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We consider non-negative, weak solutions to the doubly nonlinear parabolic equation $$ \partial_t u^q-\mbox{div}(|Du|^{p-2}Du)=0 $$ in the super-critical fast diffusion regime $0<p-1<q<\frac{N(p-1)}{(N-p)_+}$. We show that when solutions…

In this paper we analyze the asymptotic behaviour as $p\to 1^+$ of solutions $u_p$ to $$ \left\{ \begin{array}{rclr} -\Delta_pu&=&\lambda|\nabla u|^{p-2}\nabla u\cdot\frac{x}{|x|^2}+ f&\quad \mbox{ in } \Omega,\\ u_p&=&0 &\quad \mbox{ on…

We consider the Cauchy problem for the nonlinear Schr\"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ) u= \pm \partial (\overline{u}^m)$ on $\R ^d$, $d \ge 1$, with random initial data, where $\partial$ is a first…

We study the existence/nonexistence and qualitative properties of the positive solutions to the problem \begin{align*} (-\Delta)^s u -\theta\frac{u}{|x|^{2s}}&=u^p - u^q \quad\text{in }\,\, \mathbb{R}^N,\quad u > 0 \quad\text{in }\,\,…

We construct asymptotically self-similar global solutions to the Hardy-H\'enon parabolic equation $\partial_t u - \Delta u = \pm |x|^{\gamma} |u|^{\alpha-1} u$, $\alpha>1$, $\gamma \in \mathbb{R}$ for a large class of initial data belonging…

We study the long-time asymptotics of solutions of the uniformly parabolic equation \[ u_t + F(D^2u) = 0 \quad {in} \R^n\times \R_+, \] for a positively homogeneous operator $F$, subject to the initial condition $u(x,0) = g(x)$, under the…

We study the paralinearised weakly dispersive Burgers type equation: $$\partial_t u+T_u \partial_xu+\partial_x |D|^{\alpha-1}u=0,\ \alpha \in ]1,2[,$$ which contains the main non linear "worst interaction" terms, that is low-high…

We deal with existence and non-existence of non-negative entire solutions that blow-up at infinity for a quasilinear problem depending on a non-negative real parameter. Our main objectives in this paper are to provide far more general…

In the present paper, we study the Cauchy problem for the wave equation with a time-dependent scale invariant damping $\frac{2}{1+t}\partial_t v$ and a cubic convolution $(|x|^{-\gamma}*v^2)v$ with $\gamma\in \left(-\frac{1}{2},3\right)$ in…

This paper concerns solutions to the Hardy-H\'enon equation \[ -\Delta u = |x|^\sigma u^p \] in $\mathbf R ^n$ with $n \geq 1$ and arbitrary $p, \sigma \in \mathbf R$. This equation was proposed by H\'enon in 1973 as a model to study…

Solutions to the Cauchy problem for the one-dimensional cubic nonlinear Schr\"odinger equation on the real line are studied in Sobolev spaces $H^s$, for $s$ negative but close to 0. For smooth solutions there is an {\em a priori} upper…

In this paper, we study the large-time behavior of solutions to a class of partially dissipative linear hyperbolic systems with applications in velocity-jump processes in several dimensions. Given integers $n,d\ge 1$, let $\mathbf…

We construct nonnegative weak solutions to the singular parabolic free boundary problem \[ \partial_t u - \Delta u = - \frac{\mathrm{d}}{\mathrm{d} u} u_+^\gamma , \] where $\gamma \in (0,1]$, $u_+ := \max\{u,0\}$, and the term in the…

We study anti-symmetric solutions about the hyperplane $\{x_n=0\}$ to the following fractional Hardy-H\'{e}non system $$ \left\{\begin{aligned} &(-\Delta)^{s_1}u(x)=|x|^\alpha v^p(x),\ \ x\in\mathbb{R}_+^n, \\&(-\Delta)^{s_2}v(x)=|x|^\beta…

By introducing a class of new function spaces $B^{\sigma,s}_{p,q}$ as the resolution spaces, we study the Cauchy problem for the nonlinear Klein-Gordon equation (NLKG) in all spatial dimensions $d \geqslant 1$, $$ \partial^2_t u + u- \Delta…

We deal with solutions of the Cauchy problem to linear both homogeneous and nonhomogeneous parabolic second order equations with real constant coefficients in the layer ${\mathbb R}^{n+1}_T={\mathbb R}^n\times (0, T)$, where $n\geq 1$ and…

We establish both extinction and non-extinction self-similar profiles for the following fast diffusion equation with a weighted source term $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed for $(x,t)\in\real^N\times(0,\infty)$, $N\geq3$,…

This paper studies the properties of solutions for a double nonlinear time-dependent parabolic equation with variable density, not in divergence form with a source or absorption. The problem is formulated as a partial differential equation…

In this paper we consider the Cauchy problem for the semilinear damped wave equation $u_{tt}-\Delta u + u_t = h(u);\qquad u(0;x) = f(x); \quad u_t(0;x) = g(x);$ where $h(s) = |s|^{1+2/n}\mu(|s|)$. Here n is the space dimension and $\mu$ is…

Motivated by Girsanov's nonuniqueness examples for SDEs, we prove nonuniqueness for the parabolic stochastic partial differential equation (SPDE) \[\frac{\partial u}{\partial t}=\frac{\Delta}{2}u(t,x)…