we prove the existence of a $\tau$-periodic solution for the perturbed generalized Li\'enard equation u''+\phi(u,u')u'+\psi(u)=\epsilon\omega(\frac{t}{\tau},u,u') with periodic forcing term. Also we…
One proves the existence and uniqueness of a generalized (mild) solution for the nonlinear Fokker-Planck equation (FPE) \begin{align*} &u_t-\Delta (\beta(u))+{\mathrm{ div}}(D(x)b(u)u)=0, \quad t\geq0,\ x\in\mathbb{R}^d,\ d\ne2, \\…
We prove two results that together strongly suggest that obtaining a positive answer to the Navier-Stokes global regularity question requires more than a refinement of partial regularity theory. First we prove that there exists a class of…