Related papers: Existence and uniqueness of very singular solution…

We consider a scalar parabolic equation in one spatial dimension. The equation is constituted by a convective term, a reaction term with one or two equilibria, and a positive diffusivity which can however vanish. We prove the existence and…

We study the limit behaviour of solutions of a class of solutions of nonlinear parabolic equations with a degenerate strong absorption. We prove that two types of phenomena can occur: the pointwise singularity or the formation of razor…

We focus on open questions regarding the uniqueness of distributional solutions of the fast diffusion equation (FDE) with a given source term. When the source is sufficiently smooth, the uniqueness follows from standard results. Assuming…

Self-similar large time behaviour of weak solutions of the fourth-order parabolic thin film equations with absorption is studued.

We prove that the equation \begin{eqnarray*} -\Delta_p u =\lambda\Big( \frac{1} {u^\delta} + u^q + f(u)\Big)\;\text{ in } \, B_R(0) u =0 \,\text{ on} \; \partial B_R(0), \quad u>0 \text{ in } \, B_R(0) \end{eqnarray*} admits a weak radially…

We study the dynamics of the following porous medium equation with strong absorption $$\partial_t u=\Delta u^m-|x|^{\sigma}u^q,$$ posed for $(t, x) \in (0,\infty) \times \mathbb{R}^N$, with $m > 1$, $q \in (0, 1)$ and $\sigma >…

In this paper we establish the uniqueness of a solution to a stationary convection-diffusion equation in divergence form with an exponentially summable generalized divergence-free drift.

We establish existence, uniqueness as well as quantitative estimates for solutions to the fractional nonlinear diffusion equation, $\partial_t u +{\mathcal L}_{s,p} (u)=0$, where ${\mathcal L}_{s,p}=(-\Delta)_p^s$ is the standard fractional…

We study the local properties of positive solutions of the equation $-\Delta u+ae^{bu}=m|\nabla u|^q$ in a punctured domain $\Omega\setminus\{0\}$ of $\bf R^2$ where $m,a,b$ are positive parameters and $q>1$. We study particularly the…

Let $\lambda^{*}>0$ denote the largest possible value of $\lambda$ such that $$ \{{array}{lllllll} \Delta^{2}u=\lambda(1+u)^{p} & {in}\ \ \B, %0<u\leq 1 & {in}\ \ \B, u=\frac{\partial u}{\partial n} =0 & {on}\ \ \partial \B {array}. $$ has…

Solutions in self-similar form, either global in time or presenting finite time blow-up, to the supercritical fast diffusion equation with spatially inhomogeneous source $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, \quad…

We study the existence and nonexistence of singular solutions to the equation $u_t-\Delta u - \frac{\kappa}{|x|^2}u+|x|^\alpha u|u|^{p-1}=0$, $p>1$, in $\R^N\times[0,\infty)$, $N\ge 3$, with a singularity at the point $(0,0)$, that is,…

In this paper we are concerned with the stochastic partial differential equations of super-fast diffusion processes describing behavior of plasma dX(t)-{\Delta}ln(X(t)+1)dt=\surd(Q)dW(t), in (0,T)\timesO, where O is a bounded open subset of…

In this paper, we consider the following Choquard type equation \begin{equation} \left\{\begin{aligned} &-\Delta u+\lambda u=\gamma(\Phi_N(|x|)\ast|u|^p)u \ \ \mbox{in $\mathbb{R}^N$}, \\ &\lim\limits_{|x|\to\infty}u(x)=0,\\…

The phenomenon of finite time extinction of bounded and non-negative solutions to the diffusion equation with strong absorption $$\partial_t u-\Delta u^m+|x|^{\sigma}u^q=0, \qquad (t,x)\in(0,\infty)\times\mathbb{R}^N,$$ with $m\geq1$,…

This paper deals with the long-time behavior of solutions of nonlinear reaction-diffusion equations describing formation of morphogen gradients, the concentration fields of molecules acting as spatial regulators of cell differentiation in…

We show that the gradient of the $m$-power of a solution to a singular parabolic equation of porous medium-type (also known as fast diffusion equation), satisfies a reverse H\"older inequality in suitable intrinsic cylinders. Relying on an…

We prove existence and uniqueness of the branch of the so-called \emph{anomalous eternal solutions} in exponential self-similar form for the subcritical fast-diffusion equation with a weighted reaction term $$ \partial_tu=\Delta…

If $\Omega$ is a bounded domain in $\mathbb R^N$ and $f$ a continuous increasing function satisfying a super linear growth condition at infinity, we study the existence and uniqueness of solutions for the problem (P): $\partial_tu-\Delta…

We study positive solutions of equation (E1) $-\Delta u + u^p|\nabla u|^q= 0$ ($0\leq p$, $0\leq q\leq 2$, $p+q>1$) and (E2) $-\Delta u + u^p + |\nabla u|^q =0$ ($p>1$, $1<q\leq 2$) in a smooth bounded domain $\Omega \subset \mathbb{R}^N$.…